Quantum Steampunk: Quantum Information, Thermodynamics, Their Intersection, and Applications Thereof Across Physics Citation Yunger Halpern, Nicole (2018) Quantum Steampunk: Quantum Information, Thermodynamics, Their Intersection, and Applications Thereof Across Physics. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/96EJ-N815. https://resolver.caltech.edu/CaltechTHESIS:05262018-061607919 Abstract Combining quantum information theory (QIT) with thermodynamics unites 21st-century technology with 19th-century principles. The union elucidates the spread of information, the flow of time, and the leveraging of energy. This thesis contributes to the theory of quantum thermodynamics, particularly to QIT thermodynamics. The thesis also contains applications of the theory, wielded as a toolkit, across physics. Fields touched on include atomic, molecular, and optical physics; nonequilibrium statistical mechanics; condensed matter; high-energy physics; and chemistry. I propose the name quantum steampunk for this program. The term derives from the steampunk genre of literature, art, and cinema that juxtaposes futuristic technologies with 19th-century settings. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: Quantum information, thermodynamics, theoretical physics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Awards: Barbara Groce Graduate Fellowship (PMA Division, Caltech, Mar.–June. 2018). KITP Graduate Fellowship (Kavli Institute for Theoretical Physics, U. of California at Santa Barbara, July–Dec. 2017). Burke Graduate Student Fellowship (Walter Burke Institute for Theoretical Physics, Caltech, Jan.–June 2017). IQIM Graduate Fellowship (Institute for Quantum Information and Matter, Caltech, NSF, and Gordon and Betty Moore Foundation, Oct. 2014–Oct. 2015). Virginia Gilloon Fellowship (Caltech, Oct. 2013–Oct. 2014). Thesis Availability: Public (worldwide access) Research Advisor(s): Preskill, John P. Group: Institute for Quantum Information and Matter Thesis Committee: Preskill, John P. (chair) Brandao, Fernando Chen, Xie Endres, Manuel A. Defense Date: 25 May 2018 Non-Caltech Author Email: nicoleyh.11 (AT) gmail.com Additional Information: The author's last (family) name is "Yunger Halpern." Funders: Funding Agency Grant Number Institute for Quantum Information and Matter, Caltech PHY-0803371, PHY-1125565, GBMF-2644 Walter Burke Institute for Theoretical Physics, Caltech UNSPECIFIED Kavli Institute for Theoretical Physics PHY-1125915 Foundational Questions Institute (FQXi) Large Grant for "Time and the Structure of Quantum Theory" Projects: Quantum equilibration in many-body systems, Athermality as a resource in work extraction from quantum many-body systems: MBL-mobile, Athermality as a resource in information processing: Quantum information in quantum cognition, Non-Abelian thermal state, Resource theories for thermodynamics, Fluctuation relations and one-shot information theory Record Number: CaltechTHESIS:05262018-061607919 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:05262018-061607919 DOI: 10.7907/96EJ-N815 Related URLs: URL URL Type Description https://doi.org/10.1103/PhysRevA.95.012120 DOI Paper version of Ch. 1 https://doi.org/10.1103/PhysRevA.97.042105 DOI Paper version of Ch. 2 https://arxiv.org/abs/1707.07008 arXiv Paper version of Ch. 3 https://doi.org/10.1038/ncomms12051 DOI Paper version of Ch. 4 https://doi.org/10.1088/1367-2630/17/9/095003 DOI Paper "Introducing one-shot work into fluctuation relations" https://doi.org/10.1103/PhysRevE.93.022126 DOI Paper "Beyond heat baths: Generalized resource theories for small-scale thermodynamics" https://doi.org/10.1088/1751-8121/aaa62f DOI Paper "Beyond heat baths II: Framework for generalized thermodynamic resource theories" https://doi.org/10.1103/PhysRevE.93.052144 DOI Paper "Number of trials required to estimate a free-energy difference, using fluctuation relations" https://arxiv.org/abs/1711.04801 arXiv Paper "Quantum information in quantum cognition" https://doi.org/10.1103/PhysRevA.95.062306 DOI Paper "Quantum voting and violation of Arrow's impossibility theorem" https://doi.org/10.1103/PhysRevE.97.052135 DOI Paper "Maximum one-shot dissipated work from Rényi divergences" https://doi.org/10.1007/978-3-319-43760-6_8 DOI Book chapter "Toward Physical Realizations of Thermodynamic Resource Theories" https://arxiv.org/abs/1802.01587 arXiv Paper "Resilience of scrambling measurements" https://doi.org/10.1088/1367-2630/aa62ba DOI Paper "Entropic equality for worst-case work at any protocol speed" https://arxiv.org/abs/1805.00667 arXiv Paper "Strengthening weak measurement of qubit out-of-time-order correlators" https://doi.org/10.48550/arXiv.1807.09786 DOI Thesis in ArXiv ORCID: Author ORCID Yunger Halpern, Nicole 0000-0001-8670-6212 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. 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Quantum steampunk: Quantum information, thermodynamics, their intersection, and applications thereof across physics Thesis by Nicole Yunger Halpern In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Physics CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2018 Defended May 25, 2018 ii c 2018 Nicole Yunger Halpern ORCID: 0000-0001-8670-6212 All rights reserved except where otherwise noted iii ACKNOWLEDGEMENTS I am grateful to need to acknowledge many contributors. I thank my parents for the unconditional support and love, and for the sacrifices, that enabled me to arrive here. Thank you for communicating values that include diligence, discipline, love of education, and security in one’s identity. For a role model who embodies these virtues, I thank my brother. I thank my advisor, John Preskill, for your time, for mentorship, for the communication of scientific values and scientific playfulness, and for investing in me. I have deeply appreciated the time and opportunity that you’ve provided to learn and create. Advice about “thinking big”; taking risks; prioritizing; embracing breadth and exhibiting nimbleness in research; and asking, “Are you having fun?” will remain etched in me. Thank you for bringing me to Caltech. Thank you to my southern-California family for welcoming me into your homes and for sharing holidays and lunches with me. You’ve warmed the past five years. The past five years have seen the passing of both my grandmothers: Dr. Rosa Halpern during year one and Mrs. Miriam Yunger during year four. Rosa Halpern worked as a pediatrician until in her 80s. Miriam Yunger yearned to attend college but lacked the opportunity. She educated herself, to the point of erudition on Russian and American history, and amassed a library. I’m grateful for these role models who shared their industriousness, curiosity, and love. I’m grateful to my research collaborators for sharing time, consideration, and expertise: Ning Bao, Daniel Braun, Lincoln Carr, Mahn-Soo Choi, Elizabeth Crosson, Oscar Dahlsten, Justin Dressel, Philippe Faist, Andrew Garner, José Raúl Gonzalez Alonso, Sarang Gopalakrishnan, Logan Hillberry, Andrew Keller, Chris Jarzynski, Jonathan Oppenheim, Patrick Rall, Gil Refael, Joe Renes, Brian Swingle, Vlatko Vedral, Mordecai Waegell, Sara Walker, Christopher White, and Andreas Winter. I’m grateful to informal advisors for sharing experiences and guidance: Michael Beverland, Sean Carroll, Ian Durham, Alexey Gorshkov, Daniel Harlow, Dave Kaiser, Shaun Maguire, Spiros Michalakis, Jenia Mozgunov, Renato Renner, Barry Sanders, many of my research collaborators, and many other colleagues and peers. Learning and laughing with my quantum-information/-thermodynamics colleagues has been a pleasure and a privilege: Álvaro Martín Alhambra, Lídia del Río, John Goold, David Jennings, Matteo Lostaglio, Nelly Ng, Mischa Woods, aforemen- iv tioned collaborators, and many others. I’m grateful to Caltech’s Institute for Quantum Information and Matter (IQIM) for conversations, collaborations, financial support, an academic and personal home, and more. I thank especially Fernando Brandão, Xie Chen, Manuel Endres, David Gosset, Stacey Jeffery, Alexei Kitaev, Alex Kubica, Roger Mong, Oskar Painter, Fernando Pastawski, Kristan Temme, and the aforementioned IQIM members. Thanks to my administrators for logistical assistance, for further logistical assistance, for hallway conversations that counterbalanced the rigors of academic life, for your belief in me, and for more logistical assistance: Marcia Brown, Loly Ekmekjian, Ann Harvey, Bonnie Leung, Jackie O’Sullivan, and Lisa Stewart. For more such conversations, and for weekend lunches in the sun on Beckman Lawn, I’m grateful to too many friends to name. Thank you for your camaraderie, candidness, and sincerity. Also too many to name are the mentors and teachers I encountered before arriving at Caltech. I recall your guidance and encouragement more often than you realize. Time ranks amongst the most valuable resources a theorist can hope for. I deeply appreciate the financial support that has offered freedom to focus on research. Thanks to Caltech’s Graduate Office; the IQIM; the Walter Burke Institute; the Kavli Institute for Theoretical Physics (KITP); and Caltech’s Division of Physics, Mathematics, and Astronomy for a Virginia Gilloon Fellowship, an IQIM Fellowship, a Walter Burke Graduate Fellowship, a KITP Graduate Fellowship, and a Barbara Groce Fellowship. Thanks to John Preskill and Gil Refael for help with securing funding. Thanks to many others (especially the Foundational Questions Institute’s Large Grant for "Time and the Structure of Quantum Theory", Jon Barrett, and Oscar Dahlsten) for financial support for research visits. NSF grants PHY-0803371, PHY-1125565, and PHY-1125915 have supported this research. The IQIM is an NSF Physics Frontiers Center with support from the Gordon and Betty Moore Foundation (GBMF-2644). v ABSTRACT Combining quantum information theory (QIT) with thermodynamics unites 21stcentury technology with 19th-century principles. The union elucidates the spread of information, the flow of time, and the leveraging of energy. This thesis contributes to the theory of quantum thermodynamics, particularly to QIT thermodynamics. The thesis also contains applications of the theory, wielded as a toolkit, across physics. Fields touched on include atomic, molecular, and optical physics; nonequilibrium statistical mechanics; condensed matter; high-energy physics; and chemistry. I propose the name quantum steampunk for this program. The term derives from the steampunk genre of literature, art, and cinema that juxtaposes futuristic technologies with 19th-century settings. vi PUBLISHED CONTENT The following publications form the basis for this thesis. The multi-author papers resulted from collaborations to which all parties contributed equally. [1] N. Yunger Halpern, A. J. P. Garner, O. C. O. Dahlsten, and V. Vedral, New Journal of Physics 17, 095003 (2015), 10.1088/1367-2630/17/9/095003. [2] N. Yunger Halpern and J. M. Renes, 10.1103/PhysRevE.93.022126. Phys. Rev. E 93, 022126 (2016), [3] N. Yunger Halpern, Journal of Physics A: Mathematical and Theoretical 51, 094001 (2018), 10.1088/1751-8121/aaa62f. [4] N. Bao and N. Yunger Halpern, 10.1103/PhysRevA.95.062306. Phys. Rev. A 95, 062306 (2017), [5] O. C. O. Dahlsten et al., New Journal of Physics 19, 043013 (2017), 10.1088/1367-2630/aa62ba. [6] N. Yunger Halpern, A. J. P. Garner, O. C. O. Dahlsten, and V. Vedral, Phys. Rev. E 97, 052135 (2018). [7] N. Yunger Halpern, Toward physical realizations of thermodynamic resource theories, in Information and Interaction: Eddington, Wheeler, and the Limits of Knowledge, edited by I. T. Durham and D. Rickles, Frontiers Collection, Springer, 2017, 10.1007/978-3-319-43760-6. [8] N. Yunger Halpern and C. Jarzynski, 10.1103/PhysRevE.93.052144. Phys. Rev. E 93, 052144 (2016), [9] N. Yunger Halpern, P. Faist, J. Oppenheim, and A. Winter, Nature Communications 7, 12051 (2016), 10.1038/ncomms12051. [10] N. Yunger Halpern, RevA.95.012120. Phys. Rev. A 95, 012120 (2017), 10.1103/Phys- [11] N. Yunger Halpern, B. Swingle, and J. Dressel, Phys. Rev. A 97, 042105 (2018), 10.1103/PhysRevA.97.042105. [12] N. Yunger Halpern, C. D. White, S. Gopalakrishnan, and G. Refael, ArXiv e-prints (2017), 1707.07008. [13] N. Yunger Halpern and E. Crosson, ArXiv e-prints (2017), 1711.04801. [14] B. Swingle and N. Yunger Halpern, ArXiv e-prints (in press), 1802.01587, accepted by Phys. Rev. E. vii [15] J. Dressel, J. Raúl González Alonso, M. Waegell, and N. Yunger Halpern, ArXiv e-prints (2018), 1805.00667. viii TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Published Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Chapter II: Jarzynski-like equality for the out-of-time-ordered correlator . . . 11 2.1 Set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 Chapter III: The quasiprobability behind the out-of-time-ordered correlator . 28 3.1 Technical introduction . . . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 Experimentally measuring à ρ and the coarse-grained A˜ρ . . . . . . . 53 3.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.4 Calculation of A˜ρ averaged over Brownian circuits . . . . . . . . . . 74 3.5 Theoretical study of à ρ . . . . . . . . . . . . . . . . . . . . . . . . 81 3.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Chapter IV: MBL-Mobile: Many-body-localized engine . . . . . . . . . . . . 119 4.1 Thermodynamic background . . . . . . . . . . . . . . . . . . . . . 121 4.2 The MBL Otto cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 122 4.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . 138 4.4 Order-of-magnitude estimates . . . . . . . . . . . . . . . . . . . . . 141 4.5 Formal comparisons with competitor engines . . . . . . . . . . . . . 143 4.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Chapter V: Non-Abelian thermal state: The thermal state of a quantum system with noncommuting charges . . . . . . . . . . . . . . . . . . . . . . 153 5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 5.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 Appendix A: Appendices for “Jarzynski-like equality for the out-of-timeordered correlator” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A.1 Weak measurement of the combined quantum amplitude à ρ . . . . . 169 A.2 Interference-based measurement of the combined quantum amplitude à ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 ix Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 Appendix B: Appendices for “The quasiprobability behind the out-of-timeordered correlator” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 B.1 Mathematical properties of P(W,W 0 ) . . . . . . . . . . . . . . . . . 175 B.2 Retrodiction about the symmetrized composite observable Γ̃ := i(K . . . A− A . . . K ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 Appendix C: Appendices for “MBL-Mobile: Many-body-localized engine” . 180 C.1 Quantitative assessment of the mesoscopic MBL Otto engine . . . . 180 C.2 Phenomenological model for the macroscopic MBL Otto engine . . . 200 C.3 Constraint 2 on cold thermalization: Suppression of high-order-inthe-coupling energy exchanges . . . . . . . . . . . . . . . . . . . . 205 C.4 Optimization of the MBL Otto engine . . . . . . . . . . . . . . . . . 206 C.5 Numerical simulations of the MBL Otto engine . . . . . . . . . . . . 214 C.6 Comparison with competitor Otto engines: Details and extensions . . 220 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 Appendix D: Appendices for “Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges”224 D.1 Microcanonical derivation of the NATS’s form . . . . . . . . . . . . 224 D.2 Dynamical considerations . . . . . . . . . . . . . . . . . . . . . . . 234 D.3 Derivation from complete passivity and resource theory . . . . . . . 235 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 1 Chapter 1 INTRODUCTION The steampunk movement has invaded literature, film, and art over the past three decades.1 Futuristic technologies mingle, in steampunk works, with Victorian and wild-west settings. Top hats, nascent factories, and grimy cities counterbalance time machines, airships, and automata. The genre arguably originated in 1895, with the H.G. Wells novel The Time Machine. Recent steampunk books include the best-selling The Invention of Hugo Cabret; films include the major motion picture Wild Wild West; and artwork ranges from painting to jewelry to sculpture. Steampunk captures the romanticism of fusing the old with the cutting-edge. Technologies proliferated during the Victorian era: locomotives, Charles Babbage’s analytical engine, factories, and more. Innovation facilitated exploration. Add time machines, and the spirit of adventure sweeps you away. Little wonder that fans flock to steampunk conventions, decked out in overcoats, cravats, and goggles. What steampunk fans dream, quantum-information thermodynamicists live. Thermodynamics budded during the late 1800s, when steam engines drove the Industrial Revolution. Sadi Carnot, Ludwig Boltzmann, and other thinkers wondered how efficiently engines could operate. Their practical questions led to fundamental insights—about why time flows; how much one can know about a physical system; and how simple macroscopic properties, like temperature, can capture complex behaviors, like collisions by steam particles. An idealization of steam—the classical ideal gas—exemplifies the conventional thermodynamic system. Such systems contain many particles, behave classically, and are often assumed to remain in equilibrium. But thermodynamic concepts—such as heat, work, and equilibrium—characterize small scales, quantum systems, and out-of-equilibrium processes. Today’s experimentalists probe these settings, stretching single DNA strands with optical tweezers [4], cooling superconducting qubits to build quantum computers [5, 6], and extracting work from single-electron boxes [7]. These settings demand reconcilia1 Parts this introduction were adapted from [1–3]. 2 tion with 19th-century thermodynamics. We need a toolkit for fusing the old with the new. Quantum information (QI) theory provides such a toolkit. Quantum phenomena serve as resources for processing information in ways impossible with classical systems. Quantum computers can solve certain computationally difficult problems quickly; quantum teleportation transmits information as telephones cannot; quantum cryptography secures messages; and quantum metrology centers on highprecision measurements. These applications rely on entanglement (strong correlations between quantum systems), disturbances by measurements, quantum uncertainty, and discreteness. Technological promise has driven fundamental insights, as in thermodynamics. QI theory has blossomed into a mathematical toolkit that includes entropies, uncertainty relations, and resource theories. These tools are reshaping fundamental science, in applications across physics, computer science, and chemistry. QI is being used to update thermodynamics, in the field of quantum thermodynamics (QT) [8, 9]. QT features entropies suited to small scales; quantum engines; the roles of coherence in thermalization and transport; and the transduction of information into work, à la Maxwell’s demon [10]. This thesis (i) contributes to the theory of QI thermodynamics and (ii) applies the theory, as a toolkit, across physics. Spheres touched on include atomic, molecular, and optical (AMO) physics; nonequilibrium statistical mechanics; condensed matter; chemistry; and high-energy physics. I propose the name quantum steampunk for this program. The thesis contains samples of the research performed during my PhD. See [11–24] for a complete catalog. Three vertebrae form this research statement’s backbone. I overview the contributions here; see the chapters for more context, including related literature. First, the out-of-time-ordered correlator signals the scrambling of information in quantum many-body systems that thermalize internally. Second, athermal systems serve as resources in thermodynamic tasks, such as work extraction and information storage. Examples include many-body-localized systems, for which collaborators and I designed a quantum many-body engine cycle. Third, consider a small quantum system thermalizing with a bath. The systems could exchange quantities, analogous to heat and particles, that fail to commute with each other. The small system would approach a non-Abelian thermal state. 3 Related PhD research is mentioned where relevant. One paper has little relevance to thermodynamics, so I will mention it here: Quantum voting illustrates the power of nonclassical resources, in the spirit of quantum game theory, through elections [14]. Information scrambling and quantum thermalization: Chaotic evolution scrambles information stored in quantum many-body systems, such as spin chains and black holes. QI spreads throughout many degrees of freedom via entanglement. The out-of-time-ordered correlator (OTOC) registers this spread—loosely speaking, the equilibration of QI [25]. Chaos and information scrambling smack of time’s arrow and the second law of thermodynamics. So do fluctuation relations in nonequilibrium statistical mechanics. The best-known fluctuation relations include Jarzynski’s equality, he− βW i = e− β∆F [26]. W represents the work required to perform a protocol, such as pushing an electron onto a charged island in a circuit [27]. h.i denotes an average over nonequilibrium pushing trials; β denotes the inverse temperature at which the electron begins; and ∆F denotes a difference between equilibrium free energies. Chemists and biologists use ∆F; but measuring ∆F proves difficult. Jarzynski’s equality suggests a measurement scheme: One measures the work W in each of many finite-time trials (many pushings of the electron onto the charged island). One averages e− βW over trials, substitutes into the equation’s left-hand side, and solves for ∆F. Like ∆F, the OTOC is useful but proves difficult to measure. I developed a fluctuation relation, analogous to Jarzynski’s equality, for the OTOC [20] (Ch. 2). The relation has three significances. First, the equality unites two disparate, yet similar-in-spirit concepts: the OTOC of AMO, condensed matter, and high energy with fluctuation relations of nonequilibrium statistical mechanics. Second, the equality suggests a scheme for inferring the OTOC experimentally. The scheme hinges on weak measurements, which fail to disturb the measured system much. Third, the equality unveils a quantity more fundamental than the OTOC: a quasiprobability. Quasiprobability distributions represent quantum states as phase-space densities represent classical statistical-mechanical states. But quasiprobabilities assume nonclassical values (e.g., negative and nonreal values) that signal nonclassical physics (e.g., the capacity for superclassical computation [28]). Many classes of quasiprobabilities exist. Examples include the well-known Wigner function and its obscure little sibling, the Kirkwood-Dirac (KD) quasiprobability. 4 An extension of the KD quasiprobability, I found, underlies the OTOC [20]. Collaborators and I characterized this quasiprobability in [21] (Ch. 3). We generalized KD theory, proved mathematical properties of the OTOC quasiprobability, enhanced the weak-measurement scheme, and calculated the quasiprobability numerically and analytically in examples. The quasiprobability, we found, strengthens the parallel between OTOCs and chaos: Plots of the quasiprobability bifurcate, as in classicalchaos pitchfork diagrams. QI scrambling, the plots reveal, breaks a symmetry in the quasiprobability. The Jarzynski-like equality for the OTOC (Ch. 2) broadens my earlier work on fluctuation relations. Collaborators and I merged fluctuation relations with two QI toolkits: resource theories (QI-theoretic models, discussed below, including for thermodynamics) and one-shot information theory (a generalization of Shannon theory to small scales) [11, 15, 16]. We united mathematical tools from distinct disciplines, nonequilibrium statistical mechanics and QI. The union describes smallscale thermodynamics, such as DNA strands and ion traps. I applied our results with Christopher Jarzynski [18]. We bounded, in terms of an entropy, the number of trials required to estimate ∆F with desired precision. Our work harnesses QI for experiments. Experimental imperfections can devastate OTOC-measurement schemes (e.g., [20, 21, 29–31]). Many schemes require experimentalists to effectively reverse time, to negate a Hamiltonian H. An attempted negation could map H to −H + ε for some small perturbation ε. Also, environments can decohere quantum systems. Brian Swingle and I proposed a scheme for mitigating such errors [24]. The measured OTOC signal is renormalized by data from easier-to-implement trials. The scheme improves the weak-measurement scheme and other OTOC-measurement schemes [29–31], for many classes of Hamiltonians. The weak-measurement scheme was improved alternatively in [32]. Collaborators and I focused on observables O j that square to the identity operator: (O j ) 2 = 1. Examples include qubit Pauli operators. Consider time-evolving such an observable in the Heisenberg picture, forming O j (t j ). Define a correlator C = hO1 (t 1 )O2 (t 2 ) . . . Om (t m )i from m observables. C can be inferred from a sequence of measurements interspersed with time evolutions. Each measurement requires an ancilla qubit coupled to the system locally. The measurements can be of arbitrary strengths, we showed, “strengthening” the weak-measurement protocol. 5 Athermal states as resources in thermodynamic tasks: work extraction and information processing Many-body localization (MBL) defines a phase of quantum many-body systems. The phase can be realized with ultracold atoms, trapped ions, and nitrogen-vacancy centers. MBL behaves athermally: Consider measuring the positions of MBL particles. The particles stay fixed for a long time afterward. For contrast, imagine measuring the positions of equilibrating gas particles. The particles thereafter random-walk throughout their container. Athermal systems serve as resources in thermodynamic tasks: Consider a hot bath in a cool environment. The hot bath is athermal relative to the atmosphere. You can connect the hot bath to the cold, let heat flow, and extract work. As work has thermodynamic value, so does athermality. MBL’s athermality facilitates thermodynamic tasks, I argued with collaborators [22] (Ch. 4). We illustrated by formulating an engine cycle for a quantum many-body system. The engine is tuned between deep MBL and a “thermal” regime. “Thermal” Hamiltonians exhibit level repulsion: Any given energy gap has a tiny probability of being small. Energy levels tend to lie far apart. MBL energy spectra lack level repulsion. The athermality of MBL energy spectra curbs worst-case trials, in which the engine would output net negative work Wtot < 0; constrains fluctuations in Wtot ; and offers flexibility in choosing the engine’s size, from mesoscale to macroscopic. We calculated the engine’s power and efficiency; numerically simulated a spin-chain engine; estimated diabatic corrections to results, using adiabatic perturbation theory; and modeled interactions with a bosonic bath. This project opens MBL—a newly characterized phase realized recently in experiments— to applications. Possible applications include engines, energy-storing ratchets, and dielectrics. These opportunities should point to new physics. For example, formulating an engine cycle led us to define and calculate heat and work quantities that, to our knowledge, had never been defined for MBL. Just as quantum thermodynamics provided a new lens onto MBL, MBL fed back on QT. Quantum states ρ , e− βH /Z are conventionally regarded as athermal resources. Also gap statistics, we showed, offer athermal tools. The benefits of athermality may extend to biomolecules. Matthew Fisher recently proposed that Posner biomolecules store QI protected from thermalization for long times [33]. Elizabeth Crosson and I assessed how efficiently these molecules could 6 process QI [23]. We abstracted out the logical operations from Fisher’s physics, defining the model of Posner quantum computation. Operations in the model, we showed, can be used to teleport QI imperfectly. We also identified quantum error-detecting codes that could flag whether the molecules’ QI has degraded. Additionally, we identified molecular states that can serve as universal resources in measurement-based quantum computation [34]. Finally, we established a framework for quantifying Fisher’s conjecture that entanglement can influence molecularbinding rates. This work opens the door to the QI-theoretic analysis and applications of Posner molecules. Non-Abelian thermal state: Consider a small quantum system S equilibrating with a bath B. S exchanges quantities, such as heat, with B. Each quantity is conserved globally; so it may be called a charge. If exchanging just heat and particles, S equilibrates to a grand canonical ensemble e− β(H−µN ) /Z. S can exchange also electric charge, angular momentum, etc.: m observables Q1 , . . . Q m . Renes and I incorporated thermodynamic exchanges of commuting quantities into resource theories [12, 13]. What if the Q j ’s fail to commute? Can S thermalize? What form would the thermal state γ have? These questions concern truly quantum thermodynamics [13]. Collaborators and I used QI to characterize γ, which we dubbed the non-Abelian thermal state (NATS) [19] (Ch. 5). Parallel analyses took place in [35, 36]. We derived the form of γ in three ways. First, we invoked typical subspaces, a QI tool used to quantify data compression. Second, thermal states are the fixed points of ergodic dynamics. We modeled ergodic dynamics with a random unitary. Randomly evolved states have been characterized with another QI tool, canonical typicality [37–40]. We applied canonical typicality to our system’s time-evolved Pm state. The state, we concluded, lies close to the expected e− j=1 µ j Q j /Z. Third, thermal states are completely passive: Work cannot be extracted even from infinitely many copies of a thermal state [41]. We proved the complete passivity of Pm e− j=1 µ j Q j /Z, using a thermodynamic resource theory. Resource theories are QI models for agents who transform quantum states, using a restricted set of operations. The first law of thermodynamics and the ambient temperature T restrict thermodynamic operations. Restrictions prevent agents from preparing certain states, e.g., pure nonequilibrium states. Scarce states have value, as work can be extracted from nonequilibrium systems. Resource theories help us 7 to quantify states’ usefulness, to identify allowed and forbidden transformations between states, and to quantify the efficiencies with which tasks (e.g., work extraction) can be performed outside the large-system limit (e.g., [42–45]. The efficiencies are quantified with quantum entropies for small scales [46]. Most of my PhD contributions were mentioned above [11–13, 19]. Such theoretical results require testing. I outlined experimental challenges and opportunities in [17]. 8 BIBLIOGRAPHY [1] N. Yunger Halpern, Steampunk quantum, Quantum Frontiers, 2013. [2] N. Yunger Halpern, Quantum steampunk: Quantum information applied to thermodynamics, Colloquium, Cal State LA, 2016. [3] N. Yunger Halpern, Bringing the heat to Cal State LA, Quantum Frontiers, 2016. [4] A. Mossa, M. Manosas, N. Forns, J. M. Huguet, and F. Ritort, Journal of Statistical Mechanics: Theory and Experiment 2009, P02060 (2009). [5] J. M. Gambetta, J. M. Chow, and M. Steffen, npj Quantum Information 3, 2 (2017). [6] C. Neill et al., ArXiv e-prints (2017), 1709.06678. [7] O.-P. Saira et al., Phys. Rev. Lett. 109, 180601 (2012). [8] J. Goold, M. Huber, A. Riera, L. del Río, and P. Skrzypczyk, Journal of Physics A: Mathematical and Theoretical 49, 143001 (2016). [9] S. Vinjanampathy and J. Anders, Contemporary Physics 57, 545 (2016). [10] K. Maruyama, F. Nori, and V. Vedral, Rev. Mod. Phys. 81, 1 (2009). [11] N. Yunger Halpern, A. J. P. Garner, O. C. O. Dahlsten, and V. Vedral, New Journal of Physics 17, 095003 (2015), 10.1088/1367-2630/17/9/095003. [12] N. Yunger Halpern and J. M. Renes, 10.1103/PhysRevE.93.022126. Phys. Rev. E 93, 022126 (2016), [13] N. Yunger Halpern, Journal of Physics A: Mathematical and Theoretical 51, 094001 (2018), 10.1088/1751-8121/aaa62f. [14] N. Bao and N. Yunger Halpern, 10.1103/PhysRevA.95.062306. Phys. Rev. A 95, 062306 (2017), [15] O. C. O. Dahlsten et al., New Journal of Physics 19, 043013 (2017), 10.1088/1367-2630/aa62ba. [16] N. Yunger Halpern, A. J. P. Garner, O. C. O. Dahlsten, and V. Vedral, Phys. Rev. E 97, 052135 (2018). [17] N. Yunger Halpern, Toward physical realizations of thermodynamic resource theories, in Information and Interaction: Eddington, Wheeler, and the Limits of Knowledge, edited by I. T. Durham and D. Rickles, Frontiers Collection, Springer, 2017, 10.1007/978-3-319-43760-6. 9 [18] N. Yunger Halpern and C. Jarzynski, 10.1103/PhysRevE.93.052144. Phys. Rev. E 93, 052144 (2016), [19] N. Yunger Halpern, P. Faist, J. Oppenheim, and A. Winter, Nature Communications 7, 12051 (2016), 10.1038/ncomms12051. [20] N. Yunger Halpern, RevA.95.012120. Phys. Rev. A 95, 012120 (2017), 10.1103/Phys- [21] N. Yunger Halpern, B. Swingle, and J. Dressel, Phys. Rev. A 97, 042105 (2018), 10.1103/PhysRevA.97.042105. [22] N. Yunger Halpern, C. D. White, S. Gopalakrishnan, and G. Refael, ArXiv e-prints (2017), 1707.07008. [23] N. Yunger Halpern and E. Crosson, ArXiv e-prints (2017), 1711.04801. [24] B. Swingle and N. Yunger Halpern, ArXiv e-prints (in press), 1802.01587, accepted by Phys. Rev. E. [25] A. Kitaev, A simple model of quantum holography, 2015. [26] C. Jarzynski, Physical Review Letters 78, 2690 (1997). [27] O.-P. Saira et al., Phys. Rev. Lett. 109, 180601 (2012). [28] R. W. Spekkens, Phys. Rev. Lett. 101, 020401 (2008). [29] B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hayden, Phys. Rev. A 94, 040302 (2016). [30] N. Y. Yao et al., ArXiv e-prints (2016), 1607.01801. [31] G. Zhu, M. Hafezi, and T. Grover, Phys. Rev. A 94, 062329 (2016). [32] J. Dressel, J. Raúl González Alonso, M. Waegell, and N. Yunger Halpern, ArXiv e-prints (2018), 1805.00667. [33] M. P. A. Fisher, Annals of Physics 362, 593 (2015). [34] R. Raussendorf, D. E. Browne, and H. J. Briegel, Phys. Rev. A 68, 022312 (2003). [35] M. Lostaglio, D. Jennings, and T. Rudolph, New Journal of Physics 19, 043008 (2017). [36] Y. Guryanova, S. Popescu, A. J. Short, R. Silva, and P. Skrzypczyk, Nature Communications 7, 12049 EP (2016), Article. [37] S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zanghí, Physical review letters 96, 050403 (2006). 10 [38] J. Gemmer, M. Michel, and G. Mahler, 18 Equilibrium Properties of Model Systems (Springer, 2004). [39] S. Popescu, A. Short, and A. Winter, Nature Physics 2, 754 (2006). [40] N. Linden, S. Popescu, A. J. Short, and A. Winter, Phys. Rev. E 79, 061103 (2009). [41] W. Pusz and S. Woronowicz, Communications in Mathematical Physics 58, 273 (1978). [42] E. H. Lieb and J. Yngvason, Physics Reports 310, 1 (1999). [43] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and T. Beth, Int. J. Theor. Phys. 39, 2717 (2000). [44] F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Physical Review Letters 111, 250404 (2013). [45] M. Horodecki and J. Oppenheim, Nat. Commun. 4, 1 (2013). [46] M. Tomamichel, Quantum Information Processing with Finite ResourcesSpringerBriefs in Mathematical Physics (Springer, 2016). 11 Chapter 2 JARZYNSKI-LIKE EQUALITY FOR THE OUT-OF-TIME-ORDERED CORRELATOR This chapter was published as [1]. The out-of-time-ordered correlator (OTOC) F (t) diagnoses the scrambling of quantum information [2–7]: Entanglement can grow rapidly in a many-body quantum system, dispersing information throughout many degrees of freedom. F (t) quantifies the hopelessness of attempting to recover the information via local operations. Originally applied to superconductors [8], F (t) has undergone a revival recently. F (t) characterizes quantum chaos, holography, black holes, and condensed matter. The conjecture that black holes scramble quantum information at the greatest possible rate has been framed in terms of F (t) [7, 9]. The slowest scramblers include disordered systems [10–14]. In the context of quantum channels, F (t) is related to the tripartite information [15]. Experiments have been proposed [16–18] and performed [19, 20] to measure F (t) with cold atoms and ions, with cavity quantum electrodynamics, and with nuclear-magnetic-resonance quantum simulators. F (t) quantifies sensitivity to initial conditions, a signature of chaos. Consider a quantum system S governed by a Hamiltonian H. Suppose that S is initialized to a pure state |ψi and perturbed with a local unitary operator V . S then evolves forward in time under the unitary U = e−iHt for a duration t, is perturbed with a local unitary operator W, and evolves backward under U † . The state |ψ 0i := U † WUV |ψi = W (t)V |ψi results. Suppose, instead, that S is perturbed with V not at the sequence’s beginning, but at the end: |ψi evolves forward under U, is perturbed with W, evolves backward under U † , and is perturbed with V . The state |ψ 00i := VU † WU |ψi = V W (t)|ψi results. The overlap between the two possible D E final states equals the correlator: F (t) := W † (t) V † W (t) V = hψ 00 |ψ 0i. The decay of F (t) reflects the growth of [W (t), V ] [21, 22]. Forward and reverse time evolutions, as well as information theory and diverse applications, characterize not only the OTOC, but also fluctuation relations. Fluctuation relations have been derived in quantum and classical nonequilibrium statistical mechanics [23–26]. Consider a Hamiltonian H (t) tuned from Hi to H f at a finite speed. For example, electrons may be driven within a circuit [27]. Let 12 ∆F := F (H f ) − F (Hi ) denote the difference between the equilibrium free energies at the inverse temperature β:1 F (H` ) = − 1β ln Z β,` , wherein the partition function is Z β,` := Tr(e− βH` ) and ` = i, f . The free-energy difference has applications in chemistry, biology, and pharmacology [28]. One could measure ∆F, in principle, by measuring the work required to tune H (t) from Hi to H f while the system remains in equilibrium. But such quasistatic tuning would require an infinitely long time. ∆F has been inferred in a finite amount of time from Jarzynski’s fluctuation relation, D E e− βW = e− β∆F . The left-hand side can be inferred from data about experiments in which H (t) is tuned from Hi to H f arbitrarily quickly. The work required to tune H (t) during some particular trial (e.g., to drive the electrons) is denoted by W . W varies from trial to trial because the tuning can eject the system arbitrarily far from equilibrium. The expectation value h . i is with respect to the probability distribution P(W ) associated with any particular trial’s requiring an amount W of work. Nonequilibrium experiments have been combined with fluctuation relations to estimate ∆F [27, 29–36]: ∆F = − D E 1 log e− βW . β (2.1) Jarzynski’s Equality, with the exponential’s convexity, implies hW i ≥ ∆F. The average work hW i required to tune H (t) according to any fixed schedule equals at least the work ∆F required to tune H (t) quasistatically. This inequality has been regarded as a manifestation of the Second Law of Thermodynamics. The Second Law governs information loss [37], similarly to the OTOC’s evolution. I derive a Jarzynski-like equality, analogous to Eq. (2.1), for F (t) (Theorem 1). The equality unites two powerful tools that have diverse applications in quantum information, high-energy physics, statistical mechanics, and condensed matter. The union sheds new light on both fluctuation relations and the OTOC, similar to the light shed when fluctuation relations were introduced into “one-shot” statistical mechanics [38–43]. The union also relates the OTOC, known to signal quantum behavior in high energy and condensed matter, to a quasiprobability, known to signal quantum behavior in optics. The Jarzynski-like equality suggests a platformnonspecific protocol for measuring F (t) indirectly. The protocol can be implemented with weak measurements or with interference. The time evolution need not 1 F (H ) denotes the free energy in statistical mechanics, while F (t) denotes the OTOC in high ` energy and condensed matter. 13 be reversed in any interference trial. First, I present the set-up and definitions. I then introduce and prove the Jarzynski-like equality for F (t). 2.1 Set-up Let S denote a quantum system associated with a Hilbert space H of dimensionality d. The simple example of a spin chain [17–20] informs this paper: Quantities will be summed over, as spin operators have discrete spectra. Integrals replace the sums if operators have continuous spectra. P P Let W = w` ,α w` w` |w` , α w` ihw` , α w` | and V = v` ,λ v` v` |v` , λ v` ihv` , λ v` | denote local unitary operators. The eigenvalues are denoted by w` and v` ; the degeneracy parameters, by α w` and λ v` . W and V may commute. They need not be Hermitian. Examples include single-qubit Pauli operators localized at opposite ends of a spin chain. We will consider measurements of eigenvalue-and-degeneracy-parameter tuples (w` , α w` ) and (v` , λ v` ). Such tuples can be measured as follows. A Hermitian operator P GW = w` ,α w` g(w` )|w` , α w` ihw` , α w` | generates the unitary W. The generator’s eigenvalues are labeled by the unitary’s eigenvalues: w = eig(w` ) . Additionally, there exists a Hermitian operator that shares its eigenbasis with W but whose specP trum is nondegenerate: G̃W = w` ,α w` g̃(α w` )|w` , α w` ihw` , α w` |, wherein g̃(α w` ) denotes a real one-to-one function. I refer to a collective measurement of GW and G̃W as a W̃ measurement. Analogous statements concern V . If d is large, measuring W̃ and Ṽ may be challenging but is possible in principle. Such measurements may be reasonable if S is small. Schemes for avoiding measurements of the α w` ’s and λ v` ’s are under investigation [44]. Let H denote a time-independent Hamiltonian. The unitary U = e−iHt evolves S forward in time for an interval t. Heisenberg-picture operators are defined as W (t) := U † WU and W † (t) = [W (t)]† = U † W †U. The OTOC is conventionally evaluated on a Gibbs state e−H/T /Z, wherein T denotes   − H /T a temperature: F (t) = Tr e Z W † (t)V † W (t)V . Theorem 1 generalizes beyond P e−H/T /Z to arbitrary density operators ρ = j p j | jih j | ∈ D (H ). [D (H ) denotes the set of density operators defined on H .] 2.2 Definitions Jarzynski’s Equality concerns thermodynamic work, W . W is a random variable calculated from measurement outcomes. The out-of-time-ordering in F (t) requires 14 two such random variables. I label these variables W and W 0. Two stepping stones connect W and V to W and W 0. First, I define a complex probability amplitude A ρ (w2 , α w2 ; v1 , λ v1 ; w1 , α w1 ; j) associated with a quantum protocol. I combine amplitudes A ρ into a à ρ inferable from weak measurements and from interference. à ρ resembles a quasiprobability, a quantum generalization of a probability. In terms of the w` ’s and v` ’s in à ρ , I define the measurable random variables W and W 0. Jarzynski’s Equality involves a probability distribution P(W ) over possible values of the work. I define a complex analog P(W,W 0 ). These definitions are designed to parallel expressions in [45]. Talkner, Lutz and Hänggi cast Jarzynski’s Equality in terms of a time-ordered correlation function. Modifying their derivation will lead to the OTOC Jarzynski-like equality. Quantum probability amplitude A ρ The probability amplitude A ρ is defined in terms of the following protocol, P: 1. Prepare ρ. 2. Measure the eigenbasis of ρ, {| jih j |}. 3. Evolve S forward in time under U. 4. Measure W̃. 5. Evolve S backward in time under U † . 6. Measure Ṽ . 7. Evolve S forward under U. 8. Measure W̃. An illustration appears in Fig. 3.2a. Consider implementing P in one trial. The complex probability amplitude associated with the measurements’ yielding j, then (w1 , α w1 ), then (v1 , λ v1 ), then (w2 , α w2 ) is A ρ (w2 , α w2 ; v1 , λ v1 ; w1 , α w1 ; j) := hw2 , α w2 |U |v1 , λ v1 i √ × hv1 , λ v1 |U † |w1 , α w1 ihw1 , α w1 |U | ji p j . (2.2) 15 The square modulus | A ρ (.)| 2 equals the joint probability that these measurements yield these outcomes. Suppose that [ρ, H] = 0. For example, suppose that S occupies the thermal state ρ = e−H/T /Z. (I set Boltzmann’s constant to one: k B = 1.) Protocol P and Eq. (2.2) simplify: The first U can be eliminated, because [ρ, U] = 0. Why [ρ, U] = 0 obviates the unitary will become apparent when we combine A ρ ’s into à ρ . The protocol P defines A ρ ; P is not a prescription measuring A ρ . Consider implementing P many times and gathering statistics about the measurements’ outcomes. From the statistics, one can infer the probability | A ρ | 2 , not the probability amplitude A ρ . P merely is the process whose probability amplitude equals A ρ . One must calculate combinations of A ρ ’s to calculate the correlator. These combinations, labeled à ρ , can be inferred from weak measurements and interference. (w1 , ↵w1 ) (w3 , ↵w3 ) (w2 , ↵w2 ) Measure W̃. Measure W̃. Measure W̃. -t -t U† U U U Experiment time 0 Prepare ⇢. (w2 , ↵w2 ) Measure W̃. Measure {|jihj|}. j Measure Ṽ . (v1 , v1 ) (a) U† U Experiment time 0 Prepare ⇢. Measure {|jihj|}. j Measure Ṽ . (v2 , v2 ) (b) Figure 2.1: Quantum processes described by the complex amplitudes in the Jarzynski-like equality for the out-of-time-ordered correlator (OTOC): Theorem 1 shows that the OTOC depends on a complex distribution P(W,W 0 ). This P(W,W 0 ) parallels the probability distribution over possible values of thermodynamic work in Jarzynski’s Equality. P(W,W 0 ) results from summing products A∗ρ (.) Aρ (.). Each Aρ (.) denotes a probability amplitude [Eq. (2.2)], so each product resembles a probability. But the amplitudes’ arguments differ, due to the OTOC’s out-of-time ordering: The amplitudes correspond to different quantum processes. Figure 3.2a illustrates the process associated with the Aρ (.); and Fig. 3.2b, the process associated with the A∗ρ (.). Time runs from left to right. Each P process begins with the preparation of the state ρ = j p j | jih j | and a measurement of the state’s eigenbasis. Three evolutions (U, U † , U) then alternate with three measurements of observables (W̃, Ṽ , W̃). If the initial state commutes with the Hamiltonian H (e.g., if ρ = e −H /T /Z), the first U can be omitted. Figures 3.2a and 3.2b are used to define P(W,W 0 ), rather than illustrating protocols for measuring P(W,W 0 ). P(W,W 0 ) can be inferred from weak measurements and from interferometry. Combined quantum amplitude à ρ Combining quantum amplitudes A ρ yields a quantity à ρ that is nearly a probability but that differs due to the OTOC’s out-of-time ordering. I first define à ρ , which 16 resembles the Kirkwood-Dirac quasiprobability [44, 46–48]. We gain insight into à ρ by supposing that [ρ, W] = 0, e.g., that ρ is the infinite-temperature Gibbs state 1/d. à ρ can reduce to a probability in this case, and protocols for measuring à ρ simplify. I introduce weak-measurement and interference schemes for inferring à ρ experimentally. Definition of the combined quantum amplitude à ρ Consider measuring the probability amplitudes A ρ associated with all the possible measurement outcomes. Consider fixing an outcome septuple (w2 , α w2 ; v1 , λ v1 ; w1 , α w1 ; j). The amplitude A ρ (w2 , α w2 ; v1 , λ v1 ; w1 , α w1 ; j) describes one realization, illustrated in Fig. 3.2a, of the protocol P. Call this realization a. Consider the P realization, labeled b, illustrated in Fig. 3.2b. The initial and final measurements yield the same outcomes as in a [outcomes j and (w2 , α w2 )]. Let (w3 , α w3 ) and (v2 , λ v2 ) denote the outcomes of the second and third measurements in b. Realization b corresponds to the probability amplitude A ρ (w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ; j). Let us complex-conjugate the b amplitude and multiply by the a amplitude. We marginalize over j and over (w1 , α w1 ), forgetting about the corresponding measurement outcomes: X à ρ (w, v, α w , λ v ) := A∗ρ (w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ; j) j,(w1 ,α w1 ) × A ρ (w2 , α w2 ; v1 , λ v1 ; w1 , α w1 ; j) . (2.3) The shorthand w encapsulates the list (w2 , w3 ). The shorthands v, α w and λ v are defined analogously. Let us substitute in from Eq. (2.2) and invoke hA|Bi∗ = hB| Ai. The sum over (w1 , α w1 ) evaluates to a resolution of unity. The sum over j evaluates to ρ: à ρ (w, v, α w , λ v ) = hw3 , α w3 |U |v2 , λ v2 ihv2 , λ v2 |U † |w2 , α w2 i × hw2 , α w2 |U |v1 , λ v1 ihv1 , λ v1 | ρU † |w3 , α w3 i . (2.4) This à ρ resembles the Kirkwood-Dirac quasiprobability [44, 48]. Quasiprobabilities surface in quantum optics and quantum foundations [49, 50]. Quasiprobabilities generalize probabilities to quantum settings. Whereas probabilities remain between 0 and 1, quasiprobabilities can assume negative and nonreal values. Nonclassical values signal quantum phenomena such as entanglement. The best-known 17 quasiprobabilities include the Wigner function, the Glauber-Sudarshan P representation, and the Husimi Q representation. Kirkwood and Dirac defined another quasiprobability in 1933 and in 1945 [46, 47]. Interest in the Kirkwood-Dirac quasiprobability has revived recently. The distribution can assume nonreal values, obeys Bayesian updating, and has been measured experimentally [51–54]. The Kirkwood-Dirac distribution for a state σ ∈ D(H ) has the form h f |aiha|σ| f i, wherein {| f ih f |} and {|aiha|} denote bases for H [48]. Equation (2.4) has the same form except contains more outer products. Marginalizing à ρ over every variable except one w` [or one v` , one (w` , α w` ), or one (v` , λ v` )] yields a probability, as does marginalizing the Kirkwood-Dirac distribution over every variable except one. The precise nature of the relationship between à ρ and the Kirkwood-Dirac quasiprobability is under investigation [44]. For now, I harness the similarity to formulate a weak-measurement scheme for à ρ in Sec. 2.2. à ρ is nearly a probability: à ρ results from multiplying a complex-conjugated probability amplitude A∗ρ by a probability amplitude A ρ . So does the quantum mechanical probability density p(x) = ψ ∗ (x)ψ(x). Hence the quasiprobability resembles a probability. Yet the argument of the ψ ∗ equals the argument of the ψ. The argument of the A∗ρ does not equal the argument of the A ρ . This discrepancy stems from the OTOC’s out-of-time ordering. à ρ can be regarded as like a probability, differing due to the out-of-time ordering. à ρ reduces to a probability under conditions discussed in Sec. 2.2. The reduction reinforces the parallel between Theorem 1 and the fluctuation-relation work [45], which involves a probability distribution that resembles à ρ . Simple case, reduction of à ρ to a probability P Suppose that ρ shares the W̃ (t) eigenbasis: ρ = ρW (t) := w` ,α w` pw` ,α w` U † |w` , α w` ihw` , α w` |U. For example, ρ may be the infinite-temperature Gibbs state 1/d. Equation (2.4) becomes à ρW (t ) (w, v, α w , λ v ) = hw3 , α w3 |U |v2 , λ v2 i × hv2 , λ v2 |U † |w2 , α w2 ihw2 , α w2 |U |v1 , λ v1 i × hv1 , λ v1 |U † |w3 , α w3 i pw3 ,α w3 . The weak-measurement protocol simplifies, as discussed in Sec. 2.2. (2.5) 18 Equation (2.5) reduces to a probability if (w3 , α w3 ) = (w2 , α w2 ) or if (v2 , λ v2 ) = (v1 , λ v1 ). For example, suppose that (w3 , α w3 ) = (w2 , α w2 ): à ρW (t ) ((w2 , w2 ), v, (α w2 , α w2 ), λ v ) = |hv2 , λ v2 |U † |w2 , α w2 i| 2 × |hv1 , λ v1 |U † |w2 , α w2 i| 2 pw2 ,α w2 = p(v2 , λ v2 |w2 , α w2 ) p(v1 , λ v1 |w2 , α w2 ) pw2 ,α w2 . (2.6) (2.7) The pw2 ,α w2 denotes the probability that preparing ρ and measuring W̃ will yield (w2 , α w2 ). Each p(v` , λ v` |w2 , α w2 ) denotes the conditional probability that preparing |w2 , α w2 i, backward-evolving under U † , and measuring Ṽ will yield (v` , λ v` ). Hence the combination à ρ of probability amplitudes is nearly a probability: à ρ reduces to a probability under simplifying conditions. Equation (2.7) strengthens the analogy between Theorem 1 and the fluctuation relation in [45]. Equation (10) in [45] contains a conditional probability p(m,t f |n) multiplied by a probability pn . These probabilities parallel the p(v1 , λ v1 |w1 , α w1 ) and pw1 ,α w1 in Eq. (2.7). Equation (2.7) contains another conditional probability, p(v2 , λ v2 |w1 , α w1 ), due to the OTOC’s out-of-time ordering. Weak-measurement scheme for the combined quantum amplitude à ρ à ρ is related to the Kirkwood-Dirac quasiprobability, which has been inferred from weak measurements [51–56]. I sketch a weak-measurement scheme for inferring à ρ . Details appear in Appendix A.1. Let Pweak denote the following protocol: 1. Prepare ρ. 2. Couple the system’s Ṽ weakly to an ancilla A a . Measure A a strongly. 3. Evolve S forward under U. 4. Couple the system’s W̃ weakly to an ancilla A b . Measure A b strongly. 5. Evolve S backward under U † . 6. Couple the system’s Ṽ weakly to an ancilla A c . Measure A c strongly. 7. Evolve S forward under U. 8. Measure W̃ strongly (e.g., projectively). 19 Consider performing Pweak many times. From the measurement statistics, one can infer the form of à ρ (w, v, α w , λ v ). Pweak offers an experimental challenge: Concatenating weak measurements raises the number of trials required to infer a quasiprobability. The challenge might be realizable with modifications to existing set-ups (e.g., [57, 58]). Additionally, Pweak simplifies in the case discussed in Sec. 2.2—if ρ shares the W̃ (t) eigenbasis, e.g., if ρ = 1/d. The number of weak measurements reduces from three to two. Appendix A.1 contains details. Interference-based measurement of à ρ à ρ can be inferred not only from weak measurement, but also from interference. In certain cases—if ρ shares neither the W̃ (t) nor the Ṽ eigenbasis—also quantum state tomography is needed. From interference, one infers the inner products ha|U |bi in à ρ . Eigenstates of W̃ and Ṽ are labeled by a and b; and U = U,U † . The matrix element hv1 , λ v1 | ρU † |w3 , α w3 i is inferred from quantum state tomography in certain cases. The interference scheme proceeds as follows. An ancilla A is prepared in a superposition √1 (|0i + |1i). The system S is prepared in a fiducial state | f i. The ancilla 2 controls a conditional unitary on S: If A is in state |0i, S is rotated to U |bi. If A is in |1i, S is rotated to |ai. The ancilla’s state is rotated about the x-axis [if the imaginary part =(ha|U |bi) is being inferred] or about the y-axis [if the real part <(ha|U |bi) is being inferred]. The ancilla’s σ z and the system’s {|ai} are measured. The outcome probabilities imply the value of ha|U |bi. Details appear in Appendix A.2. The time parameter t need not be negated in any implementation of the protocol. The absence of time reversal has been regarded as beneficial in OTOC-measurement schemes [17, 18], as time reversal can be difficult to implement. Interference and weak measurement have been performed with cold atoms [59], which have been proposed as platforms for realizing scrambling and quantum chaos [16, 17, 60]. Yet cold atoms are not necessary for measuring à ρ . The measurement schemes in this paper are platform-nonspecific. Measurable random variables W and W 0 The combined quantum amplitude à ρ is defined in terms of two realizations of the protocol P. The realizations yield measurement outcomes w2 , w3 , v1 , and v2 . 20 Consider complex-conjugating two outcomes: w3 7→ w3∗ , and v2 7→ v2∗ . The four values are combined into W := w3∗ v2∗ and W 0 := w2 v1 . (2.8) Suppose, for example, that W and V denote single-qubit Paulis. (W,W 0 ) can equal (1, 1), (1, −1), (−1, 1), or (−1, −1). W and W 0 function analogously to the thermodynamic work in Jarzynski’s Equality: W , W 0, and work are random variables calculable from measurement outcomes. Complex distribution function P(W,W 0 ) Jarzynski’s Equality depends on a probability distribution P(W ). I define an analog P(W,W 0 ) in terms of the combined quantum amplitude à ρ . Consider fixing W and W 0. For example, let (W,W 0 ) = (1, −1). Consider the set of all possible outcome octuples (w2 , α w2 ; w3 , α w3 ; v1 , λ v1 ; v2 , λ v2 ) that satisfy the constraints W = w3∗ v2∗ and W 0 = w2 v1 . Each octuple corresponds to a set of combined quantum amplitudes à ρ (w, v, α w , λ v ). These à ρ ’s are summed, subject to the constraints: X P(W,W 0 ) := à ρ (w, v, α w , λ v ) w,v,α w ,λ v × δW (w3∗ v2∗ ) δW 0 (w2 v1 ) . (2.9) The Kronecker delta is denoted by δ ab . The form of Eq. (2.9) is analogous to the form of the P(W ) in [45] [Eq. (10)], as à ρ is nearly a probability. Equation (2.9), however, encodes interference of quantum probability amplitudes. P(W,W 0 ) resembles a joint probability distribution. Summing any function f (W,W 0 ) with weights P(W,W 0 ) yields the average-like quantity X f (W,W 0 ) := f (W,W 0 ) P(W,W 0 ) . (2.10) W,W 0 2.3 Result The above definitions feature in the Jarzynski-like equality for the OTOC. Theorem 1. The out-of-time-ordered correlator obeys the Jarzynski-like equality D E ∂2 −( βW + β 0 W 0 ) F (t) = e , (2.11) β,β 0 =0 ∂ β ∂ β0 wherein β, β0 ∈ R. 21 Proof. The derivation of Eq. (2.11) is inspired by [45]. Talkner et al. cast Jarzynski’s Equality in terms of a time-ordered correlator of two exponentiated Hamiltonians. Those authors invoke the characteristic function Z G(s) := dW eisW P(W ) , (2.12) the Fourier transform of the probability distribution P(W ). The integration variable s is regarded as an imaginary inverse temperature: is = − β. We analogously invoke the (discrete) Fourier transform of P(W,W 0 ): X X 0 0 G(s, s0 ) := eisW eis W P(W,W 0 ) , (2.13) W0 W wherein is = − β and is0 = − β0. P(W,W 0 ) is substituted in from Eqs. (2.9) and (2.4). The delta functions are summed over: X 0 ∗ ∗ 0 eisw3 v2 eis w2 v1 hw3 , α w3 |U |v2 , λ v2 i G(s, s ) = w,v,α w ,λ v × hv2 , λ v2 |U † |w2 , α w2 ihw2 , α w2 |U |v1 , λ v1 i × hv1 , λ v1 |U † ρ(t)|w3 , α w3 i . (2.14) The ρU † in Eq. (2.4) has been replaced with U † ρ(t), wherein ρ(t) := U ρU † . The sum over (w3 , α w3 ) is recast as a trace. Under the trace’s protection, ρ(t) is shifted to the argument’s left-hand side. The other sums and the exponentials are distributed across the product: " X 0 G(s, s ) = Tr ρ(t) |w3 , α w3 ihw3 , α w3 | w3 ,α w3 ×U X isw3 ∗ v2∗ e # |v2 , λ v2 ihv2 , λ v2 |U † v2 ,λ v2 × " X |w2 , α w2 ihw2 , α w2 | w2 ,α w2 ×U X v1 ,λ v1 is0 w2 v1 e #! |v1 , λ v1 ihv1 , λ v1 | U † . (2.15) 22 The v` and λ v` sums are eigendecompositions of exponentials of unitaries: " X # 0 isw3∗ V † † G(s, s ) = Tr ρ(t) |w3 , α w3 ihw3 , α w3 |U e U w3 ,α w3 × " X is0 w2 V |w2 , α w2 ihw2 , α w2 |U e #! U . † (2.16) w2 ,α w2 The unitaries time-evolve the V ’s: " X # 0 isw2∗ V † (−t) G(s, s ) = Tr ρ(t) |w3 , α w3 ihw3 , α w3 |e w3 ,α w3 × " X is0 w2 V (−t) |w2 , α w2 ihw2 , α w2 |e #! . (2.17) w2 ,α w2 We differentiate with respect to is0 = − β0 and with respect to is = − β. Then, we take the limit as β, β0 → 0:
∂2 G i β,i β0 0 β,β 0 =0 ∂β ∂β " X # ∗ † = Tr ρ(t) w3 |w3 , α w3 ihw3 , α w3 |V (−t) (2.18) (2.19) w3 ,α w3 × " X #! w2 |w2 , α w2 ihw2 , α w2 |V (−t) w2 ,α w2 = Tr( ρ(t) W V † (−t) W V (−t)) . † (2.20) Recall that ρ(t) := U ρU † . Time dependence is transferred from ρ(t), V (−t) = UV †U † , and V † (t) = UVU † to W † and W, under the trace’s cyclicality:   ∂2 0 † † G(i β,i β ) = Tr ρ W (t) V W (t) V 0 β,β =0 ∂ β ∂ β0 D E = W † (t) V † W (t) V = F (t) . (2.21) (2.22) By Eqs. (2.10) and (2.13), the left-hand side equals ∂ 2 D −( βW + β 0W 0 ) E e . β,β 0 =0 ∂ β ∂ β0 (2.23)  Theorem 1 resembles Jarzynski’s fluctuation relation in several ways. Jarzynski’s Equality encodes a scheme for measuring the difficult-to-calculate ∆F from realizable nonequilibrium trials. Theorem 1 encodes a scheme for measuring the 23 difficult-to-calculate F (t) from realizable nonequilibrium trials. ∆F depends on just a temperature and two Hamiltonians. Similarly, the conventional F (t) (defined with respect to ρ = e−H/T /Z) depends on just a temperature, a Hamiltonian, and two unitaries. Jarzynski relates ∆F to the characteristic function of a probability distribution. Theorem 1 relates F (t) to (a moment of) the characteristic function of a (complex) distribution. The complex distribution, P(W,W 0 ), is a combination of probability amplitudes à ρ related to quasiprobabilities. The distribution in Jarzynski’s Equality is a combination of probabilities. The quasiprobability-vs.-probability contrast fittingly arises from the OTOC’s out-of-time ordering. F (t) signals quantum behavior (noncommutation), as quasiprobabilities signal quantum behaviors (e.g., entanglement). Timeordered correlators similar to F (t) track only classical behaviors and are moments of (summed) classical probabilities [44]. OTOCs that encode more time reversals than F (t) are moments of combined quasiprobability-like distributions lengthier than à ρ [44]. 2.4 Conclusions The Jarzynski-like equality for the out-of-time correlator combines an important tool from nonequilibrium statistical mechanics with an important tool from quantum information, high-energy theory, and condensed matter. The union opens all these fields to new modes of analysis. For example, Theorem 1 relates the OTOC to a combined quantum amplitude à ρ . This à ρ is closely related to a quasiprobability. The OTOC and quasiprobabilities have signaled nonclassical behaviors in distinct settings—in high-energy theory and condensed matter and in quantum optics, respectively. The relationship between OTOCs and quasiprobabilities merits study: What is the relationship’s precise nature? How does à ρ behave over time scales during which F (t) exhibits known behaviors (e.g., until the dissipation time or from the dissipation time to the scrambling time [16])? Under what conditions does à ρ behave nonclassically (assume negative or nonreal values)? How does a chaotic system’s à ρ look? These questions are under investigation [44]. As another example, fluctuation relations have been used to estimate the free-energy difference ∆F from experimental data. Experimental measurements of F (t) are possible for certain platforms, in certain regimes [16–20]. Theorem 1 expands the set of platforms and regimes. Measuring quantum amplitudes, as via weak mea- 24 surements [51–54], now offers access to F (t). Inferring small systems’ à ρ ’s with existing platforms [57] might offer a challenge for the near future. Finally, Theorem 1 can provide a new route to bounding F (t). A Lyapunov exponent λ L governs the chaotic decay of F (t). The exponent has been bounded, including with Lieb-Robinson bounds and complex analysis [7, 61, 62]. The righthand side of Eq. (2.11) can provide an independent bounding method that offers new insights. 25 BIBLIOGRAPHY [1] N. Yunger Halpern, Phys. Rev. A 95, 012120 (2017). [2] S. H. Shenker and D. Stanford, Journal of High Energy Physics 3, 67 (2014). [3] S. H. Shenker and D. Stanford, Journal of High Energy Physics 12, 46 (2014). [4] S. H. Shenker and D. Stanford, Journal of High Energy Physics 5, 132 (2015). [5] D. A. Roberts, D. Stanford, and L. Susskind, Journal of High Energy Physics 3, 51 (2015). [6] D. A. Roberts and D. Stanford, Physical Review Letters 115, 131603 (2015). [7] J. Maldacena, S. H. Shenker, and D. Stanford, 1503.01409. ArXiv e-prints (2015), [8] A. Larkin and Y. N. Ovchinnikov, Soviet Journal of Experimental and Theoretical Physics 28 (1969). [9] Y. Sekino and L. Susskind, Journal of High Energy Physics 2008, 065 (2008). [10] Y. Huang, Y.-L. Zhang, and X. Chen, ArXiv e-prints (2016), 1608.01091. [11] B. Swingle and D. Chowdhury, ArXiv e-prints (2016), 1608.03280. [12] R. Fan, P. Zhang, H. Shen, and H. Zhai, ArXiv e-prints (2016), 1608.01914. [13] R.-Q. He and Z.-Y. Lu, ArXiv e-prints (2016), 1608.03586. [14] Y. Chen, ArXiv e-prints (2016), 1608.02765. [15] P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Journal of High Energy Physics 2, 4 (2016), 1511.04021. [16] B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hayden, ArXiv e-prints (2016), 1602.06271. [17] N. Y. Yao et al., ArXiv e-prints (2016), 1607.01801. [18] G. Zhu, M. Hafezi, and T. Grover, ArXiv e-prints (2016), 1607.00079. [19] J. Li et al., ArXiv e-prints (2016), 1609.01246. [20] M. Gärttner et al., ArXiv e-prints (2016), 1608.08938. [21] J. Maldacena and D. Stanford, ArXiv e-prints (2016), 1604.07818. 26 [22] J. Polchinski and V. Rosenhaus, Journal of High Energy Physics 4, 1 (2016), 1601.06768. [23] C. Jarzynski, Physical Review Letters 78, 2690 (1997). [24] G. E. Crooks, Physical Review E 60, 2721 (1999). [25] H. Tasaki, arXiv e-print (2000), cond-mat/0009244. [26] J. Kurchan, eprint arXiv:cond-mat/0007360 (2000), cond-mat/0007360. [27] O.-P. Saira et al., Phys. Rev. Lett. 109, 180601 (2012). [28] C. Chipot and A. Pohorille, editors, Free Energy Calculations: Theory and Applications in Chemistry and Biology, Springer Series in Chemical Physics Vol. 86 (Springer-Verlag, 2007). [29] D. Collin et al., Nature 437, 231 (2005). [30] F. Douarche, S. Ciliberto, A. Petrosyan, and I. Rabbiosi, EPL (Europhysics Letters) 70, 593 (2005). [31] V. Blickle, T. Speck, L. Helden, U. Seifert, and C. Bechinger, Phys. Rev. Lett. 96, 070603 (2006). [32] N. C. Harris, Y. Song, and C.-H. Kiang, Phys. Rev. Lett. 99, 068101 (2007). [33] A. Mossa, M. Manosas, N. Forns, J. M. Huguet, and F. Ritort, Journal of Statistical Mechanics: Theory and Experiment 2009, P02060 (2009). [34] M. Manosas, A. Mossa, N. Forns, J. M. Huguet, and F. Ritort, Journal of Statistical Mechanics: Theory and Experiment 2009, P02061 (2009). [35] T. B. Batalhão et al., Physical Review Letters 113, 140601 (2014), 1308.3241. [36] S. An et al., Nature Physics 11, 193 (2015). [37] K. Maruyama, F. Nori, and V. Vedral, Rev. Mod. Phys. 81, 1 (2009). [38] J. Åberg, Nature Communications 4, 1925 (2013), 1110.6121. [39] N. Yunger Halpern, A. J. P. Garner, O. C. O. Dahlsten, and V. Vedral, New Journal of Physics 17, 095003 (2015). [40] S. Salek and K. Wiesner, ArXiv e-prints (2015), 1504.05111. [41] N. Yunger Halpern, A. J. P. Garner, O. C. O. Dahlsten, and V. Vedral, Phys. Rev. E 97, 052135 (2018). [42] O. C. O. Dahlsten et al., New Journal of Physics 19, 043013 (2017). 27 [43] A. M. Alhambra, L. Masanes, J. Oppenheim, and C. Perry, Phys. Rev. X 6, 041017 (2016). [44] J. Dressel, B. Swingle, and N. Yunger Halpern, in prep. [45] P. Talkner, E. Lutz, and P. Hänggi, Phys. Rev. E 75, 050102 (2007). [46] J. G. Kirkwood, Physical Review 44, 31 (1933). [47] P. A. M. Dirac, Reviews of Modern Physics 17, 195 (1945). [48] J. Dressel, Phys. Rev. A 91, 032116 (2015). [49] H. J. Carmichael, Statistical Methods in Quantum Optics I: Master Equations and Fokker-Planck Equations (Springer-Verlag, 2002). [50] C. Ferrie, Reports on Progress in Physics 74, 116001 (2011). [51] J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011). [52] J. S. Lundeen and C. Bamber, Phys. Rev. Lett. 108, 070402 (2012). [53] C. Bamber and J. S. Lundeen, Phys. Rev. Lett. 112, 070405 (2014). [54] M. Mirhosseini, O. S. Magaña Loaiza, S. M. Hashemi Rafsanjani, and R. W. Boyd, Phys. Rev. Lett. 113, 090402 (2014). [55] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, Rev. Mod. Phys. 86, 307 (2014). [56] A. G. Kofman, S. Ashhab, and F. Nori, Physics Reports 520, 43 (2012), Nonperturbative theory of weak pre- and post-selected measurements. [57] T. C. White et al., npj Quantum Information 2 (2016). [58] J. Dressel, T. A. Brun, and A. N. Korotkov, Phys. Rev. A 90, 032302 (2014). [59] G. A. Smith, S. Chaudhury, A. Silberfarb, I. H. Deutsch, and P. S. Jessen, Phys. Rev. Lett. 93, 163602 (2004). [60] I. Danshita, M. Hanada, and M. Tezuka, ArXiv e-prints (2016), 1606.02454. [61] N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden, Journal of High Energy Physics 2013, 22 (2013). [62] A. Kitaev, A simple model of quantum holography, 2015. 28 Chapter 3 THE QUASIPROBABILITY BEHIND THE OUT-OF-TIME-ORDERED CORRELATOR This chapter was published as [1]. Two topics have been flourishing independently: the out-of-time-ordered correlator (OTOC) and the Kirkwood-Dirac (KD) quasiprobability distribution. The OTOC signals chaos, and the dispersal of information through entanglement, in quantum many-body systems [2–7]. Quasiprobabilities represent quantum states as phasespace distributions represent statistical-mechanical states [8]. Classical phase-space distributions are restricted to positive values; quasiprobabilities are not. The bestknown quasiprobability is the Wigner function. The Wigner function can become negative; the KD quasiprobability, negative and nonreal [9–15]. Nonclassical values flag contextuality, a resource underlying quantum-computation speedups [15– 21]. Hence the KD quasiprobability, like the OTOC, reflects nonclassicality. Yet disparate communities use these tools: The OTOC F (t) features in quantum information theory, high-energy physics, and condensed matter. Contexts include black holes within AdS/CFT duality [2, 22–24], weakly interacting field theories [25– 28], spin models [2, 29], and the Sachdev-Ye-Kitaev model [30, 31]. The KD distribution features in quantum optics. Experimentalists have inferred the quasiprobability from weak measurements of photons [11–14, 32–35] and superconducting qubits [36, 37]. The two tools were united in [38]. The OTOC was shown to equal a moment of a summed quasiprobability, à ρ : F (t) = ∂ 2 D −( βW + β 0W 0 ) E e . ∂ β ∂ β0 β,β 0 =0 (3.1) W and W 0 denote measurable random variables analogous to thermodynamic work; and β, β0 ∈ R. The average h.i is with respect to a sum of quasiprobability values à ρ (.). Equation (3.1) resembles Jarzynski’s Equality, a fluctuation relation in nonequilibrium statistical mechanics [39]. Jarzynski cast a useful, difficult-tomeasure free-energy difference ∆F in terms of the characteristic function of a probability. Equation (3.1) casts the useful, difficult-to-measure OTOC in terms of the 29 characteristic function of a summed quasiprobability.1 The OTOC has recently been linked to thermodynamics also in [40, 41]. Equation (3.1) motivated definitions of quantities that deserve study in their own right. The most prominent quantity is the quasiprobability à ρ . à ρ is more fundamental than F (t): à ρ is a distribution that consists of many values. F (t) equals a combination of those values—a derived quantity, a coarse-grained quantity. à ρ contains more information than F (t). This paper spotlights à ρ and related quasiprobabilities “behind the OTOC.” à ρ , we argue, is an extension of the KD quasiprobability. Weak-measurement tools used to infer KD quasiprobabilities can be applied to infer à ρ from experiments [38]. Upon measuring à ρ , one can recover the OTOC. Alternative OTOCmeasurement proposals rely on Lochshmidt echoes [42], interferometry [38, 42– 44], clocks [45], particle-number measurements of ultracold atoms [44, 46, 47], and two-point measurements [40]. Initial experiments have begun the push toward characterizing many-body scrambling: OTOCs of an infinite-temperature four-site NMR system have been measured [48]. OTOCs of symmetric observables have been measured with infinite-temperature trapped ions [49] and in nuclear spin chains [50]. Weak measurements offer a distinct toolkit, opening new platforms and regimes to OTOC measurements. The weak-measurement scheme in [38] is expected to provide a near-term challenge for superconducting qubits [36, 51–56], trapped ions [57– 63], ultracold atoms [64], cavity quantum electrodynamics (QED) [65, 66], and perhaps NMR [67, 68]. We investigate the quasiprobability à ρ that “lies behind” the OTOC. The study consists of three branches: We discuss experimental measurements, calculate (a coarse-grained) à ρ , and explore mathematical properties. Not only does quasiprobability theory shed new light on the OTOC. The OTOC also inspires questions about quasiprobabilities and motivates weak-measurement experimental challenges. The paper is organized as follows. In a technical introduction, we review the KD quasiprobability, the OTOC, the OTOC quasiprobability à ρ , and schemes for measuring à ρ . We also introduce our set-up and notation. All the text that follows the technical introduction is new (never published before, to our knowledge). Next, we discuss experimental measurements. We introduce a coarse-graining A˜ρ of à ρ . The coarse-graining involves a “projection trick” that decreases, expo1 For a thorough comparison of Eq. (3.1) with Jarzynski’s equality, see the two paragraphs that follow the proof in [38]. 30 nentially in system size, the number of trials required to infer F (t) from weak measurements. We evaluate pros and cons of the quasiprobability-measurement schemes in [38]. We also compare our schemes with alternative F (t)-measurement schemes [42, 43, 45]. We then present a circuit for weakly measuring a qubit system’s A˜ρ . Finally, we show how to infer the coarse-grained A˜ρ from alternative OTOC-measurement schemes (e.g., [42]). Sections 3.3 and 3.4 feature calculations of A˜ρ . First, we numerically simulate a transverse-field Ising model. A˜ρ changes significantly, we find, over time scales relevant to the OTOC. The quasiprobability’s behavior distinguishes nonintegrable from integrable Hamiltonians. The quasiprobability’s negativity and nonreality remains robust with respect to substantial quantum interference. We then calculate an average, over Brownian circuits, of A˜ρ . Brownian circuits model chaotic dynamics: The system is assumed to evolve, at each time step, under random two-qubit couplings [69–72]. A final “theory” section concerns mathematical properties and physical interpretations of à ρ . à ρ shares some, though not all, of its properties with the KD distribution. The OTOC motivates a generalization of a Bayes-type theorem obeyed by the KD distribution [15, 73–76]. The generalization exponentially shrinks the memory required to compute weak values, in certain cases. The OTOC also motivates a generalization of decompositions of quantum states ρ. This decomposition property may help experimentalists assess how accurately they prepared the desired initial state when measuring F (t). A time-ordered correlator FTOC (t) analogous to F (t), we show next, depends on a quasiprobability that can reduce to a probability. The OTOC quasiprobability lies farther from classical probabilities than the TOC quasiprobability, as the OTOC registers quantum-information scrambling that FTOC (t) does not. Finally, we recall that the OTOC encodes three time reversals. OTOCs that encode more are moments of sums of “longer” quasiprobabilities. We conclude with theoretical and experimental opportunities. We invite readers to familiarize themselves with the technical review, then to dip into the sections that interest them most. The technical review is intended to introduce condensed-matter, high-energy, and quantum-information readers to the KD quasiprobability and to introduce quasiprobability and weak-measurement readers to the OTOC. Armed with the technical review, experimentalists may wish to focus on Sec. 3.2 and perhaps Sec. 3.3. Adherents of abstract theory may prefer Sec. 3.5. The computationally minded may prefer Sections 3.3 and 3.4. The paper’s modules 31 (aside from the technical review) are independently accessible. 3.1 Technical introduction This review consists of three parts. In Sec. 3.1, we overview the KD quasiprobability. Section 3.1 introduces our set-up and notation. In Sec. 3.1, we review the OTOC and its quasiprobability à ρ . We overview also the weak-measurement and interference schemes for measuring à ρ and F (t). The quasiprobability section (3.1) provides background for quantum-information, high-energy, and condensed-matter readers. The OTOC section (3.1) targets quasiprobability and weak-measurement readers. We encourage all readers to study the set-up (3.1), as well as à ρ and the schemes for measuring à ρ (3.1). The KD quasiprobability in quantum optics The Kirkwood-Dirac quasiprobability is defined as follows. Let S denote a quantum system associated with a Hilbert space H . Let {|ai} and {| f i} denote orthonormal bases for H . Let B(H ) denote the set of bounded operators defined on H , and let O ∈ B(H ). The KD quasiprobability (1) ÃO (a, f ) := h f |aiha|O| f i , (3.2) regarded as a function of a and f , contains all the information in O, if ha| f i , 0 for all a, f . Density operators O = ρ are often focused on in the literature and in this (1) paper. This section concerns the context, structure, and applications of ÃO (a, f ). We set the stage with phase-space representations of quantum mechanics, alternative quasiprobabilities, and historical background. Equation (3.2) facilitates retrodiction, or inference about the past, reviewed in Sec. 3.1. How to decompose an operator O in terms of KD-quasiprobability values appears in Sec. 3.1. The quasiprobability has mathematical properties reviewed in Sec. 3.1. Much of this section parallels Sec. 3.5, our theoretical investigation of the OTOC quasiprobability. More background appears in [15]. Phase-space representations, alternative quasiprobabilities, and history Phase-space distributions form a mathematical toolkit applied in Liouville mechanics [77]. Let S denote a system of 6N degrees of freedom (DOFs). An example system consists of N particles, lacking internal DOFs, in a three-dimensional space. We index the particles with i and let α = x, y, z. The α th component qiα of particle 32 i’s position is conjugate to the α th component piα of the particle’s momentum. The variables qiα and piα label the axes of phase space. Suppose that the system contains many DOFs: N  1. Tracking all the DOFs is difficult. Which phase-space point S occupies, at any instant, may be unknown. The probability that, at time t, S occupies an infinitesimal volume element localized at (q1x , . . . , pNz ) is ρ({qiα }, {piα }; t) d 3N q d 3N p. The phase-space distribution ρ({qiα }, {piα }; t) is a probability density. qiα and piα seem absent from quantum mechanics (QM), prima facie. Most introductions to QM cast quantum states in terms of operators, Dirac kets |ψi, and wave functions ψ(x). Classical variables are relegated to measurement outcomes and to the classical limit. Wigner, Moyal, and others represented QM in terms of phase space [8]. These representations are used most in quantum optics. In such a representation, a quasiprobability density replaces the statistical-mechanical probability density ρ.2 Yet quasiprobabilities violate axioms of probability [17]. Probabilities are nonnegative, for example. Quasiprobabilities can assume negative values, associated with nonclassical physics such as contextuality [15–19, 21], and nonreal values. Relaxing different axioms leads to different quasiprobabilities. Different quasiprobabilities correspond also to different orderings of noncommutative operators [10]. The best-known quasiprobabilities include the Wigner function, the Glauber-Sudarshan P representation, and the Husimi Q function [8]. The KD quasiprobability resembles a little brother of theirs, whom hardly anyone has heard of [78]. Kirkwood and Dirac defined the quasiprobability independently in 1933 [9] and 1945 [10]. Their finds remained under the radar for decades. Rihaczek rediscovered the distribution in 1968, in classical-signal processing [79, 80]. (The KD quasiprobability is sometimes called “the Kirkwood-Rihaczek distribution.”) The quantum community’s attention has revived recently. Reasons include experimental measurements, mathematical properties, and applications to retrodiction and state decompositions. 2 We will focus on discrete quantum systems, motivated by a spin-chain example. Discrete systems are governed by quasiprobabilities, which resemble probabilities. Continuous systems are governed by quasiprobability densities, which resemble probability densities. Our quasiprobabilities can be replaced with quasiprobability densities, and our sums can be replaced with integrals, in, e.g., quantum field theory. 33 Bayes-type theorem and retrodiction with the KD quasiprobability Prediction is inference about the future. Retrodiction is inference about the past. One uses the KD quasiprobability to infer about a time t 0, using information about an event that occurred before t 0 and information about an event that occurred after t 0. This forward-and-backward propagation evokes the OTOC’s out-of-time ordering. We borrow notation from, and condense the explanation in, [15]. Let S denote a discrete quantum system. Consider preparing S in a state |ii at time t = 0. Suppose that S evolves under a time-independent Hamiltonian that generates the family Ut of P unitaries. Let F denote an observable measured at time t 00 > 0. Let F = f f | f ih f | be the eigendecomposition, and let f denote the outcome. P Let A = a a|aiha| be the eigendecomposition of an observable that fails to commute with F. Let t 0 denote a time in (0,t 00 ). Which value can we most reasonably attribute to the system’s time-t 0 A, knowing that S was prepared in |ii and that the final measurement yielded f ? Propagating the initial state forward to time t 0 yields |i0i := Ut 0 |ii. Propagating the final state backward yields | f 0i := Ut†00 −t 0 | f i. Our best guess about A is the weak value [37, 74–76, 81–83] ! h f 0 |A|i0i Aweak (i, f ) := < . (3.3) h f 0 |i0i The real part of a complex number z is denoted by <(z). The guess’s accuracy is quantified with a distance metric (Sec. 3.5) and with comparisons to weakmeasurement data. Aharonov et al. discovered weak values in 1988 [73]. Weak values be anomalous, or strange: Aweak can exceed the greatest eigenvalue amax of A and can dip below the least eigenvalue amin . Anomalous weak values concur with negative quasiprobabilities and nonclassical physics [15, 18, 19, 84, 85]. Debate has surrounded weak values’ role in quantum mechanics [86–92]. The weak value Aweak , we will show, depends on the KD quasiprobability. We replace the A in Eq. (3.3) with its eigendecomposition. Factoring out the eigenvalues yields ! X h f 0 |aiha|i0i . (3.4) Aweak (i, f ) = a< h f 0 |i0i a 34 The weight <(.) is a conditional quasiprobability. It resembles a conditional probability—the likelihood that, if |ii was prepared and the measurement yielded f , a is the value most reasonably attributable to A. Multiplying and dividing the argument by hi0 | f 0i yields p̃(a|i, f ) := < (h f 0 |aiha|i0ihi0 | f 0i) . |h f 0 |i0i| 2 (3.5) Substituting into Eq. (3.4) yields Aweak (i, f ) = X a p̃(a|i, f ) . (3.6) a Equation (3.6) illustrates why negative quasiprobabilities concur with anomalous weak values. Suppose that p̃(a|i, f ) ≥ 0 ∀a. The triangle inequality, followed by the Cauchy-Schwarz inequality, implies |Aweak (i, f )| ≤ ≤ X a X a p̃(a|i, f ) (3.7) |a| · | p̃(a|i, f )| (3.8) a ≤ |amax | X | p̃(a|i, f )| (3.9) p̃(a|i, f ) (3.10) a = |amax | X a = |amax | . (3.11) The penultimate equality follows from p̃(a|i, f ) ≥ 0. Suppose, now, that the quasiprobability contains a negative value p̃(a− |i, f ) < 0. The distribution remains normalized. Hence the rest of the p̃ values sum to > 1. The RHS of (3.9) exceeds |amax |. The numerator of Eq. (3.5) is the Terletsky-Margenau-Hill (TMH) quasiprobability [74, 93–95]. The TMH distribution is the real part of a complex number. That complex generalization, h f 0 |aiha|i0ihi0 | f 0i , (3.12) is the KD quasiprobability (3.2). We can generalize the retrodiction argument to arbitrary states ρ [96]. Let D (H ) denote the set of density operators (unit-trace linear positive-semidefinite operators) 35 defined on H . Let ρ = i pi |iihi| ∈ D (H ) be a density operator’s eigendecomposition. Let ρ0 := Ut 0 ρUt†0 . The weak value Eq. (3.3) becomes ! h f 0 |A ρ0 | f 0i Aweak ( ρ, f ) := < . (3.13) h f 0 | ρ0 | f 0i P Let us eigendecompose A and factor out a a. The eigenvalues are weighted by the conditional quasiprobability P p̃(a| ρ, f ) = < (h f 0 |aiha| ρ0 | f 0i) . h f 0 | ρ0 | f 0i (3.14) The numerator is the TMH quasiprobability for ρ. The complex generalization 0 0 0 Ã(1) ρ (a, f ) = h f |aiha| ρ | f i (3.15) is the KD quasiprobability (3.2) for ρ.3 We rederive (3.15), via an operator decomposition, next. Decomposing operators in terms of KD-quasiprobability coefficients The KD distribution can be interpreted not only in terms of retrodiction, but also in terms of operation decompositions [11, 12]. Quantum-information scientists decompose qubit states in terms of Pauli operators. Let σ = σ x x̂ + σ y ŷ + σ z ẑ denote a vector of the one-qubit Paulis. Let n̂ ∈ R3 denote a unit vector. Let ρ denote any state of a qubit, a two-level quantum system. ρ can be expressed as ρ = 21 (1 + n̂ · σ) . The identity operator is denoted by 1. The n̂ components n` constitute decomposition coefficients. The KD quasiprobability consists of coefficients in a more general decomposition. Let S denote a discrete quantum system associated with a Hilbert space H . Let {| f i} and {|ai} denote orthonormal bases for H . Let O ∈ B(H ) denote a bounded operator defined on H . Consider operating on each side of O with a resolution of unity:     X X O = 1O 1 =  |aiha|  O  | f ih f |  (3.16)   a f X = |aih f | ha|O| f i . (3.17) a, f 3 The A in the quasiprobability à ρ should not be confused with the observable A. 36 Suppose that every element of {|ai} has a nonzero overlap with every element of {| f i}: h f |ai , 0 ∀a, f . (3.18) Each term in Eq. (3.17) can be multiplied and divided by the inner product: X |aih f | O= h f |aiha|O| f i . (3.19) h f |ai a, f n f|o Under condition (3.18), |aih h f |ai forms an orthonormal basis for B(H ) . [The orthonormality is with respect to the Hilbert-Schmidt inner product. Let O1 , O2 ∈ B(H ). The operators have the Hilbert-Schmidt inner product (O1 , O2 ) = Tr(O1† O2 ).] The KD quasiprobability h f |aiha|O| f i consists of the decomposition coefficients. Condition (3.18) is usually assumed to hold [11, 12, 35]. In [11, 12], for example, {|aiha|} and {| f ih f |} manifest as the position and momentum eigenbases {|xi} and {|pi}. Let |ψi denote a pure state. Let ψ(x) and ψ̃(p) represent |ψi relative to the position and momentum eigenbases. The KD quasiprobability for ρ = |ψihψ| has the form Ã(1) (p, x) = hx|pihp|ψihψ|xi |ψihψ| e−ixp/~ ψ̃(p) ψ ∗ (x) . = √ 2π~ (3.20) (3.21) The OTOC motivates a relaxation of condition (3.18) (Sec. 3.5). [Though assumed in the operator decomposition (3.19), and assumed often in the literature, condition (3.18) need not hold in arbitrary KD-quasiprobability arguments.] Properties of the KD quasiprobability The KD quasiprobability shares some, but not all, of its properties with other quasiprobabilities. The notation below is defined as it has been throughout Sec. 3.1. (1) Property 1. The KD quasiprobability ÃO (a, f ) maps B(H ) × {a} × { f } to C . The domain is a composition of the set B(H ) of bounded operators and two sets of real numbers. The range is the set C of complex numbers, not necessarily the set R of real numbers. The Wigner function assumes only real values. Only by dipping below zero can the Wigner function deviate from classical probabilistic behavior. The KD distribution’s negativity has the following physical significance: Imagine projectively 37 measuring two (commuting) observables, A and B, simultaneously. The measurement has some probability p(a; b) of yielding the values a and b. Now, suppose that A does not commute with B. No joint probability distribution p(a; b) exists. Infinitely precise values cannot be ascribed to noncommuting observables simultaneously. Negative quasiprobability values are not observed directly: Observable phenomena are modeled by averages over quasiprobability values. Negative values are visible only on scales smaller than the physical coarse-graining scale. But negativity causes observable effects, visible in sequential measurements. Example effects include anomalous weak values [15, 18, 19, 73, 84, 85] and violations of Leggett-Garg inequalities [97, 98]. Unlike the Wigner function, the KD distribution can assume nonreal values. Consider measuring two noncommuting observables sequentially. How much does the first measurement affect the second measurement’s outcome? This disturbance is encoded in the KD distribution’s imaginary component [99–102]. Property 2. Summing Ã(1) ρ (a, f ) over a yields a probability distribution. So does (1) summing à ρ (a, f ) over f . Consider substituting O = ρ into Eq. (3.2). Summing over a yields h f | ρ| f i. This inner product equals a probability, by Born’s Rule. Property 3. The KD quasiprobability is defined as in Eq. (3.2) regardless of whether {a} and { f } are discrete. The KD distribution and the Wigner function were defined originally for continuous systems. Discretizing the Wigner function is less straightforward [17, 21]. Property 4. The KD quasiprobability obeys an analog of Bayes’ Theorem, Eq. (3.5). Bayes’ Theorem governs the conditional probability p( f |i) that an event f will occur, given that an event i has occurred. p( f |i) is expressed in terms of the conditional probability p(i| f ) and the absolute probabilities p(i) and p( f ): p( f |i) = p(i| f ) p( f ) . p(i) (3.22) Equation (3.22) can be expressed in terms of jointly conditional distributions. Let p(a|i, f ) denote the probability that an event a will occur, given that an event i occurred and that f occurred subsequently. p(a, f |i) is defined similarly. What is 38 the joint probability p(i, f , a) that i, f , and a will occur? We can construct two expressions: p(i, f , a) = p(a|i, f ) p(i, f ) = p(a, f |i) p(i) . (3.23) The joint probability p(i, f ) equals p( f |i) p(i). This p(i) cancels with the p(i) on the right-hand side of Eq. (3.23). Solving for p(a|i, f ) yields Bayes’ Theorem for jointly conditional probabilities, p(a|i, f ) = p(a, f |i) . p( f |i) (3.24) Equation (3.5) echoes Eq. (3.24). The KD quasiprobability’s Bayesian behavior [13, 101] has been applied to quantum state tomography [11, 12, 14, 102–105] and to quantum foundations [99]. Having reviewed the KD quasiprobability, we approach the extended KD quasiprobability behind the OTOC. We begin by concretizing our set-up, then reviewing the OTOC. Set-up This section concerns the set-up and notation used throughout the rest of this paper. Our framework is motivated by the OTOC, which describes quantum many-body systems. Examples include black holes [2, 31], the Sachdev-Ye-Kitaev model [30, 31], other holographic systems [22–24] and spin chains. We consider a system S associated with a Hilbert space H of dimensionality d. The system evolves under a Hamiltonian H that might be nonintegrable or integrable. H generates the timeevolution operator U := e−iHt . We will have to sum or integrate over spectra. For concreteness, we sum, supposing that H is discrete. A spin-chain example, discussed next, motivates our choice. Our sums can be replaced with integrals unless, e.g., we evoke spin chains explicitly. We will often illustrate with a one-dimensional (1D) chain of spin- 21 degrees of freedom. Figure 3.1 illustrates the chain, simulated numerically in Sec. 3.3. Let N denote the number of spins. This system’s H has dimensionality d = 2 N . We will often suppose that S occupies, or is initialized to, a state X ρ= p j | jih j | ∈ D(H ) . j (3.25) 39 1 W= <latexit sha1_base64="xIwUqL0Q0lbyvzWWjji9SI+p3S4=">AAACKXicbZBNS8NAEIY3ftb6FfXoZbEI9WBJRFAPQsWLXqSCsYWmLZvtpl26yYbdjVBDfo8X/4qXHvy6+kfctEG0dWDg4Z0ZZub1IkalsqwPY25+YXFpubBSXF1b39g0t7bvJY8FJg7mjIuGhyRhNCSOooqRRiQICjxG6t7gMqvXH4iQlId3ahiRVoB6IfUpRkpLHfPCDZDqY8SSegrhOYSupL0AtR81cUUDIjVkLZ6f2Gk7+VHLN/AQ2gcw7Zglq2KNA86CnUMJ5FHrmCO3y3EckFBhhqRs2lakWgkSimJG0qIbSxIhPEA90tQYIr2ulYxfTeG+VrrQ50JnqOBY/T2RoEDKYeDpzuxqOV3LxP9qzVj5p62EhlGsSIgni/yYQcVh5hvsUkGwYkMNCAuqb4W4jwTCSrtb1CbY0y/PgnNUOatYt8el6nXuRgHsgj1QBjY4AVVwBWrAARg8gRfwCt6MZ2NkvBufk9Y5I5/ZAX/C+PoGNQGk2g==</latexit> z 2 … ⌦ 1⌦(N 1) V = 1⌦(N <latexit sha1_base64="L3a1oTxwe6k+eTnl+Y2LuEfTfTI=">AAACH3icbVDLSgMxFM3UV62vqks3F4tQF5YZEaoLoeBGN1LBaQudtmTSTBuaeZBkxDLMp7jxV9y4UBF3/RvTB6KtBwKHc+7l5hw34kwq0xwZmaXlldW17HpuY3Nreye/u1eTYSwItUnIQ9FwsaScBdRWTHHaiATFvstp3R1cjf36AxWShcG9Gka05eNewDxGsNJSJ1+uAVwCOD5WfddLrLSdgBMq5lMJxVs4AesYUviRwJGs5+P2YydfMEvmBLBIrBkpoBmqnfyX0w1J7NNAEY6lbFpmpFoJFooRTtOcE0saYTLAPdrUNMD6XCuZBEzhSCtd8EKhX6Bgov7eSLAv5dB39eQ4iJz3xuJ/XjNW3nkrYUEUKxqQ6SEv5qBCGLcFXSYoUXyoCSaC6b8C6WOBidKd5nQJ1nzkRWKfli5K5t1ZoXIzayOLDtAhKiILlVEFXaMqshFBT+gFvaF349l4NT6Mz+loxpjt7KM/MEbfWUygRQ==</latexit> N 1) ⌦ x Figure 3.1: Spin-chain example: A spin chain exemplifies the quantum many-body systems characterized by the out-of-time-ordered correlator (OTOC). We illustrate with a one-dimensional chain of N spin- 21 degrees of freedom. The vertical red bars mark the sites. The dotted red arrows illustrate how spins can point in arbitrary directions. The OTOC is defined in terms of local unitary or Hermitian operators W and V . Example operators include single-qubit Paulis σ x and σ z that act nontrivially on opposite sides of the chain. The set of density operators defined on H is denoted by D (H ), as in Sec. 3.1. Orthonormal eigenstates are indexed by j; eigenvalues are denoted by p j . Much literature focuses on temperature-T thermal states e−H/T /Z. (The partition function Z normalizes the state.) We leave the form of ρ general, as in [38]. The OTOC is defined in terms of local operators W and V . In the literature, W and V are assumed to be unitary and/or Hermitian. Unitarity suffices for deriving the results in [38], as does Hermiticity. Unitarity and Hermiticity are assumed there, and here, for convenience.4 In our spin-chain example, the operators manifest as one-qubit Paulis that act nontrivially on opposite sides of the chain, e.g., W = σ z ⊗ 1⊗(N −1) , and V = 1⊗(N −1) ⊗ σ x . In the Heisenberg Picture, W evolves as W (t) := U † WU . The operators eigendecompose as X W= w` |w` , α w` ihw` , α w` | (3.26) w` ,α w ` and V= X v` |v` , λ v` ihv` , λ v` | . (3.27) v` ,λ v ` The eigenvalues are denoted by w` and v` . The degeneracy parameters are denoted by α w` and λ v` . Recall that W and V are local. In our example, W acts nontrivially on just one of N  1 qubits. Hence W and V are exponentially degenerate in N. The degeneracy parameters can be measured: Some nondegenerate Hermitian 4 Measurements of W and V are discussed in [38] and here. Hermitian operators G W and GV generate W and V . If W and V are not Hermitian, G W and GV are measured instead of W and V . 40 operator W̃ has eigenvalues in a one-to-one correspondence with the α w` ’s. A measurement of W and W̃ outputs a tuple (w` , α w` ). We refer to such a measurement as “a W̃ measurement,” for conciseness. Analogous statements concern V and a Hermitian operator Ṽ . Section 3.2 introduces a trick that frees us from bothering with degeneracies. The out-of-time-ordered correlator Given two unitary operators W and V , the out-of-time-ordered correlator is defined as F (t) := hW † (t)V † W (t)V i ≡ Tr( ρW † (t)V † W (t)V ) . (3.28) This object reflects the degree of noncommutativity of V and the Heisenberg operator W (t). More precisely, the OTOC appears in the expectation value of the squared magnitude of the commutator [W (t),V ], C(t) := h[W (t),V ]† [W (t),V ]i = 2 − 2<(F (t)) . (3.29) Even if W and V commute, the Heisenberg operator W (t) generically does not commute with V at sufficiently late times. An analogous definition involves Hermitian W and V . The commutator’s square magnitude becomes C(t) = −h[W (t),V ]2 i. (3.30) This squared commutator involves TOC (time-ordered-correlator) and OTOC terms. The TOC terms take the forms hV W (t)W (t)V i and hW (t)VV W (t)i. [Technically, hV W (t)W (t)V i is time-ordered. hW (t)VV W (t)i behaves similarly.] The basic physical process reflected by the OTOC is the spread of Heisenberg operators with time. Imagine starting with a simple W, e.g., an operator acting nontrivially on just one spin in a many-spin system. Time-evolving yields W (t). The operator has grown if W (t) acts nontrivially on more spins than W does. The operator V functions as a probe for testing whether the action of W (t) has spread to the spin on which V acts nontrivially. Suppose W and V are unitary and commute. At early times, W (t) and V approximately commute. Hence F (t) ≈ 1, and C(t) ≈ 0. Depending on the dynamics, at later times, W (t) may significantly fail to commute with V . In a chaotic quantum 41 system, W (t) and V generically do not commute at late times, for most choices of W and V . The analogous statement for Hermitian W and V is that F (t) approximately equals the TOC terms at early times. At late times, depending on the dynamics, the commutator can grow large. The time required for the TOC terms to approach their equilibrium values is called the dissipation time t d . This time parallels the time required for a system to reach local thermal equilibrium. The time scale on which the commutator grows to be order-one is called the scrambling time t ∗ . The scrambling time parallels the time over which a drop of ink spreads across a container of water. Why consider the commutator’s square modulus? The simpler object h[W (t),V ]i often vanishes at late times, due to cancellations between states in the expectation value. Physically, the vanishing of h[W (t),V ]i signifies that perturbing the system with V does not significantly change the expectation value of W (t). This physics is expected for a chaotic system, which effectively loses its memory of its initial conditions. In contrast, C(t) is the expectation value of a positive operator (the magnitude-squared commutator). The cancellations that zero out h[W (t),V ]i canD E not zero out |[W (t),V ]| 2 . Mathematically, the diagonal elements of the matrix that represents [W (t),V ] relative to the energy eigenbasis can be small. h[W (t),V ]i, evaluated on a thermal state, would be small. Yet the matrix’s off-diagonal elements can boost the operap
tor’s Frobenius norm, Tr |[W (t),V ]| 2 , which reflects the size of C(t). We can gain intuition about the manifestation of chaos in F (t) from a simple quantum system that has a chaotic semiclassical limit. Let W = q and V = p for some position q and momentum p: C(t) = −h[q(t), p]2 i ∼ ~2 e2λ L t . (3.31) This λ L is a classical Lyapunov exponent. The final expression follows from the Correspondence Principle: Commutators are replaced with i~ times the corresponding Poisson bracket. The Poisson bracket of q(t) with p equals the derivative of the final position with respect to the initial position. This derivative reflects the butterfly effect in classical chaos, i.e., sensitivity to initial conditions. The growth of C(t), and the deviation of F (t) from the TOC terms, provide a quantum generalization of the butterfly effect. Within this simple quantum system, the analog of the dissipation time may be regarded as t d ∼ λ L−1 . The analog of the scrambling time is t ∗ ∼ λ L−1 ln Ω~ . The 42 Ω denotes some measure of the accessible phase-space volume. Suppose that the phase space is large in units of ~. The scrambling time is much longer than the dissipation time: t ∗  t d . Such a parametric separation between the time scales characterizes the systems that interest us most. In more general chaotic systems, the value of t ∗ depends on whether the interactions are geometrically local and on W and V . Consider, as an example, a spin chain governed by a local Hamiltonian. Suppose that W and V are local operators that act nontrivially on spins separated by a distance `. The scrambling time is generically proportional to `. For this class of local models, `/t ∗ defines a velocity vB called the butterfly velocity. Roughly, the butterfly velocity reflects how quickly initially local Heisenberg operators grow in space. Consider a system in which t d is separated parametrically from t ∗ . The rate of change of F (t) [rather, a regulated variation on F (t)] was shown to obey a nontrivial bound. Parameterize the OTOC as F (t) ∼ TOC −  e λ L t . The parameter   1 encodes the separation of scales. The exponent λ L obeys λ L ≤ 2πkBT in thermal equilibrium at temperature T [7]. kB denotes Boltzmann’s constant. Black holes in the AdS/CFT duality saturate this bound, exhibiting maximal chaos [2, 31]. More generally, λ L and vB control the operators’ growth and the spread of chaos. The OTOC has thus attracted attention for a variety of reasons, including (but not limited to) the possibilities of nontrivial bounds on quantum dynamics, a new probe of quantum chaos, and a signature of black holes in AdS/CFT. Introducing the quasiprobability behind the OTOC F (t) was shown, in [38], to equal a moment of a summed quasiprobability. We review this result, established in four steps: A quantum probability amplitude A ρ is reviewed in Sec. 3.1 . Amplitudes are combined to form the quasiprobability à ρ in Sec. 3.1. Summing à ρ (.) values, with constraints, yields a complex distribution P(W,W 0 ) in Sec. 3.1. Differentiating P(W,W 0 ) yields the OTOC. à ρ can be inferred experimentally from a weak-measurement scheme and from interference. We review these schemes in Sec. 3.1. A third quasiprobability is introduced in Sec. 3.2, the coarse-grained quasiprobability A˜ρ . A˜ρ follows from summing values of à ρ . A˜ρ has a more concise description than à ρ . Also, measuring A˜ρ requires fewer resources (e.g., trials) than measuring à ρ . Hence Sections 3.2-3.4 will spotlight A˜ρ . à ρ returns to prominence in the proofs of Sec. 3.5 and in opportunities detailed in Sec. 4.6. Different distri- 43 butions suit different investigations. Hence the presentation of three distributions in this thorough study: à ρ , A˜ρ , and P(W,W 0 ). Quantum probability amplitude A ρ The OTOC quasiprobability à ρ is defined in terms of probability amplitudes A ρ . The A ρ ’s are defined in terms of the following process, P A: (1) Prepare ρ. (2) Measure the ρ eigenbasis, {| jih j |}. (3) Evolve S forward in time under U. (4) Measure W̃. (5) Evolve S backward under U † . (6) Measure Ṽ . (7) Evolve S forward under U. (8) Measure W̃. Suppose that the measurements yield the outcomes j, (w1 , α w1 ), (v1 , λ v1 ), and (w2 , α w2 ). Figure 3.2a illustrates this process. The process corresponds to the probability amplitude5 A ρ ( j; w1 , α w1 ; v1 , λ v1 ; w2 , α w2 ) := hw2 , α w2 |U |v1 , λ v1 i √ × hv1 , λ v1 |U † |w1 , α w1 ihw1 , α w1 |U | ji p j . (3.32) We do not advocate for performing P A in any experiment. P A is used to define A ρ and to interpret A ρ physically. Instances of A ρ are combined into à ρ . A weakmeasurement protocol can be used to measure à ρ experimentally. An interference protocol can be used to measure A ρ (and so à ρ ) experimentally. 5 We order the arguments of A ρ differently than in [38]. Our ordering here parallels our later ordering of the quasiprobability’s argument. Weak-measurement experiments motivate the quasiprobability arguments’ ordering. This motivation is detailed in Footnote 7. 44 (w1 , ↵w1 ) (w3 , ↵w3 ) (w2 , ↵w2 ) Measure W̃. Measure W̃. -t U U † U U Experiment time 0 (a) Measure W̃. Measure W̃. -t Prepare ⇢. (w2 , ↵w2 ) Measure {|jihj|}. j Measure Ṽ . (v1 , U† U Experiment time 0 v1 ) Prepare ⇢. Measure {|jihj|}. j Measure Ṽ . (v2 , v2 ) (b) Figure 3.2: Quantum processes described by the probability amplitudes Aρ in the out-of-time-ordered correlator (OTOC): These figures, and parts of this caption, appear in [38]. The OTOC quasiprobability Ãρ results from summing products A∗ρ (.) Aρ (.). Each Aρ (.) denotes a probability amplitude [Eq. (3.32)], so each product resembles a probability. But the amplitudes’ arguments differ—the amplitudes correspond to different quantum processes—because the OTOC operators W (t) and V fail to commute, typically. Figure 3.2a illustrates the process described by the Aρ (.); and Fig. 3.2b, the process described by the A∗ρ (.). Time, as measured by a laboratory clock, increases from left to P right. Each process begins with the preparation of the state ρ = j p j | jih j | and a measurement of the state’s eigenbasis. Three evolutions (U, U † , and U) then alternate with three measurements of observables (W̃, Ṽ , and W̃). Figures 3.2a and 3.2b are used to define Ãρ , rather than showing protocols for measuring Ãρ . The fine-grained OTOC quasiprobability à ρ The quasiprobability’s definition is constructed as follows. Consider a realization of P A that yields the outcomes j, (w3 , α w3 ), (v2 , λ v2 ), and (w2 , α w2 ). Figure 3.2b illustrates this realization. The initial and final measurements yield the same outcomes as in the (3.32) realization. We multiply the complex conjugate of the second realization’s amplitude by the first realization’s probability amplitude. Then, we sum 45 over j and (w1 , α w1 ):6, 7 à ρ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ) X := A∗ρ ( j; w3 , α w3 ; v2 , λ v2 ; w2 , α w2 ) j,(w1 ,α w1 ) × A ρ ( j; w1 , α w1 ; v1 , λ v1 ; w2 , α w2 ) . (3.33) Equation (3.33) resembles a probability but differs due to the noncommutation of W (t) and V . We illustrate this relationship in two ways. Consider a 1D quantum system, e.g., a particle on a line. We represent the system’s state with a wave function ψ(x). The probability density at point x equals ψ ∗ (x) ψ(x). The A∗ρ A ρ in Eq. (3.33) echoes ψ ∗ ψ. But the argument of the ψ ∗ equals the argument of the ψ. The argument of the A∗ρ differs from the argument of the A ρ , because W (t) and V fail to commute. Substituting into Eq. (3.33) from Eq. (3.32) yields à ρ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ) = hw3 , α w3 |U |v2 , λ v2 ihv2 , λ v2 |U † |w2 , α w2 i × hw2 , α w2 |U |v1 , λ v1 ihv1 , λ v1 | ρU † |w3 , α w3 i . (3.34) A simple example illustrates how à ρ nearly equals a probability. Suppose that an n o  eigenbasis of ρ coincides with |v` , λ v` ihv` , λ v` | or with U † |w` , α w` ihw` , α w` |U . Suppose, for example, that X ρ = ρV := pv` ,λ v` |v` , λ v` ihv` , λ v` | . (3.35) v` ,λ v ` One such ρ is the infinite-temperature Gibbs state 1/d. Another example is easier to prepare: Suppose that S consists of N spins and that V = σ Nx . One ρV equals 6 Familiarity with tensors might incline one to sum over the (w , α ) shared by the trajec2 w2 tories. But we are not invoking tensors. More importantly, summing over (w2 , α w2 ) introduces a δ v1 v2 δ λ v1 λ v2 that eliminates one (v` , λ v ` ) degree of freedom. The resulting quasiprobability would not “lie behind” the OTOC. One could, rather than summing over (w1 , α w1 ), sum over (w3 , α w3 ). Either way, one sums over one trajectory’s first W̃ outcome. We sum over (w1 , α w1 ) to maintain consistency with [38]. 7 In [38], the left-hand side’s arguments are ordered differently and are condensed into the shorthand (w, v, α w , λ v ). Experiments motivate our reordering: Consider inferring Ãρ (a, b, c, d) from experimental measurements. In each trial, one (loosely speaking) weakly measures a, then b, then c; and then measures d strongly. As the measurements are ordered, so are the arguments. 46 a product of N σ x eigenstates. Let (v2 , λ v2 ) = (v1 , λ v1 ). [An analogous argument follows from (w3 , α w3 ) = (w2 , α w2 ).] Equation (3.34) reduces to |hw2 , α w2 |U |v1 , λ v1 i| 2 |hw3 , α w3 |U |v1 , λ v1 i| 2 pv1 ,λ v1 . (3.36) Each square modulus equals a conditional probability. pv1 ,λ v1 equals the probability  that, if ρ is measured with respect to |v` , λ v` ihv` , λ v` | , outcome (v1 , λ v1 ) obtains. In this simple case, certain quasiprobability values equal probability values—the quasiprobability values that satisfy (v2 , λ v2 ) = (v1 , λ v1 ) or (w3 , α w3 ) = (w2 , α w2 ). When both conditions are violated, typically, the quasiprobability value does not equal a probability value. Hence not all the OTOC quasiprobability’s values reduce to probability values. Just as a quasiprobability lies behind the OTOC, quasiprobabilities lie behind time-ordered correlators (TOCs). Every value of a TOC quasiprobability reduces to a probability value in the same simple case (when ρ equals, e.g., a V eigenstate) (Sec. 3.5). Complex distribution P(W,W 0 ) à ρ is summed, in [38], to form a complex distribution P(W,W 0 ). Let W := w3∗ v2∗ and W 0 := w2 v1 denote random variables calculable from measurement outcomes. If W and V are Paulis, (W,W 0 ) can equal (1, 1), (1, −1), (−1, 1), or (−1, −1). W and W 0 serve, in the Jarzynski-like equality (3.1), analogously to thermodynamic work Wth in Jarzynski’s equality. Wth is a random variable, inferable from experiments, that fluctuates from trial to trial. So are W and W 0. One infers a value of Wth by performing measurements and processing the outcomes. The two-point measurement scheme (TPMS) illustrates such protocols most famously. The TPMS has been used to derive quantum fluctuation relations [106]. One prepares the system in a thermal state, measures the Hamiltonian, Hi , projectively; disconnects the system from the bath; tunes the Hamiltonian to H f ; and measures H f projectively. Let Ei and E f denote the measurement outcomes. The work invested in the Hamiltonian tuning is defined as Wth := E f − Ei . Similarly, to infer W and W 0, one can measure W and V as in Sec. 3.1, then multiply the outcomes. Consider fixing the value of (W,W 0 ). For example, let (W,W 0 ) = (1, −1). Consider the octuples (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ) that satisfy the constraints W = w3∗ v2∗ and W 0 = w2 v1 . Each octuple corresponds to a quasiprobability value à ρ (.). Sum- 47 ming these quasiprobability values yields X P(W,W 0 ) := (3.37) (v1 ,λ v1 ),(w2 ,α w2 ),(v2 ,λ v2 ),(w3 ,α w3 ) à ρ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ) δW (w3∗ v2∗ ) δW 0 (w2 v1 ) . The Kronecker delta is represented by δ ab . P(W,W 0 ) functions analogously to the probability distribution, in the fluctuation-relation paper [39], over values of thermodynamic work. The OTOC equals a moment of P(W,W 0 ) [Eq. (3.1)], which equals a constrained sum over à ρ [38]. Hence our labeling of à ρ as a “quasiprobability behind the OTOC.” Equation (3.37) expresses the useful, difficult-to-measure F (t) in terms of a characteristic function of a (summed) quasiprobability, as Jarzynski [39] expresses a useful, difficult-to-measure free-energy difference ∆F in terms of a characteristic function of a probability. Quasiprobabilities reflect nonclassicality (contextuality) as probabilities do not; so, too, does F (t) reflect nonclassicality (noncommutation) as ∆F does not. The definition of P involves arbitrariness: The measurable random variables, and P, may be defined differently. Alternative definitions, introduced in Sec. 3.5, extend more robustly to OTOCs that encode more time reversals. All possible definitions share two properties: (i) The arguments W , etc. denote random variables inferable from measurement outcomes. (ii) P results from summing à ρ (.) values subject to constraints δ ab . P(W,W 0 ) resembles a work distribution constructed by Solinas and Gasparinetti (S&G) [107, 108]. They study fluctuation-relation contexts, rather than the OTOC. S&G propose a definition for the work performed on a quantum system [109, 110]. The system is coupled weakly to detectors at a protocol’s start and end. The couplings are represented by constraints like δW (w3∗ v2∗ ) and δW 0 (w2 v1 ) . Suppose that the detectors measure the system’s Hamiltonian. Subtracting the measurements’ outcomes yields the work performed during the protocol. The distribution over possible work values is a quasiprobability. Their quasiprobability is a Husimi Qfunction, whereas the OTOC quasiprobability is a KD distribution [110]. Related frameworks appear in [111–113]. The relationship between those thermodynamics frameworks and our thermodynamically motivated OTOC framework merits exploration. 48 Weak-measurement and interference schemes for inferring à ρ à ρ can be inferred from weak measurements and from interference, as shown in [38]. Section 3.2 shows how to infer a coarse-graining of à ρ from other OTOCmeasurement schemes (e.g., [42]). We focus mostly on the weak-measurement scheme here. The scheme is simplified in Sec. 3.2. First, we briefly review the interference scheme. The interference scheme in [38] differs from other interference schemes for measuring F (t) [42–44]: From the [38] interference scheme, one can infer not only F (t), but also à ρ . Time need not be inverted (H need not be negated) in any trial. The scheme is detailed in Appendix B of [38]. The system is coupled to an ancilla prepared in a superposition √1 (|0i + |1i). A unitary, conditioned on the ancilla, 2 rotates the system’s state. The ancilla and system are measured projectively. From many trials’ measurement data, one infers ha|U |bi, wherein U = U or U † and a, b = (w` , α w` ), (vm , λ vm ). These inner products are multiplied together to form à ρ [Eq. (3.34)]. If ρ shares neither the Ṽ nor the W̃ (t) eigenbasis, quantum-state tomography is needed to infer hv1 , λ v1 | ρU † |w3 , α w3 i. The weak-measurement scheme is introduced in Sec. II B 3 of [38]. A simple case, in which ρ = 1/d, is detailed in Appendix A of [38]. Recent weak measurements [11–14, 32–36], some used to infer KD distributions, inspired our weak à ρ measurement proposal. We review weak measurements, a Kraus-operator model for measurements, and the à ρ -measurement scheme. Review of weak measurements: Measurements can alter quantum systems’ states. A weak measurement barely disturbs the measured system’s state. In exchange, the measurement provides little information about the system. Yet one can infer much by performing many trials and processing the outcome statistics. Extreme disturbances result from strong measurements [114]. The measured system’s state collapses onto a subspace. For example, let ρ denote the initial state. Let P A = a a|aiha| denote the measured observable’s eigendecomposition. A strong measurement has a probability ha| ρ|ai of projecting ρ onto |ai. P One can implement a measurement with an ancilla. Let X = x x|xihx| denote an ancilla observable. One correlates A with X via an interaction unitary. Von Neumann modeled such unitaries with Vint := e−i g̃ A⊗X [15, 115]. The parameter g̃ P signifies the interaction strength.8 An ancilla observable—say, Y = y y|yihy|—is 8 A and X are dimensionless: To form them, we multiply dimensionful observables by natural 49 measured strongly. The greater the g̃, the stronger the correlation between A and Y . A is measured strongly if it is correlated with Y maximally, if a one-to-one mapping interrelates the y’s and the a’s. Suppose that the Y measurement yields y. We say that an A measurement has yielded some outcome a y . Suppose that g̃ is small. A is correlated imperfectly with Y . The Y -measurement outcome, y, provides incomplete information about A. The value most reasonably attributable to A remains a y . But a subsequent measurement of A would not necessarily yield a y . In exchange for forfeiting information about A, we barely disturb the system’s initial state. We can learn more about A by measuring A weakly in each of many trials, then processing measurement statistics. Kraus-operator model for measurement: Kraus operators [114] model the systemof-interest evolution induced by a weak measurement. Let us choose the following P P form for A. Let V = v` ,λ v` v` |v` , λ v` ihv` , λ v` | = v` v` ΠvV` denote an observable of the system. ΠvV` projects onto the v` eigenspace. Let A = |v` , λ v` ihv` , λ v` |. Let ρ denote the system’s initial state, and let |Di denote the detector’s initial state. Suppose that the Y measurement yields y. The system’s state evolves under the Kraus operator My = hy|Vint |Di as ρ 7→ (3.38)
= hy| exp −i g̃|v` , λ v` ihv` , λ v` | ⊗ X |Di (3.39) = hy|Di 1   + hy| e−i g̃X − 1 |Di |v` , λ v` ihv` , λ v` | (3.40) My ρMy†   . The third equation follows from Taylor-expanding the exponenTr My ρMy† tial, then replacing the projector’s square with the projector.9 We reparameterize the scales of the subsystems. These scales are incorporated into g̃. 9 Suppose that each detector observable (each of X and Y ) has at least as many eigenvalues as V . For example, let Y represent a pointer’s position and X represent the momentum. Each X eigenstate can be coupled to one V eigenstate. A will equal V , and Vint will have the form e −i g̃V ⊗ X . Such a coupling makes efficient use of the detector: Every possible final pointer position y correlates with some (v` , λ v ` ). Different |v` , λ v ` ihv` , λ v ` |’s need not couple to different detectors. Since a weak measurement of V provides information about one (v` , λ v ` ) as well as a weak measurement of |v` , λ v ` ihv` , λ v ` | does, we will sometimes call a weak measurement of |v` , λ v ` ihv` , λ v ` | “a weak measurement of V ,” for conciseness. The efficient detector use trades off against mathematical simplicity, if A is not a projector: Eq. (3.38) fails to simplify to Eq. (3.40). Rather, Vint should be approximated to some order in g̃. The approximation is (i) first-order if a KD quasiprobability is being inferred and (ii) third-order if 50 coefficients as hy|Di ≡ p(y) eiφ , wherein p(y) := |hy|Di|, and hy| e−i g̃X − 1 |Di ≡ g(y) eiφ . An unimportant global phase is denoted by eiφ . We remove this global phase from the Kraus operator, redefining My as  My = p p(y) 1 + g(y) |v` , λ v` ihv` , λ v` | .  (3.42) The coefficients have the following significances. Suppose that the ancilla did not couple to the system. The Y measurement would have a baseline probability p(y) of outputting y. The dimensionless parameter g(y) ∈ C is derived from g̃. We can roughly interpret My statistically: In any given trial, the coupling has a probability p(y) of failing to disturb the system (of evolving ρ under 1) and a probability |g(y)| 2 of projecting ρ onto |v` , λ v` ihv` , λ v` |. Weak-measurement scheme for inferring the OTOC quasiprobability à ρ : Weak measurements have been used to measure KD quasiprobabilities [11–14, 32, 33, 35, 36]. These experiments’ techniques can be applied to infer à ρ and, from à ρ , the OTOC. Our scheme involves three sequential weak measurements per trial (if ρ is arbitrary) or two [if ρ shares the Ṽ or the W̃ (t) eigenbasis, e.g., if ρ = 1/d]. The weak measurements alternate with time evolutions and precede a strong measurement. We review the general and simple-case protocols. A projection trick, introduced in Sec. 3.2, reduces exponentially the number of trials required to infer about à ρ and F (t). The weak-measurement and interference protocols are analyzed in Sec. 3.2. A circuit for implementing the weak-measurement scheme appears in Sec. 3.2. Suppose that ρ does not share the Ṽ or the W̃ (t) eigenbasis. One implements the following protocol, P: (1) Prepare ρ. the OTOC quasiprobability is being inferred. If A is a projector, Eq. (3.38) simplifies to Eq. (3.40) even if A is degenerate, e.g., A = ΠV v ` . Such an A assignment will prove natural in Sec. 3.2: Weak measurements of eigenstates |v` , λ v ` ihv` , λ v ` | are replaced with less-resource-consuming weak measurements of ΠV v ` ’s. Experimentalists might prefer measuring Pauli operators σ α (for α = x, y, z) to measuring projectors Π explicitly. Measuring Paulis suffices, as the eigenvalues of σ α map, bijectively and injectively, onto the eigenvalues of Π (Sec. 3.2). Paulis square to the identity, rather than to themselves: (σ α ) 2 = 1. Hence Eq. (3.40) becomes hy| cos (g̃X ) |Di 1 − ihy| sin (g̃X ) |Di σ α . (3.41) 51 (2) Measure Ṽ weakly. (Couple the system’s Ṽ weakly to some observable X of a clean ancilla. Measure X strongly.) (3) Evolve the system forward in time under U. (4) Measure W̃ weakly. (Couple the system’s W̃ weakly to some observable Y of a clean ancilla. Measure Y strongly.) (5) Evolve the system backward under U † . (6) Measure Ṽ weakly. (Couple the system’s Ṽ weakly to some observable Z of a clean ancilla. Measure Z strongly.) (7) Evolve the system forward under U. (8) Measure W̃ strongly. X, Y , and Z do not necessarily denote Pauli operators. Each trial yields three ancilla eigenvalues (x, y, and z) and one W̃ eigenvalue (w3 , α w3 ). One implements P many times. From the measurement statistics, one infers the probability Pweak (x; y; z; w3 , α w3 ) that any given trial will yield the outcome quadruple (x; y; z; w3 , α w3 ). From this probability, one infers the quasiprobability à ρ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ). The probability has the form Pweak (x; y; z; w3 , α w3 ) = hw3 , α w3 |U Mz U † My U Mx × ρMx†U † My†U Mz†U † |w3 , α w3 i . (3.43) We integrate over x, y, and z, to take advantage of all measurement statistics. We substitute in for the Kraus operators from Eq. (3.42), then multiply out. The result appears in Eq. (A7) of [38]. Two terms combine into ∝ =( à ρ (.)). The other terms form independently measurable “background” terms. To infer <( à ρ (.)), one performs P many more times, using different couplings (equivalently, measuring different detector observables). Details appear in Appendix A of [38]. To infer the OTOC, one multiplies each quasiprobability value à ρ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ) by the eigenvalue product v1 w2 v2∗ w3∗ . Then, one sums over the eigenvalues and the degeneracy parameters: X F (t) = v1 w2 v2∗ w3∗ (3.44) (v1 ,λ v1 ),(w2 ,α w2 ),(v2 ,λ v2 ),(w3 ,α w3 ) × Ã ρ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ) . 52 Equation (3.44) follows from Eq. (3.1). Hence inferring the OTOC from the weakmeasurement scheme—inspired by Jarzynski’s equality—requires a few steps more than inferring a free-energy difference ∆F from Jarzynski’s equality [39]. Yet such quasiprobability reconstructions are performed routinely in quantum optics. W and V are local. Their degeneracies therefore scale with the system size. If S consists of N spin- 12 degrees of freedom, |α w` |, |λ v` | ∼ 2 N . Exponentially many à ρ (.) values must be inferred. Exponentially many trials must be performed. We sidestep this exponentiality in Sec. 3.2: One measures eigenprojectors of the degenerate W and V , rather than of the nondegenerate W̃ and Ṽ . The one-dimensional |v` , λ v` ihv` , λ v` | of Eq. (3.40) is replaced with ΠvV` . From the weak measurements, P one infers the coarse-grained quasiprobability degeneracies à ρ (.) =: A˜ρ (.). Summing A˜ρ (.) values yields the OTOC: X F (t) = v1 w2 v2∗ w3∗ A˜ρ (v1 , w2 , v2 , w3 ) . (3.45) v1 ,w2 ,v2 ,w3 Equation (3.45) follows from performing the sums over the degeneracy parameters α and λ in Eq. (3.44). Suppose that ρ shares the Ṽ or the W̃ (t) eigenbasis. The number of weak measurements reduces to two. For example, suppose that ρ is the infinite-temperature Gibbs state 1/d. The protocol P becomes (1) Prepare a W̃ eigenstate |w3 , α w3 i. (2) Evolve the system backward under U † . (3) Measure Ṽ weakly. (4) Evolve the system forward under U. (5) Measure W̃ weakly. (6) Evolve the system backward under U † . (7) Measure Ṽ strongly. In many recent experiments, only one weak measurement is performed per trial [11, 13, 32]. A probability Pweak must be approximated to first order in the coupling constant g(x). Measuring à ρ requires two or three weak measurements per trial. 53 We must approximate Pweak to second or third order. The more weak measurements performed sequentially, the more demanding the experiment. Yet sequential weak measurements have been performed recently [33–35]. The experimentalists aimed to reconstruct density matrices and to measure non-Hermitian operators. The OTOC measurement provides new applications for their techniques. 3.2 Experimentally measuring à ρ and the coarse-grained A˜ρ Multiple reasons motivate measurements of the OTOC quasiprobability à ρ . à ρ is more fundamental than the OTOC F (t), F (t) results from combining values of à ρ . à ρ exhibits behaviors not immediately visible in F (t), as shown in Sections 3.3 and 3.4. à ρ therefore holds interest in its own right. Additionally, à ρ suggests new schemes for measuring the OTOC. One measures the possible values of à ρ (.), then combines the values to form F (t). Two measurement schemes are detailed in [38] and reviewed in Sec. 3.1. One scheme relies on weak measurements; one, on interference. We simplify, evaluate, and augment these schemes. First, we introduce a “projection trick”: Summing over degeneracies turns onedimensional projectors (e.g., |w` , α w` ihw` , α w` |) into projectors onto degenerate eigenspaces (e.g., ΠwW` ). The coarse-grained OTOC quasiprobability A˜ρ results. This trick decreases exponentially the number of trials required to infer the OTOC from weak measurements.10 Section 3.2 concerns pros and cons of the weak-measurement and interference schemes for measuring à ρ and F (t). We also compare those schemes with alternative schemes for measuring F (t). Section 3.2 illustrates a circuit for implementing the weak-measurement scheme. Section 3.2 shows how to infer A˜ρ not only from the measurement schemes in Sec. 3.1, but also with alternative OTOCmeasurement proposals (e.g., [42]) (if the eigenvalues of W and V are ±1). The coarse-grained OTOC quasiprobability A˜ρ and a projection trick W and V are local. They manifest, in our spin-chain example, as one-qubit Paulis that nontrivially transform opposite ends of the chain. The operators’ degeneracies grows exponentially with the system size N: |α w` |, |λ vm | ∼ 2 N . Hence the number of à ρ (.) values grows exponentially. One must measure exponentially many numbers to calculate F (t) precisely via à ρ . We circumvent this inconvenience by summing over the degeneracies in à ρ (.), forming the coarse-grained quasiproba10 The summation preserves interesting properties of the quasiprobability—nonclassical nega- tivity and nonreality, as well as intrinsic time scales. We confirm this preservation via numerical simulation in Sec. 3.3. 54 bility A˜ρ (.). A˜ρ (.) can be measured in numerical simulations, experimentally via weak measurements, and (if the eigenvalues of W and V are ±1) experimentally with other F (t)-measurement set-ups (e.g., [42]). The coarse-grained OTOC quasiprobability results from marginalizing à ρ (.) over its degeneracies: X A˜ρ (v1 , w2 , v2 , w3 ) := λ v1 ,α w2 ,λ v2 ,α w3 à ρ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ) . (3.46) Equation (3.46) reduces to a more practical form. Consider substituting into Eq. (3.46) for à ρ (.) from Eq. (3.34). The right-hand side of Eq. (3.34) equals a trace. Due to the trace’s cyclicality, the three rightmost factors can be shifted leftward: X  Tr ρU † |w3 , α w3 ihw3 , α w3 |U A˜ρ (v1 , w2 , v2 , w3 ) = λ v1 ,α w2 , λ v2 ,α w3  × |v2 , λ v2 ihv2 , λ v2 |U † |w2 , α w2 ihw2 , α w2 |U |v1 , λ v1 ihv1 , λ v1 | . (3.47) The sums are distributed throughout the trace: " X # † A˜ρ (v1 , w2 , v2 , w3 ) = Tr ρ U |w3 , α w3 ihw3 , α w3 |U α w3 × "X #" X # † |v2 , λ v2 ihv2 , λ v2 | U |w2 , α w2 ihw2 , α w2 |U α w2 λ v2 × "X #! |v1 , λ v1 ihv1 , λ v1 | . (3.48) λ v1 Define ΠwW` := X |w` , α w` ihw` , α w` | (3.49) α w` as the projector onto the w` eigenspace of W, ΠwW` (t) := U † ΠwW` U as the projector onto the w` eigenspace of W (t), and X ΠvV` := |v` , λ v` ihv` , λ v` | λ v` (3.50) (3.51) 55 as the projector onto the v` eigenspace of V . We substitute into Eq. (3.48), then invoke the trace’s cyclicality:   A˜ρ (v1 , w2 , v2 , w3 ) = Tr ΠwW3 (t) ΠvV2 ΠwW2 (t) ΠvV1 ρ . (3.52) Asymmetry distinguishes Eq. (3.52) from Born’s Rule and from expectation values. Imagine preparing ρ, measuring V strongly, evolving S forward under U, measuring W strongly, evolving S backward under U † , measuring V strongly, evolving S forward under U, and measuring W. The probability of obtaining the outcomes v1 , w2 , v2 , and w3 , in that order, is   Tr ΠwW3 (t) ΠvV2 ΠwW2 (t) ΠvV1 ρΠvV1 ΠwW2 (t) ΠvV2 ΠwW3 (t) . (3.53) The operator ΠwW3 (t) ΠvV2 ΠwW2 (t) ΠvV1 conjugates ρ symmetrically. This operator multiplies ρ asymmetrically in Eq. (3.52). Hence A˜ρ does not obviously equal a probability. Nor does A˜ρ equal an expectation value. Expectation values have the form Tr( ρA), wherein A denotes a Hermitian operator. The operator leftward of the ρ in Eq. (3.52) is not Hermitian. Hence A˜ρ lacks two symmetries of familiar quantum objects: the symmetric conjugation in Born’s Rule and the invariance, under Hermitian conjugation, of the observable A in an expectation value. The right-hand side of Eq. (3.52) can be measured numerically and experimentally. We present numerical measurements in Sec. 3.3. The weak-measurement scheme follows from Appendix A of [38], reviewed in Sec. 3.1: Section 3.1 features projectors onto one-dimensional eigenspaces, e.g., |v1 , λ v1 ihv1 , λ v1 |. Those projectors are replaced with Π’s onto higher-dimensional eigenspaces. Section 3.2 details how A˜ρ can be inferred from alternative OTOC-measurement schemes. Analysis of the quasiprobability-measurement schemes and comparison with other OTOC-measurement schemes Section 3.1 reviews two schemes for inferring à ρ : a weak-measurement scheme and an interference scheme. From à ρ measurements, one can infer the OTOC F (t). We evaluate our schemes’ pros and cons. Alternative schemes for measuring F (t) have been proposed [40, 42–47], and two schemes have been realized [48, 49]. We compare our schemes with alternatives, as summarized in Table 3.3. For specificity, we focus on [42, 43, 45]. 5614 WeakYunger Halpern measurement interferometry Key tools Weak measurement What’s inferable (1) F (t), Ã⇢ , from the mea& ⇢ or surement? (2) F (t) & A˜⇢ Generality Arbitrary of ⇢ ⇢ 2 D(H) Ancilla Yes needed? Ancilla coupNo ling global? How long must 1 weak ancilla stay measurement coherent? # time 2 reversals # copies of ⇢ 1 needed / trial Signal-toTo be deternoise ratio mined [115] Restrictions Hermitian or on W & V unitary Interference ¯ F (K ) (t), ÃK ⇢, & ⇢ 8K Arbitrary ⇢ 2 D(H) Yes Swingle et al. Yao et al. Zhu et al. Interference, Lochschmidt echo F (t) Ramsey interfer., Rényi-entropy meas. Regulated correlator Freg (t) Thermal: e H/T /Z Yes Quantum clock F (t) Arbitrary ⇢ 2 D(H) Yes No Yes Yes Arbitrary ⇢ 2 D(H) Yes for <(F (t)), no for |F (t)|2 Yes Whole protocol Whole protocol Whole protocol Whole protocol 0 1 0 1 1 2 2 (implemented via ancilla) 1 To be determined [115] Unitary Constant in N Unitary (extension to Hermitian possible) ⇠e N Hermitian and unitary Constant in N Unitary TABLE I: Comparison of our measurement schemes with alternatives: This paper focuses on the Figure 3.3: Comparison our measurement schemes with alternatives: This paper weak-measurement and interferenceof schemes for measuring the OTOC quasiprobability Ã⇢ or the coarse-grained ˜⇢ . From Ã⇢ or A˜⇢ , one can infer the OTOC F (t). These schemes appear in [37], are reviewed in Sec. I D 4, quasiprobability A focuses on the weak-measurement and interference schemes for measuring the OTOC and are assessed in Sec. II B. We compare our schemes with the OTOC-measurement schemes in [41, 42, 44]. More ˜ρ . From OTOC-measurement in [39, 43, 45–48]. Each row corresponds to A a desirable quantity or to quasiprobabilityschemes Ãρ orappear the coarse-grained quasiprobability Ãρ or A˜aρ ,resource one can potentially challenging to realize experimentally. The regulated correlator Freg (t) [Eq. (108)] is expected to behave similarly infer the OTOC F (t). These schemes appear in [38], are reviewed in Sec. 3.1, and are to F (t) [6, 42]. D(H) denotes the set of density operators defined on the Hilbert space H. ⇢ denotes the initially prepared state. Target states ⇢ are never prepared perfectly; ⇢ may di↵er from ⇢ . Experimentalists can reconstruct ⇢ by target target assessed in Sec. 3.2. We compare our schemes with the OTOC-measurement schemes ¯ trivially processing data taken to infer Ã⇢ [37] (Sec. V B). F (K ) (t) denotes the K¯-fold OTOC, which encodes K = 2K¯ 1 ¯) (K ) in [42, 43, The 45].conventional More OTOC-measurement schemes appear in [40, time reversals. OTOC corresponds to K = 3. The quasiprobability behind44, F (K46–49]. (t) is Ã⇢ Each (Sec. row V E). N denotes the systemtosize, e.g., the number of qubits. et al. and Zhu et al. schemes have constant signal-to-noise corresponds a desirable quantity orThe to aSwingle resource potentially challenging to realize ratios (SNRs) in the absence of environmental decoherence. The Yao et al. scheme’s SNR varies inverse-exponentially with experimentally. Theentropy, regulated correlator Freg (t) [Eq. state (3.108)] is expected toNbehave the system’s entanglement SvN . The system occupies a thermal e H/T /Z, so SvN ⇠ log(2 ) = N. similarly to F (t) [7, 43]. D(H ) denotes the set of density operators defined on the Hilbert space H . ρ denotes the initially prepared state. Target states ρtarget are never prepared future, with superconducting 50–55], trapped motivates a new application ρofby recently realized sequenperfectly; ρ may differ qubits from [35, ρtarget . Experimentalists can reconstruct trivially ions [56–62], cavity QED [64, 65], ultracold atoms [63], tial weak¯ measurements. ) (t) denotes ¯ -fold and perhaps NMR [66,taken 67]. Circuits for weakly measuring schemes furnish not only OTOC F (t), but also processing data to infer Ãρ [38] (Sec. 3.5). Our F (K the Kthe OTOC, qubit systems have been designed [36, 50]. Initial proofmore information: ¯ − 1 time reversals. The conventional OTOC corresponds to which encodes K = 2 K of-principle experiments might not require direct access ¯ ) the weak-measurement scheme in [37], we can From to Thequasiprobability five superconductingbehind qubits available Kthe=qubits: 3. The F (K ) (t) is (1) Ã(K 3.5). N denotes the system ρinfer(Sec. the following: from IBM, via the cloud, might suffice [116]. Random size, e.g., the number of qubits. The Swingle et al. and Zhu et al. schemes have constant two-qubit unitaries could simulate chaotic Hamiltonian (A) The OTOC quasiprobability Ã⇢ . The evolution. signal-to-noise ratios (SNRs) in the absence of environmental decoherence. Yao et al. quasiprobability is more The fundamental than In many weak-measurement experiments, just one with the system’s F (t), asentanglement combining Ã⇢ (.)entropy, values yields scheme’s SNR varies inverse-exponentially SvNF. (t) weak measurement is performed per trial [10–13]. Yet [Eq. (44)]. − H /T /Z, so S N ) = N. Theweak system occupieshave a thermal stateperformed ∼ log(2 two measurements recently been vN sequentially [32–34]. Experimentalists aimed to “directly measure general quantum states” [11] and to infer about non-Hermitian observable-like operators. The OTOC (B) The OTOC F (t). (C) The form ⇢ of the state prepared. Suppose that we wish to evaluate F (t) on a target state The weak-measurement scheme augments the set of techniques and platforms with which F (t) can be measured. Alternative schemes rely on interferometry [42–44], controlled unitaries [42, 45], ultracold-atoms tools [44, 46, 47], and strong twopoint measurements [40]. Weak measurements, we have shown, belong in the OTOC-measurement toolkit. Such weak measurements are expected to be realizable, in the immediate future, with superconducting qubits [36, 51–56], trapped ions [57–63], cavity QED [65, 66], ultracold atoms [64], and perhaps NMR [67, 68]. 57 Circuits for weakly measuring qubit systems have been designed [37, 51]. Initial proof-of-principle experiments might not require direct access to the qubits: The five superconducting qubits available from IBM, via the cloud, might suffice [116]. Random two-qubit unitaries could simulate chaotic Hamiltonian evolution. In many weak-measurement experiments, just one weak measurement is performed per trial [11–14]. Yet two weak measurements have recently been performed sequentially [33–35]. Experimentalists aimed to “directly measure general quantum states” [12] and to infer about non-Hermitian observable-like operators. The OTOC motivates a new application of recently realized sequential weak measurements. Our schemes furnish not only the OTOC F (t), but also more information: (1) From the weak-measurement scheme in [38], we can infer the following: (A) The OTOC quasiprobability à ρ . The quasiprobability is more fundamental than F (t), as combining à ρ (.) values yields F (t) [Eq. (3.44)]. (B) The OTOC F (t). (C) The form ρ of the state prepared. Suppose that we wish to evaluate F (t) on a target state ρtarget . ρtarget might be difficult to prepare, e.g., might be thermal. The prepared state ρ approximates ρtarget . Consider performing the weak-measurement protocol P with ρ. One infers à ρ . Summing à ρ (.) values yields the form of ρ. We can assess the preparation’s accuracy without performing tomography independently. Whether this assessment meets experimentalists’ requirements for precision remains to be seen. Details appear in Sec. 3.5. (2) The weak-measurement protocol P is simplified later in this section. Upon implementing the simplified protocol, we can infer the following information: (A) The coarse-grained OTOC quasiprobability A˜ρ . Though less fundamental than the fine-grained à ρ , A˜ρ implies the OTOC’s form [Eq. (3.45)]. (B) The OTOC F (t). (3) Upon implementing the interferometry scheme in [38], we can infer the following information: (A) The OTOC quasiprobability à ρ . (B) The OTOC F (t). 58 (C) The form of the state ρ prepared. ¯ (D) All the K¯ -fold OTOCs F (K ) (t), which generalize the OTOC F (t). ¯ F (t) encodes three time reversals. F (K ) (t) encodes K = 2K¯ − 1 = 3, 5, . . . time reversals. Details appear in Sec. 3.5. ¯ (E) The quasiprobability Ã(ρK ) behind F (K ) (t), for all K (Sec. 3.5). We have delineated the information inferable from the weak-measurement and interference schemes for measuring à ρ and F (t). Let us turn to other pros and cons. The weak-measurement scheme’s ancillas need not couple to the whole system. One measures a system weakly by coupling an ancilla to the system, then measuring the ancilla strongly. Our weak-measurement protocol requires one ancilla per weak measurement. Let us focus, for concreteness, on an A˜ρ measurement for a general ρ. The protocol involves three weak measurements and so three ancillas. Suppose that W and V manifest as one-qubit Paulis localized at opposite ends of a spin chain. Each ancilla need interact with only one site (Fig. 3.4). In contrast, the ancilla in [45] couples to the entire system. So does the ancilla in our interference scheme for measuring à ρ . Global couplings can be engineered in some platforms, though other platforms pose challenges. Like our weak-measurement scheme, [42] and [43] require only local ancilla couplings. In the weak-measurement protocol, each ancilla’s state must remain coherent during only one weak measurement—during the action of one (composite) gate in a circuit. The first ancilla may be erased, then reused in the third weak measurement. In contrast, each ancilla in [42, 43, 45] remains in use throughout the protocol. The Swingle et al. scheme for measuring <(F (t)), too, requires an ancilla that remains coherent throughout the protocol [42]. The longer an ancilla’s “active-duty” time, the more likely the ancilla’s state is to decohere. Like the weak-measurement sheme, the Swingle et al. scheme for measuring |F (t)| 2 requires no ancilla [42]. Also in the interference scheme for measuring à ρ [38], an ancilla remains active throughout the protocol. That protocol, however, is short: Time need not be reversed in any trial. Each trial features exactly one U or U † , not both. Time can be difficult to reverse in some platforms, for two reasons. Suppose that a Hamiltonian H generates a forward evolution. A perturbation ε might lead −(H + ε) to generate the reverse evolution. Perturbations can mar long-time measurements of F (t) [45]. Second, systems interact with environments. Decoherence might not be completely 59 reversible [42]. Hence the lack of a need for time reversal, as in our interference scheme and in [43, 45], has been regarded as an advantage. Unlike our interference scheme, the weak-measurement scheme requires that time be reversed. Perturbations ε threaten the weak-measurement scheme as they threaten the Swingle et al. scheme [42]. ε’s might threaten the weak-measurement scheme more, because time is inverted twice in our scheme. Time is inverted only once in [42]. However, our error might be expected to have roughly the size of the Swingle et al. scheme’s error [117]. Furthermore, tools for mitigating the Swingle et al. scheme’s inversion error are being investigated [117]. Resilience of the Swingle et al. scheme to decoherence has been analyzed [42]. These tools may be applied to the weak-measurement scheme [117]. Like resilience, our schemes’ signal-to-noise ratios require further study. As noted earlier, as the system size N grows, the number of trials required to infer à ρ grows exponentially. So does the number of ancillas required to infer à ρ : Measuring a degeneracy parameter α w` or λ vm requires a measurement of each spin. Yet the number of trials, and the number of ancillas, required to measure the coarse-grained A˜ρ remains constant as N grows. One can infer A˜ρ from weak measurements and, alternatively, from other F (t)-measurement schemes (Sec. 3.2). A˜ρ is less fundamental than à ρ , as A˜ρ results from coarse-graining à ρ . A˜ρ , however, exhibits nonclassicality and OTOC time scales (Sec. 3.3). Measuring A˜ρ can balance the desire for fundamental knowledge with practicalities. The weak-measurement scheme for inferring A˜ρ can be rendered more convenient. Section 3.2 describes measurements of projectors Π. Experimentalists might prefer measuring Pauli operators σ α . Measuring Paulis suffices for inferring a multiqubit system’s A˜ρ : The relevant Π projects onto an eigenspace of a σ α . Measuring the σ α yields ±1. These possible outcomes map bijectively onto the possible Πmeasurement outcomes. See Footnote 9 for mathematics. Our weak-measurement and interference schemes offer the advantage of involving general operators. W and V must be Hermitian or unitary, not necessarily one or the other. Suppose that W and V are unitary. Hermitian operators GW and GV generate W and V , as discussed in Sec. 3.1. GW and GV may be measured in place of W and V . This flexibility expands upon the measurement opportunities of, e.g., [42, 43, 45], which require unitary operators. Our weak-measurement and interference schemes offer leeway in choosing not only 60 W and V , but also ρ. The state can assume any form ρ ∈ D (H ). In contrast, infinite-temperature Gibbs states ρ = 1/d were used in [48, 49]. Thermality of ρ is assumed in [43]. Commutation of ρ with V is assumed in [40]. If ρ shares a V eigenbasis or the W (t) eigenbasis, e.g., if ρ = 1/d, our weak-measurement protocol simplifies from requiring three sequential weak measurements to requiring two. Circuit for inferring A˜ρ from weak measurements Consider a 1D chain S of N qubits. A circuit implements the weak-measurement scheme reviewed in Sec. 3.1. We exhibit a circuit for measuring A˜ρ . One subcircuit implements each weak measurement. These subcircuits result from augmenting Fig. 1 of [118]. Dressel et al. use the partial-projection formalism, which we review first. We introduce notation, then review the weak-measurement subcircuit of [118]. Copies of the subcircuit are embedded into our A˜ρ -measurement circuit. Partial-projection operators Partial-projection operators update a state after a measurement that may provide incomplete information. Suppose that S begins in a state |ψi. Consider performing a measurement that could output + or −. Let Π+ and Π− denote the projectors onto the + and − eigenspaces. Parameters p, q ∈ [0, 1] quantify the correlation between the outcome and the premeasurement state. If |ψi is a + eigenstate, the measurement has a probability p of outputting +. If |ψi is a − eigenstate, the measurement has a probability q of outputting −. Suppose that outcome + obtains. We update |ψi using the partial-projection operp √ |ψi ator D+ := p Π+ + 1 − q Π− : |ψi 7→ ||DD+|ψi|| 2 . If the measurement yields −, we + p √ update |ψi with D− := 1 − p Π+ + q Π− . The measurement is strong if (p, q) = (0, 1) or (1, 0). D+ and D− reduce to projectors. The measurement collapses |ψi onto an eigenspace. The measurement is weak if p and q lie close to 21 : D± lies close to the normalized identity, 1d . Such an operator barely changes the state. The measurement provides hardly any information. We modeled measurements with Kraus operators Mx in Sec. 3.1. The polar decomposition of Mx [119] is a partial-projection operator. Consider measuring a qubit’s 61 σ z . Recall that X denotes a detector observable. Suppose that, if an X measurement yields x, a subsequent measurement of the spin’s σ z most likely yields +. The p Kraus operator Mx q = p(x) 1 + g(x) Π+ updates the system’s state. Mx is related to D+ by D+ = Ux Mx† Mx for some unitary Ux . The form of Ux depends on the system-detector coupling and on the detector-measurement outcome. The imbalance |p − q| can be tuned experimentally. Our scheme has no need for a nonzero imbalance. We assume that p equals q. Notation Let σ := σ x x̂ + σ y ŷ + σ z ẑ denote a vector of one-qubit Pauli operators. The σ z basis serves as the computational basis in [118]. We will exchange the σ z basis with the W eigenbasis, or with the V eigenbasis, in each weak-measurement subcircuit. In our spin-chain example, W and V denote one-qubit Pauli operators localized on opposite ends of the chain S: W = σ W ⊗ 1⊗(N −1) , and V = 1⊗(N −1) ⊗ σV . Unit vectors Ŵ, V̂ ∈ R3 are chosen such that σ n := σ · n̂, for n = W,V . The one-qubit Paulis eigendecompose as σ W = |+Wih+W | − |−Wih−W | and σV = |+V ih+V | − |−V ih−V |. The whole-system operators eigendecompose as W = Π+W − Π−W and V = Π+V − Π−V . A rotation operator Rn maps the σ z eigenstates to the σ n eigenstates: Rn |+zi = |+ni, and Rn |−zi = |−ni. We model weak W measurements with the partial-projection operators p √ D+W := pW Π+W + 1 − pW Π−W and p √ D−W := 1 − pW Π+W + pW Π−W . The V partial-projection operators are defined analogously: p √ D+V := pV Π+V + 1 − pV Π−V and p √ DV− := 1 − pV Π+V + pV Π−V . (3.54) (3.55) (3.56) (3.57) Weak-measurement subcircuit Figure 3.4a depicts a subcircuit for measuring n = W or V weakly. To simplify notation, we relabel pn as p. Most of the subcircuit appears in Fig. 1 of [118]. We set the imbalance parameter  to 0. We sandwich Fig. 1 of [118] between two onequbit unitaries. The sandwiching interchanges the computational basis with the n eigenbasis. 62 (a) (b) Figure 3.4: Quantum circuit for inferring the coarse-grained OTOC quasiprobability A˜ρ from weak measurements: We consider a system of N qubits prepared in a state ρ. The local operators W = σ W ⊗ 1 ⊗(N −1) and V = 1 ⊗(N −1) ⊗ σV manifest as one-qubit Paulis. Weak measurements can be used to infer the coarse-grained quasiprobability A˜ρ . Combining values of A˜ρ yields the OTOC F (t). Figure 3.4a depicts a subcircuit used to implement a weak measurement of n = W or V . An ancilla is prepared in a fiducial state |0i. A unitary Rn† rotates the qubit’s σ n eigenbasis into its σ z eigenbasis. Ry (±φ) rotates the ancilla’s state counterclockwise about the y-axis through a small angle ±φ, controlled by the system’s σ z . The angle’s smallness guarantees the measurement’s weakness. Rn rotates the system’s σ z eigenbasis back into the σ n eigenbasis. The ancilla’s σ z is measured strongly. The outcome, + or −, dictates which partial-projection operator D±n updates the state. Figure 3.4b shows the circuit used to measure A˜ρ . Three weak measurements, interspersed with three time evolutions (U, U † , and U), precede a strong measurement. Suppose that the initial state, ρ, commutes with W or V , e.g., ρ = 1/d. Figure 3.4b requires only two weak measurements. The subcircuit implements the following algorithm: (1) Rotate the n eigenbasis into the σ z eigenbasis, using Rn† . (2) Prepare an ancilla in a fiducial state |0i ≡ |+zi. (3) Entangle S with the ancilla via a Z-controlled-Y : If S is in state |0i, rotate the ancilla’s state counterclockwise (CCW) through a small angle φ  π2 about the y-axis. Let Ry (φ) denote the one-qubit unitary that implements this 63 rotation. If S is in state |1i, rotate the ancilla’s state CCW through an angle −φ, with Ry (−φ). (4) Measure the ancilla’s σ z . If the measurement yields outcome +, D+ updates the system’s state; and if −, then D− . (5) Rotate the σ z eigenbasis into the n eigenbasis, using Rn . The measurement is weak because φ is small. Rotating through a small angle precisely can pose challenges [36]. Full circuit for weak-measurement scheme Figure 3.4b shows the circuit for measuring A˜ρ . The full circuit contains three weak-measurement subcircuits. Each ancilla serves in only one subcircuit. No ancilla need remain coherent throughout the protocol, as discussed in Sec. 3.2. The ancilla used in the first V measurement can be recycled for the final V measurement. The circuit simplifies in a special case. Suppose that ρ shares an eigenbasis with V or with W (t), e.g., ρ = 1/d. Only two weak measurements are needed, as discussed in Sec. 3.1. We can augment the circuit to measure à ρ , rather than A˜ρ : During each weak measurement, every qubit will be measured. The qubits can be measured individually: The N-qubit measurement can be a product of local measurements. Consider, for concreteness, the first weak measurement. Measuring just qubit N would yield an eigenvalue v1 of V . We would infer whether qubit N pointed upward or downward along the V̂ axis. Measuring all the qubits would yield a degeneracy parameter λ v1 . We could define λ v` as encoding the V̂ -components of the other N − 1 qubits’ angular momenta. How to infer A˜ρ from other OTOC-measurement schemes F (t) can be inferred, we have seen, from the quasiprobability à ρ and from the coarse-grained A˜ρ . A˜ρ can be inferred from F (t)-measurement schemes, we show, if the eigenvalues of W and V equal ±1. We assume, throughout this section, that they do. The eigenvalues equal ±1 if W and V are Pauli operators. The projectors (3.49) and (3.51) can be expressed as ΠwW` = 1 ( 1 + w` W ) 2 and ΠvV` = 1 (1 + v` V ) . 2 (3.58) 64 Consider substituting from Eqs. (3.58) into Eq. (3.52). Multiplying out yields sixteen terms. If h.i := Tr(. .), 1 h A˜ρ (v1 , w2 , v2 , w3 ) = 1 + (w2 + w3 ) hW (t)i 16 D E D E + (v1 + v2 ) hV i + w2 w3 W 2 (t) + v1 v2 V 2 + (w2 v1 + w3 v1 + w3 v2 ) hW (t)V i + w2 v2 hV W (t)i D E D E + w2 w3 v1 W 2 (t)V + w3 v1 v2 W (t)V 2 + w2 w3 v2 hW (t)V W (t)i + w2 v1 v2 hV W (t)V i i + w2 w3 v1 v2 F (t) . (3.59) If W (t) and V are unitary, they square to 1. Equation (3.59) simplifies to 1n A˜ρ (v1 , w2 , v2 , w3 ) = (1 + w2 w3 + v1 v2 ) 16 + [w2 + w3 (1 + v1 v2 )] hW (t)i + [v1 (1 + w2 w3 ) + v2 ] hV i + (w2 v1 + w3 v1 + w3 v2 ) hW (t)V i + w2 v2 hV W (t)i + w2 w3 v2 hW (t)V W (t)i + w2 v1 v2 hV W (t)V i o + w2 w3 v1 v2 F (t) . (3.60) The first term is constant. The next two terms are single-observable expectation values. The next two terms are two-point correlation functions. hV W (t)V i and hW (t)V W (t)i are time-ordered correlation functions. F (t) is the OTOC. F (t) is the most difficult to measure. If one can measure it, one likely has the tools to infer A˜ρ . One can measure every term, for example, using the set-up in [42]. 3.3 Numerical simulations We now study the OTOC quasiprobability’s physical content in two simple models. In this section, we study a geometrically local 1D model, an Ising chain with transverse and longitudinal fields. In Sec. 3.4, we study a geometrically nonlocal model known as the Brownian-circuit model. This model effectively has a time-dependent Hamiltonian. We compare the physics of A˜ρ with that of the OTOC. The time scales inherent in A˜ρ , as compared to the OTOC’s time scales, particularly interest us. We study also nonclassical behaviors—negative and nonreal values—of A˜ρ . Finally, we find a parallel with classical chaos: The onset of scrambling breaks a symmetry. This breaking manifests in bifurcations of A˜ρ , reminiscent of pitchfork diagrams. 65 The Ising chain is defined on a Hilbert space of N spin- 12 degrees of freedom. The total Hilbert space has dimensionality d = 2 N . The single-site Pauli matrices are y labeled {σix , σi , σiz }, for i = 1, ..., N. The Hamiltonian is H = −J N −1 X i=1 z σiz σi+1 −h N X σiz − g i=1 N X σix . (3.61) i=1 The chain has open boundary conditions. Energies are measured in units of J. Times are measured in units of 1/J. The interaction strength is thus set to one, J = 1, henceforth. We numerically study this model for N = 10 by exactly diagonalizing H. This system size suffices for probing the quasiprobability’s time scales. However, N = 10 does not necessarily illustrate the thermodynamic limit. When h = 0, this model is integrable and can be solved with noninteracting-fermion variables. When h , 0, the model appears to be reasonably chaotic. These statements’ meanings are clarified in the data below. As expected, the quasiprobability’s qualitative behavior is sensitive primarily to whether H is integrable, as well as to the initial state’s form. We study two sets of parameters, Integrable: h = 0, g = 1.05 Nonintegrable: h = .5, g = 1.05 . and (3.62) We study several classes of initial states ρ, including thermal states, random pure states, and product states. For W and V , we choose single-Pauli operators that act nontrivially on just the chain’s ends. We illustrate with W = σ1x or W = σ1z and V = σ Nx or σ Nz . These operators are unitary and Hermitian. They square to the identity, enabling us to use Eq. (3.60). We calculate the coarse-grained quasiprobability directly:   A˜ρ (v1 , w2 , v2 , w3 ) = Tr ρΠwW3 (t) ΠvV2 ΠwW2 (t) ΠvV1 . (3.63) For a Pauli operator O, ΠaO = 12 (1 + aO) projects onto the a ∈ {1, −1} eigenspace. We also compare the quasiprobability with the OTOC, Eq. (3.45). F (t) deviates from one at roughly the time needed for information to propagate from one end of the chain to the other. This onset time, which up to a constant shift is also approximately the scrambling time, lies approximately between t = 4 and t = 6, according to our the data. The system’s length and the butterfly velocity vB set the scrambling time (Sec. 3.1). Every term in the Hamiltonian (3.61) is orderone. Hence vB is expected to be order-one, too. In light of our spin chain’s length, the data below are all consistent with a vB of approximately two. 66 Thermal states We consider first thermal states ρ ∝ e−H/T . Data for the infinite-temperature (T = ∞) state, with W = σ1z , V = σ Nz , and nonintegrable parameters, appear in Figures 3.5, 3.6, and 3.7. The legend is labeled such that abcd corresponds to w3 = (−1) a , v2 = (−1) b , w2 = (−1) c , and v1 = (−1) d . This labelling corresponds to the order in which the operators appear in Eq. (3.63). Three behaviors merit comment. Generically, the coarse-grained quasiprobability is a complex number: A˜ρ (.) ∈ C. However, A˜(1/d) is real. The imaginary component     = A˜(1/d) might appear nonzero in Fig. 3.7. Yet = A˜(1/d) ≤ 10−16 . This value equals zero, to within machine precision. The second feature to notice is that the time required for A˜(1/d) to deviate from its initial value equals approximately the time required for the OTOC to deviate from its initial value. Third, although A˜(1/d) is real, it is negative and hence nonclassical for some values of its arguments. What about lower temperatures? Data for the T = 1 thermal state are shown in Figures 3.8, 3.9, and 3.10. The coarse-grained quasiprobability is no longer real. Here, too, the time required for A˜ρ to deviate significantly from its initial value is comparable with the time scale of changes in F (t). This comparability characterizes the real and imaginary parts of A˜ρ . Both parts oscillate at long times. In the small systems considered here, such oscillations can arise from finite-size effects, including the energy spectrum’s discreteness. With nonintegrable parameters, this model has an energy gap ∆ N=10 = 2.92 above the ground state. The temperature T = 1 is smaller than the gap. Hence lowering T from ∞ to 1 brings the thermal state close to the ground state. What about long-time behavior? At infinite temperature, A˜(1/d) approaches a limiting form after the scrambling time but before any recurrence time. Furthermore, A˜(1/d) can approach one of only a few possible limiting values, depending on the function’s arguments. This behavior follows from the terms in Eq. (3.60). At infinite temperature, hWi = hV i = 0. Also the 3-point functions vanish, due to the trace’s cyclicity. We expect the nontrivial 2- and 4-point functions to be small at late times. (Such smallness is visible in the 4-point function in Fig. 3.5.) Hence Eq. (3.60) reduces as 1 + w2 w3 + v1 v2 . A˜ρ (v1 , w2 , v2 , w3 ) |{z} −→ 16 (3.64) t→∞ According to Eq. (3.64), the late-time values of A˜(1/d) should cluster around 3/16, 67 1/16, and −1/16. This expectation is roughly consistent with Fig. 3.6, modulo the upper lines’ bifurcation. A bifurcation of A˜ρ signals the breaking of a symmetry at the onset of scrambling. Similarly, pitchfork plots signal the breaking of a symmetry in classical chaos [120]. The symmetry’s mathematical form follows from Eq. (3.60). At early times, W (t) commutes with V , and F (t) ≈ 1. Suppose, for simplicity, that ρ = 1/d. The expectation values hW (t)i and hV i vanish, because every Pauli has a zero trace. Equation (3.60) becomes 1 h A˜ρ (v1 , w2 , v2 , w3 ) = (1 + w2 w3 + v1 v2 + w2 w3 v1 v2 ) 16 i + (w2 + w3 )(v1 + v2 ) hW (t)V i . (3.65) Suppose that w2 = −w3 and/or v1 = −v2 , as in the lower lines in Fig. 3.6. A˜ρ (.) reduces to the constant 1 (1 + w2 w3 + v1 v2 + w2 w3 v1 v2 ) 16 i 1 h (1 + w2 w3 + v1 v2 ) 2 − (w2 w3 ) 2 − (v1 v2 ) 2 + 1 . = 32 (3.66) The right-hand side depends on the eigenvalues w` and vm only through squares. A˜ρ (.) remains invariant under the interchange of w2 with w3 , under the interchange of v1 with v2 , under the simultaneous negations of w2 and w3 , and under the simultaneous negations of v1 and v2 . These symmetries have operational significances: à ρ remains constant under permutations and negations of measurement outcomes in the weak-measurement scheme (Sec. 3.1). Symmetries break as the system starts scrambling: F (t) shrinks, shrinking the final term in Eq. (3.66). A˜ρ starts depending not only on squares of w` -and-vm functions, but also on the eigenvalues individually. Whereas the shrinking of F (t) bifurcates the lower lines in Fig. 3.6, the shrinking does not bifurcate the upper lines. The reason is that each upper line corresponds to w2 w3 = v1 v2 = 1. [At early times, |F (t)| is small enough that any F (t)-dependent correction would fall within the lines’ widths.] Hence the final term in Eq. (3.65) is proportional to ± hW (t)V i. This prediction is consistent with the observed splitting. The hW (t)V i term does not split the lower lines: Each lower line satisfies w2 = −w3 and/or v1 = −v2 . Hence the hW (t)V i term vanishes. We leave as an open question whether these pitchforks can be understood in terms of equilibria, like classical-chaos pitchforks [120]. 68 1.0 0.8 0.6 Re[F] Im[F] 0.4 0.2 2 4 6 8 10 12 14 t Figure 3.5: Real and imaginary parts of F (t) as a function of time. T = ∞ thermal state. Nonintegrable parameters, N = 10, W = σ1z , V = σ zN . 0.25 0000 1000 0.20 0001 1001 0.15 0010 1010 0011 1011 0100 1100 0101 1101 0110 1110 0111 1111 0.10 0.05 2 -0.05 4 6 8 10 12 14 t Figure 3.6: Real part of A˜ρ as a function of time. T = ∞ thermal state. Nonintegrable parameters, N = 10, W = σ1z , V = σ zN . In contrast with the T = ∞ data, the T = 1 data oscillate markedly at late times (after the quasiprobability’s initial sharp change). We expect these oscillations to decay to zero at late times, if the system is chaotic, in the thermodynamic limit. Unlike at infinite temperature, W and V can have nonzero expectation values. But, if all nontrivial connected correlation functions have decayed, Eq. (3.60) still implies a simple dependence on the w` and vm parameters at late times. Finally, Figures 3.11 and 3.12 show the coarse-grained quasiprobability at infinite temperature, A˜(1/d) , with integrable parameters. The imaginary part remains zero, so we do not show it. The difference from the behavior in Figures 3.5 and 3.6 (which shows T = ∞, nonintegrable-H data) is obvious. Most dramatic is the large revival that occurs at what would, in the nonintegrable model, be a late time. Although this is not shown, the quasiprobability depends significantly on the choice of operator. This dependence is expected, since different Pauli operators have different degrees of complexity in terms of the noninteracting-fermion variables. Random states We now consider random pure states ρ ∝ |ψihψ| and nonintegrable parameters. Figures 3.13, 3.14, and 3.15 show F (t) and A˜ρ for the operator choice W = σ1z 69 2. × 10-17 2 4 6 8 10 12 t 14 -2. × 10-17 -4. × 10-17 -6. × 10-17 0000 1000 0001 1001 0010 1010 0011 1011 0100 1100 0101 1101 0110 1110 0111 1111 Figure 3.7: Imaginary part of A˜ρ as a function of time. T = ∞ thermal state. Nonintegrable parameters, N = 10, W = σ1z , V = σ zN . To within machine precision,   = Ãρ vanishes for all values of the arguments. 1.0 0.8 0.6 Re[F] Im[F] 0.4 0.2 2 4 6 8 10 12 14 t Figure 3.8: Real and imaginary parts of F (t) as a function of time. T = 1 thermal state. Nonintegrable parameters, N = 10, W = σ1z , V = σ zN . 0.6 0.4 0.2 2 4 6 8 10 12 14 t 0000 1000 0001 1001 0010 1010 0011 1011 0100 1100 0101 1101 0110 1110 0111 1111 Figure 3.9: Real part of A˜ρ as a function of time. T = 1 thermal state. Nonintegrable parameters, N = 10, W = σ1z , V = σ zN . 0.010 0.005 2 - 0.005 - 0.010 4 6 8 10 12 14 t 0000 1000 0001 1001 0010 1010 0011 1011 0100 1100 0101 1101 0110 1110 0111 1111 Figure 3.10: Imaginary part of A˜ρ as a function of time. T = 1 thermal state. Nonintegrable parameters, N = 10, W = σ1z , V = σ zN . 70 1.0 0.8 0.6 Re[F] 0.4 Im[F] 0.2 2 4 6 8 10 12 14 t Figure 3.11: Real and imaginary parts of F (t) as a function of time. T = ∞ thermal state. Integrable parameters, N = 10, W = σ1z , V = σ zN . 0.25 0000 1000 0.20 0001 1001 0.15 0010 1010 0.10 0011 1011 0100 1100 0101 1101 0110 1110 0111 1111 0.05 2 -0.05 4 6 8 10 12 14 t Figure 3.12: Real part of A˜ρ as a function of time. T = ∞ thermal state. Integrable parameters, N = 10, W = σ1z , V = σ zN . and V = σ Nz in a randomly chosen pure state. The pure state is drawn according to the Haar measure. Each figure shows a single shot (contains data from just one pure state). Broadly speaking, the features are similar to those exhibited by the infinite-temperature ρ = 1/d, with additional fluctuations. The upper branch of lines in Fig. 3.14 exhibits dynamics before the OTOC does. However, lines’ average positions move significantly (the lower lines bifurcate, and the upper lines shift downward) only after the OTOC begins to evolve. The early motion must be associated with the early dynamics of the 2- and 3-point functions in Eq. (3.60). The late-time values are roughly consistent with those for ρ = 1/d but fluctuate more pronouncedly. The agreement between random pure states and the T = ∞ thermal state is expected, due to closed-system thermalization [121, 122]. Consider assigning a temperature to a pure state by matching its energy density with the energy density of the thermal state e−H/T /Z, cast as a function of temperature. With high probability, any given random pure state corresponds to an infinite temperature. The reason is the thermodynamic entropy’s monotonic increase with temperature. Since the thermodynamic entropy gives the density of states, more states correspond to higher temperatures. Most states correspond to infinite temperature. 71 1.0 0.8 0.6 Re[F] Im[F] 0.4 0.2 2 4 6 8 10 12 14 t Figure 3.13: Real and imaginary parts of F (t) as a function of time. Random pure state. Nonintegrable parameters, N = 10, W = σ1z , V = σ zN . 0.25 0000 1000 0.20 0001 1001 0010 1010 0011 1011 0100 1100 0101 1101 0110 1110 0111 1111 0.15 0.10 0.05 2 4 6 8 10 12 14 t -0.05 Figure 3.14: Real part of A˜ρ as a function of time. Random pure state. Nonintegrable parameters, N = 10, W = σ1z , V = σ zN . 0.006 0000 1000 0.004 0001 1001 0.002 0010 1010 0011 1011 0100 1100 -0.002 0101 1101 -0.004 0110 1110 0111 1111 2 -0.006 4 6 8 10 12 14 t Figure 3.15: Imaginary part of A˜ρ as a function of time. Random pure state. Nonintegrable parameters, N = 10, W = σ1z , V = σ zN . For the random states and system sizes N considered, if H is nonintegrable, the agreement with thermal results is not complete. However, the physics appears qualitatively similar. Product states Finally, we consider the product |+xi⊗N of N copies of the +1 σ x eigenstate (Figures 3.16–3.18). We continue to use W = σ1z and V = σ Nz . For the Hamiltonian parameters chosen, this state lies far from the ground state. The state therefore should correspond to a large effective temperature. Figures 3.16, 3.17, and 3.18 show F (t) and A˜ρ for nonintegrable parameters. 72 1.0 0.8 0.6 Re[F] Im[F] 0.4 0.2 2 4 6 8 10 12 14 t Figure 3.16: Real and imaginary parts of F (t) as a function of time. Product |+xi ⊗ N of N copies of the +1 σ x eigenstate. Nonintegrable parameters, N = 10, W = σ1z , V = σ zN . 0.3 0.2 0.1 2 4 6 8 10 12 14 t 0000 1000 0001 1001 0010 1010 0011 1011 0100 1100 0101 1101 0110 1110 0111 1111 Figure 3.17: Real part of A˜ρ as a function of time. Product |+xi ⊗ N of N copies of the +1 σ x eigenstate. Nonintegrable parameters, N = 10, W = σ1z , V = σ zN . The real part of F (t) decays significantly from its initial value of one. The imaginary part of F (t) is nonzero but remains small. These features resemble the infinitetemperature features. However, the late-time F (t) values are substantially larger than in the T = ∞ case and oscillate significantly. Correspondingly, the real and imaginary components of A˜ρ oscillate significantly.   < A˜ρ exhibits dynamics before scrambling begins, as when ρ is a random pure state. The real and imaginary parts of A˜ρ differ more from their T = ∞ counterparts than F (t) differs from its counterpart. Some of this differing is apparently washed out by the averaging needed to construct F (t) [Eq. (3.45)]. We expected pure product states to behave roughly like random pure states. The data support this expectation very roughly, at best. Whether finite-size effects cause this deviation, we leave as a question for further study. Summary The main messages from this study are the following. (1) The coarse-grained quasiprobability A˜ρ is generically complex. Exceptions include the T = ∞ thermal state 1/d and states ρ that share an eigenbasis 73 0.015 0000 1000 0.010 0001 1001 0.005 0010 1010 0011 1011 0100 1100 -0.005 0101 1101 -0.010 0110 1110 0111 1111 2 4 6 8 10 12 14 -0.015 t Figure 3.18: Imaginary part of A˜ρ as a function of time. Product |+xi ⊗ N of N copies of the +1 σ x eigenstate. Nonintegrable parameters, N = 10, W = σ1z , V = σ zN . with V or with W (t) [e.g., as in Eq. (3.35)]. Recall that the KD distribution’s nonreality signals nonclassical physics (Sec. 3.1). (2) The derived quantity P(W,W 0 ) is generically complex, our results imply.11 Nonclassicality thus survives even the partial marginalization that defines P [Eq. (3.37)]. In general, marginalization can cause interference to dampen nonclassicality. (We observe such dampening in Property 6 of Sec. 3.5 and in Property 9 of Appendix B.1.) (3) Random pure states’ quasiprobabilities resemble the T = ∞ thermal state’s quasiprobability but fluctuate more. (4) Certain product states’ quasiprobabilities display anomalously large fluctuations. We expected these states to resemble random states more. (5) The A˜ρ ’s generated by integrable Hamiltonians differ markedly from the A˜ρ ’s generated by nonintegrable Hamiltonians. Both types of A˜ρ ’s achieve nonclassical values, however. We did not clearly observe a third class of behavior. (6) The time scale after which A˜ρ changes significantly is similar to the OTOC time scale. A˜ρ can display nontrivial early-time dynamics not visible in F (t). This dynamics can arise, for example, because of the 2-point function contained in the expansion of A˜ρ [see Eq. (3.60)]. (7) A˜ρ reveals that scrambling breaks a symmetry. Operationally, the symmetry consists of invariances of A˜ρ under permutations and negations of measurement outcomes in the weak-measurement scheme (Sec. 3.1). The symmetry 11 The relevant plots are not shown, so that this section maintains a coherent focus on A˜ . This ρ result merits inclusion, however, as P(W,W 0 ) plays important roles in (i) [38] and (ii) connections between the OTOC and quantum thermodynamics (Sec. 4.6). 74 breaking manifests in bifurcations of A˜ρ . These bifurcations evoke classicalchaos pitchfork diagrams, which also arise when a symmetry breaks. One equilibrium point splits into three in the classical case [120]. Perhaps the quasiprobability’s pitchforks can be recast in terms of equilibria. 3.4 Calculation of A˜ρ averaged over Brownian circuits We study a geometrically nonlocal model—the Brownian-circuit model—governed by a time-dependent Hamiltonian [72]. We access physics qualitatively different from the physics displayed in the numerics of Sec. 3.3. We also derive results for large systems and compare with the finite-size numerics. Since the two models’ locality properties differ, we do not expect agreement at early times. The late-time scrambled states, however, may be expected to share similarities. We summarize our main findings at the end of the section. We consider a system of N qubits governed by the random time-dependent Hamiltonian X X α ,α α Ji,ji j (t) σiα i σ j j . (3.67) H (t) ∝ i< j α i ,α j The couplings J are time-dependent random variables. We denote the site-i identity operator and Pauli operators by σiα , for α = 0, 1, 2, 3. According to the model’s precise formulation, the time-evolution operator U (t) is a random variable that obeys N (3.68) U (t + dt) − U (t) = − U (t)dt − i dB(t) . 2 The final term’s dB(t) has the form s XX 1 α α ,α dB(t) = σiα i σ j j dBi,ji j (t) . (3.69) 8(N − 1) i< j α ,α i j We will sometimes call Eq. (3.69) “dB.” dB is a Gaussian random variable with zero mean and with variance n α,β α0 ,β 0 o E B dBi,j dBi 0 ,j 0 = δ α,α0 δ β,β 0 δi,i 0 δ j,j 0 dt. (3.70) The expectation value E B is an average over realizations of the noise B. We demand that dt dt = 0 and dB dt = 0, in accordance with the standard Ito calculus. dB(t) is independent of U (t), i.e., of all previous dB’s. We wish to compute the average, over the ensemble defined by Eq. (3.68), of the coarse-grained quasiprobability: n o A(v1 , w2 , v2 , w3 ) = E B A˜ρ (v1 , w2 , v2 , w3 ) . (3.71) 75 Infinite-temperature thermal state 1/2 N We focus here on the infinite-temperature thermal state, ρ = 1/2 N , for two reasons. First, a system with a time-dependent Hamiltonian generically heats to infinite temperature with respect to any Hamiltonian in the ensemble. Second, the T = ∞ state is convenient for calculations. A discussion of other states follows. The ensemble remains invariant under single-site rotations, and all qubits are equivalent. Therefore, all possible choices of single-site Pauli operators for W and V are equivalent. Hence we choose W = σ1z and V = σ2z without loss of generality. Let us return to Eq. (3.59). Equation (3.59) results from substituting in for the projectors in A˜ρ . The sum contains 16 terms. To each term, each projector contributes the identity 1 or a nontrivial Pauli (W or V ). The terms are n o (1) 1111: Tr 21N = 1, (2) W 111, 1V 11, 11W 1, 111V : 0, (3) WV , W 11V , 1V W 1, 11WV :  11 σ1z (t)σ2z Tr =: G(t), 2N n o (4) W 1W 1, 1V 1V : Tr 21N = 1, (5) WV W 1, WV 1V , W 1WV , 1V WV : 0,  z z z z σ (t)σ σ (t)σ (6) WV WV : Tr 1 22N 1 2 = F (t). and These computations rely on ρ = 1/2 N . Each term that contains an odd number of Pauli operators vanishes, due to the trace’s cyclicality and to the Paulis’ tracelessness. We have introduced a 2-point function G(t). An overall factor of 1/16 comes from the projectors’ normalization. Combining all the ingredients, we can express A˜ρ in terms of G and F. The result is 16 A˜ρ (v1 , w2 , v2 , w3 ) = (1 + w2 w3 + v1 v2 ) (3.72) + (w2 + w3 )(v1 + v2 ) G + w2 w3 v1 v2 F. This result depends on ρ = 1/2 N , not on the form of the dynamics. But to compute A, we must compute G = E B {G} (3.73) 76 and F = E B {F} . (3.74) The computation of F appears in the literature [4]. F initially equals unity. It decays to zero around t ∗ = 13 log N, the scrambling time. The precise functional form of F is not crucial. The basic physics is captured in a phenomenological form inspired by AdS/CFT computations [4], ! c2 1 + c1 , (3.75) F∼ 1 + c1 e3t wherein c1 ∼ 1/N and c2 ∼ 1. To convey a sense of the physics, we review the simpler calculation of G. The two-point function evolves according to # " 1 N G(t + dt) = N Tr U (t) − U (t)dt − i dB U (t) σ1z 2 2 " #  N † † † † × U (t) − U (t) dt + i U (t) dB σ2z . (3.76) 2 Using the usual rules of Ito stochastic calculus, particularly Eq. (3.70) and dt dt = dB dt = 0, we obtain G(t + dt) − G(t) = −N dt G(t) + dt 1 8(N − 1) XX 1 n n oo αi α j z αi α j z × E Tr σ (t)σ σ σ σ σ . B i j j 1 2 i 2N i< j α ,α i (3.77) j We have applied the trace’s cyclicality in the second term. The second term’s value depends on whether i and/or j equals 2. If i and/or j P equals 2, the second term vanishes because 3α=0 σ α σ z σ α = 0. If neither i nor j α is 2, σiα i σ j j commutes with σ2z . The second term becomes proportional to G. In (N − 1)(N − 2)/2 terms, i, j , 2. An additional factor of 42 = 16 comes from the two sums over Pauli matrices. Hence G(t + dt) − G(t) = −2dt G , (3.78) dG = −2G. dt (3.79) or 77 This differential equation implies that G exponentially decays from its initial value. The initial value is zero: G(0) = G(0) = 0. Hence G(t) is identically zero. Although it does not arise when we consider A, the ensemble-average autocorrelan o tion function E B hσ1z (t)σ1z i obeys a differential equation similar to the equation obeyed by G. In particular, the equation decays exponentially with an order-one rate. By the expectation value’s linearity and the vanishing of G, A= (1 + w2 w3 + v1 v2 ) + w2 w3 v1 v2 F . 16 (3.80) This simple equation states that the ensemble-averaged quasiprobability depends only on the ensemble-averaged OTOC F (t), at infinite temperature. The time scale of F’s decay is t ∗ = 13 log N. Hence this is the time scale of changes in A. Equation (3.80) shows (as intuition suggests) that A depends only on the combinations w2 w3 and v1 v2 . At t = 0, F(0) = 1. Hence A is At=0 = 1 + w2 w3 + v1 v2 + w2 w3 v1 v2 . 16 (3.81) The cases are (1) w2 w3 = 1, v1 v2 = 1: A = 1/4, (2) w2 w3 = 1, v1 v2 = −1: A = 0, (3) w2 w3 = −1, v1 v2 = 1: A = 0, and (4) w2 w3 = −1, v1 v2 = −1: A = 0. These values are consistent with Fig. 3.6 at t = 0. These values’ degeneracies are consistent with the symmetries discussed in Sec. 3.3 and in Sec. 3.5 (Property 7). At long times, F(∞) = 0, so A is At=∞ = 1 + w2 w3 + v1 v2 . 16 The cases are (1) w2 w3 = 1, v1 v2 = 1: A = 3/16, (2) w2 w3 = 1, v1 v2 = −1: A = 1/16, (3.82) 78 (3) w2 w3 = −1, v1 v2 = 1: A = 1/16, and (4) w2 w3 = −1, v1 v2 = −1: A = −1/16. Modulo the splitting of the upper two lines, this result is broadly consistent with the long-time behavior in Fig. 3.6. As the models in Sec. 3.3 and this section differ, the long-time behaviors need not agree perfectly. However, the models appear to achieve qualitatively similar scrambled states at late times. General state Consider a general state ρ, such that A˜ρ assumes the general form in Eq. (3.59). We still assume that W = σ1z and V = σ2z . However, the results will, in general, now depend on these choices via the initial condition ρ. We still expect that, at late times, the results will not depend on the precise choices. Below, we use the notation h.i ≡ Tr( ρ .). We must consider 16 terms again. The general case involves fewer simplifications. The terms are (1) 1111: 1, (2) W 111, 1V 11, 11W 1, 111V : hσ1z (t)i , hσ2z i, (3) WV 11, W 11V , 1V W 1, 11WV : hσ1z (t) σ2z i, hσ2z σ1z (t)i, (4) W 1W 1, 1V 1V : 1, (5) WV W 1, WV 1V , W 1WV , 1V WV : hσ1z (t) σ2z σ1z (t)i, hσ1z (t) i, hσ2z i, hσ2z σ1z (t) σ2z i, and (6) WV WV : hσ1z (t) σ2z σ1z (t) σ2z i = F (t). Consider first the terms of the form qi (t) := E B {hσiz (t)i}. The time derivative is dqi = −Nqi dt XX 1 α α + E B {hσ j j σ kα k U (t)σiz U (t) † σ j j σ kα k i}. 8(N − 1) j<k α ,α j (3.83) k To simplify the second term, we use a trick. Since α α αm αn αm αn σ j j σ kα k σm σn σ j j σ kα k = ±σm σn , (3.84) 79 α we may pass the factors of σ j j σ kα k through U (t), at the cost of changing some Brownian weights. We must consider a different set of dB’s, related to the originals by minus signs. This alternative set of Brownian weights has the original set’s ensemble probability. Hence the ensemble average gives the same result. Therefore, α α E B {hσ j j σ kα k U (t) σiz U (t) † σ j j σ kα k i} α α = E B {hU (t)σ j j σ kα k σiz σ j j σ kα k U (t) † i}. (3.85) If i = j and/or i = k, the sum over α j and/or the sum over α k vanishes. If i equals neither j nor k, the Pauli operators commute. The term reduces to qi . i equals neither j nor k in (N − 1)(N − 2)/2 terms. A factor of 16 comes from the sums over α j and α k . Hence dqi = −Nqi + (N − 2)qi = −2qi . dt (3.86) Consider the terms of the form qi j (t) := hσiz (t)σ zj i. Note that hσ zj σiz (t)i = qi∗j . We may reuse the trick introduced above. [This trick fails only when more than two copies of U appear, as in F (t)]. To be precise, αm αn z αm αn σn σ j i} σn U (t)σiz U (t) † σm E B {hσm αm αn z αm αn = E B {hU (t)σm σn σi σm σn U (t) † σ zj i}. (3.87) As before, the sums over α kill the relevant term in the time derivative of qi j , unless i , m, n. Hence dqi j = −2qi j , dt (3.88) as at infinite temperature. Item (5), in the list above, concerns products of three W’s and V ’s. We must consider four expectation values of Pauli products. As seen above, two of these terms reduce to qi terms. By the trick used earlier, E B {hσ2z U (t)σ1z U (t) † σ2z } = E B {hU (t)σ2z σ1z σ2z U (t) † } = q1 (t). (3.89) The other term we must consider is E B {hσiz (t)σ zj σiz (t)i} =: fi j . Our trick will not work, because there are multiple copies of U (t) that are not all simultaneously 80 switched as operators are moved around. At early times, when σiz (t) and σ zj approximately commute, this term approximately equals hσ zj i = q j (0). At later times, including around the scrambling time, this term decays to zero. The general expression for A becomes 16 A = 1 + w3 w2 + v1 v2 + (w3 + w2 ) q1 (t) + (v1 + v2 ) q2 (0) + (w3 v2 + w3 v1 + w2 v1 ) q12 (t) + v2 w2 q12 (t) ∗ + w3 v2 w2 f12 (t) + (w3 v1 v2 + w2 v1 v2 ) q1 (t) + w3 w2 v1 q2 (0) + w3 w2 v1 v2 F(t). (3.90) All these q functions obey known differential equations. The functions decay after a time of order one. We do not have explicit expressions for the f functions that appear. They are expected to vary after a time ∼ log N. Special case: σ2z eigenstate In a concrete example, we suppose that ρ is a +1 eigenstate of σ2z . Expressions simplify: q2 (0) = 1, (3.91) q12 (t) = q1 (t) = q12 (t) ∗ , (3.92) f12 = F. (3.93) and Hermiticity of the Pauli operators implies that f12 is real. Hence the ensembleaveraged OTOC F is real for this choice of ρ. The ensemble-averaged à ρ has the form A= k 1 + k 2q1 + k 3F , 16 (3.94) wherein k1 = (1 + v1 )(1 + v2 + w3 w2 ), (3.95) k 2 = (1 + v1 )(w3 + w2 )(1 + v2 ), (3.96) 81 and k3 = (1 + v1 )w3 v2 w2 . (3.97) Equations (3.94)–(3.97) imply that A = 0 unless v1 = 1. The time scale after which q1 decays is order-one. The time required for F to decay is of order log N (although not necessarily exactly the same as for the T = ∞ state). Therefore, the late-time value of A is well approximated by At1 = k1 + k3 F . 16 (3.98) Summary This study has the following main messages. (1) In this model, the ensemble-averaged quasiprobability varies on two time scales. The first time scale is an order-one relaxation time. At later times, the OTOC controls the physics entirely. F (t) varies after a time of order log N. (2) While the late-time physics of A˜ρ is controlled entirely by the ensembleaveraged F (t), the negative values of A˜ρ show a nonclassicality that might not be obvious from F (t) alone. Furthermore, we computed only the first moment of A˜ρ . The higher moments are likely not determined by F (t) alone. (3) For T = ∞, the late-time physics is qualitatively similar to the late-time physics of the geometrically local spin chain in Sec. 3.3. (4) Nonclassicality, as signaled by negative values of A˜ρ , is extremely robust. It survives the long-time limit and the ensemble average. One might have expected thermalization and interference to stamp out nonclassicality. On the other hand, we expect the circuit average to suppress the imaginary part of A˜ρ   rapidly. We have no controlled examples in which = A˜ρ remains nonzero at long times. Finding further evidence for or against this conjecture remains an open problem. 3.5 Theoretical study of à ρ We have discussed experimental measurements, numerical simulations, and analytical calculations of the OTOC quasiprobability à ρ . We now complement these discussions with mathematical properties and physical interpretations. First, we define an extended Kirkwood-Dirac distribution exemplified by à ρ . We still denote by B(H ) the set of bounded operators defined on H . 82 Definition 1 (K -extended Kirkwood-Dirac quasiprobability). Let {|ai} , . . . , {|ki} and {| f i} denote orthonormal bases for the Hilbert space H . Let O ∈ B(H ) denote a bounded operator defined on H . A K -extended Kirkwood-Dirac quasiprobability for O is defined as12 (K ) ÃO (a, . . . , k, f ) := h f |kihk | . . . |aiha|O| f i . (3.99) This quasiprobability can be measured via an extension of the protocol in Sec. 3.1. Suppose that O denotes a density matrix. In each trial, one prepares O, weakly measures the bases sequentially (weakly measures {|ai}, and so on, until weakly measuring {|ki}), then measures | f ih f | strongly. We will focus mostly on density operators O = ρ ∈ D(H ). One infers Ã(ρK ) by performing 2K − 1 weak measurements, and one strong measurement, per trial. The order in which the bases are measured is the order in which the labels a, . . . , k, f (K ) appear in the argument of ÃO (.). The conventional KD quasiprobability is 1extended. The OTOC quasiprobability à ρ is 3-extended. Our investigation parallels the exposition, in Sec. 3.1, of the KD distribution. First, we present basic mathematical properties. à ρ , we show next, obeys an analog of Bayes’ Theorem. Our analog generalizes the known analog (3.5). Our theorem reduces exponentially (in system size) the memory needed to compute weak values, in certain cases. Third, we connect à ρ with the operator-decomposition argument in Sec. 3.1. à ρ consists of coefficients in a decomposition of an operator ρ0 that results from asymmetrically decohering ρ. Summing à ρ (.) values yields a KD representation for ρ. This sum can be used, in experimental measurements of à ρ and the OTOC, to evaluate how accurately the desired initial state was prepared. Fourth, we explore the relationship between out-of-time ordering and quasiprobabilities. Time-ordered correlators are moments of quasiprobabilities that clearly reduce to classical probabilities. Finally, we generalize beyond the OTOC, which ¯ encodes K = 3 time reversals. Let K¯ := 12 (K + 1). A K¯ -fold OTOC F (K ) (t) ¯ encodes K time reversals [123, 124]. The quasiprobability behind F (K ) (t), we find, is K -extended. 12 Time evolutions may be incorporated into the bases. For example, Eq. (3.15) features the 1-extended KD quasiprobability h f 0 |aiha| ρ0 | f 0 i. The ρ0 := Ut 0 ρUt†0 results from time-evolving P a state ρ. The | f 0 i := Ut†00 −t 0 | f i results from time-evolving an eigenket | f i of F = f f | f ih f |. (1) 0 0 We label (3.15) as Ã(1) ρ ( ρ, a, f ), rather than as Ãρ ( ρ , a, f ). Why? One would measure (3.15) by preparing ρ, evolving the system, measuring A weakly, inferring outcome a, evolving the system, measuring F, and obtaining outcome f . No outcome f 0 is obtained. Our notation is that in [15] and is consistent with the notation in [38]. 83 Recent quasiprobability advances involve out-of-time ordering, including in correlation functions [125–129]. Merging these works with the OTOC framework offers an opportunity for further research (Sec. 4.6). Mathematical properties of à ρ à ρ shares some of its properties with the KD quasiprobability (Sec. 3.1). Properties of à ρ imply properties of P(W,W 0 ), presented in Appendix B.1.  Property 5. The OTOC quasiprobability is a map à ρ : D (H ) × {v1 } × λ v1 ×    {w2 } × α w2 × {v2 } × λ v2 × {w3 } × α w3 × → C . The domain is a composition of the set D (H ) of density operators defined on H and eight sets of complex numbers. The range is not necessarily real: C ⊃ R. à ρ depends on H and t implicitly through U. The KD quasiprobability in [15] depends implicitly on time similarly (see Footnote 12). Outside of OTOC contexts, D (H ) may be replaced with B(H ). K -extended KD distributions represent bounded operators, not only quantum states. C, not necessarily R, is the range also of the K -fold generalization Ã(ρK ) . We expound upon the range’s complexity after discussing the number of arguments of à ρ . Five effective arguments of à ρ : On the left-hand side of Eq. (3.33), semicolons separate four tuples. Each tuple results from a measurement, e.g., of W̃. We coarse-grained over the degeneracies in Sections 3.2–3.4. Hence each tuple often functions as one degree of freedom. We treat à ρ as a function of four arguments (and of ρ). The KD quasiprobability has just two arguments (apart from O). The need for four arises from the noncommutation of W (t) and V . Complexity of à ρ : The ability of à ρ to assume nonreal values mirrors Property 1 of the KD distribution. The Wigner function, in contrast, is real. The OTOC quasiprobability’s real component, <( à ρ ), parallels the Terletsky-Margenau-Hill distribution. We expect nonclassical values of à ρ to reflect nonclassical physics, as nonclassical values of the KD quasiprobability do (Sec. 3.1). Equations (3.33) and (3.34) reflect the ability of à ρ to assume nonreal values. Equation (3.33) would equal a real product of probabilities if the backward-process amplitude A∗ρ and the forward-process amplitude A ρ had equal arguments. But the arguments typically do not equal each other. Equation (3.34) reveals conditions under which à ρ (.) ∈ R and < R. We illustrate the ∈ case with two examples and the < case with one example. 84 Example 1 (Real à ρ #1: t = 0, shared eigenbasis, arbitrary ρ). Consider t = 0, at which U = 1. The operators W (t) = W and V share an eigenbasis, under the   assumption that [W, V ] = 0: |w` , α w` i = |v` , λ v` i . With respect to that basis, à ρ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 )     = δw3 v2 δ α w3 λ v2 δv2 w2 δ λ v2 α w2 δw2 v1 δ α w2 λ v1 X × p j |hw3 , α w3 | ji| 2 j ∈ R. (3.100) We have substituted into Eq. (3.34). We substituted in for ρ from Eq. (3.25). Example 1 is consistent with the numerical simulations in Sec. 3.3. According to P Eq. (3.100), at t = 0, degeneracies à ρ =: A˜ρ ∈ R. In Figures 3.10, 3.15, and 3.18, the imaginary parts =(A˜ρ ) clearly vanish at t = 0. In Fig. 3.7, =(A˜ρ ) vanishes to within machine precision.13 Consider a ρ that lacks coherences relative to the shared eigenbasis, e.g., ρ = 1/d.     Example 1 implies that = Ã(1/d) at t = 0. But = Ã(1/d) remains zero for all t in the numerical simulations. Why, if time evolution deforms the W (t) eigenbasis from the V eigenbasis? The reason appears to be a cancellation, as in Example 2. Example 2 requires more notation. Let us focus on a chain of N spin- 21 degrees of freedom. Let σ α denote the α = x, y, z Pauli operator. Let |σ α , ±i denote the σ α eigenstates, such that σ α |σ α , ±i = ±|σ α , ±i. N-fold tensor products are denoted by |σ α, ±i := |σ α , ±i⊗N . We denote by σ αj the α th Pauli operator that acts nontrivially on site j. 13 The =(A˜ ) in Fig. 3.7 equals zero identically, if w ρ 2 = w3 and/or if v1 = v2 . For general arguments, =(A˜ρ (v1 , w2 , v2 , w3 )) = The final term equals h 1 h Ãρ (v1 , w2 , v2 , w3 ) 2i i − Ã∗ρ (v1 , w2 , v2 , w3 ) .  W (t ) i∗   W (t ) V W (t ) V W (t ) Tr Πw ΠV Πv1 = Tr ΠV Πv2 Πw3 v2 Πw2 v1 Πw2 3  W (t )  W (t ) V = Tr Πw ΠV Πv1 = A˜ρ (v1 , w3 , v2 , w2 ) . v2 Πw3 2 (3.101) (3.102) (3.103) The first equality follows from projectors’ Hermiticity; and the second, from the trace’s cyclicality. Substituting into Eq. (3.101) shows that A˜ρ (.) is real if w2 = w3 . A˜ρ (.) is real if v1 = v2 , by an analogous argument. 85 Example 2 (Real à ρ #2: t = 0, nonshared eigenbases, ρ = 1/d). Consider the y spin chain at t = 0, such that U = 1. Let W = σ1z and V = σ N . Two W eigen  ⊗N 1 z y z z √ (|σ , +i + i|σ , −i) states are |σ , ±i. Two V eigenstates are |σ , +i = and 2   ⊗N |σ y, −i = √1 (|σ z , +i − i|σ z , −i) . The overlaps between the W eigenstates 2 and the V eigenstates are hσ , +|σ , +i = z y hσ z, +|σ y, −i = hσ z, −|σ y, +i = hσ z, −|σ y, −i = !N 1 , √ 2 !N 1 , √ 2 !N i , and √ 2 !N −i . √ 2 (3.104) Suppose that ρ = 1/d. Ã(1/d) (.) would have a chance of being nonreal only if some |v` , λ v` i equaled |σ z, −i. That |σ z, −i would introduce an i into Eq. (3.34). But hσ z, −| would introduce another i. The product would be real. Hence Ã(1/d) (.) ∈ R. à ρ is nonreal in the following example. Example 3 (Nonreal à ρ : t = 0, nonshared eigenbases, ρ nondiagonal relative to   both). Let t, W, V , |w` , α w` i , and |vm , λ vm i be as in Example 2. Suppose that ρ has coherences relative to the W and V eigenbases. For instance, let ρ = |σ x, +ihσ x, +|. Since |σ x , +i = √1 (|σ z , +i + |σ z , −i), 2 ρ= 1 (|σ z , +ihσ z , +| + |σ z , +ihσ z , −| 2N + |σ z , −ihσ z , +| + |σ z , −ihσ z , −|) ⊗N . (3.105) Let |w3 , α w3 i = |σ z, −i, such that its overlaps with V eigenstates can contain i’s. The final factor in Eq. (3.34) becomes hv1 , λ v1 | ρ|w3 , α w3 i =   1 h z ⊗N hv , λ | |σ , +i 1 v 1 2N  i + hv1 , λ v1 | |σ z , −i⊗N . (3.106) 86  The first inner product evaluates to N  . Hence product evaluates to ± √i √1 2 N , by Eqs. (3.104). The second inner 2 hv1 , λ v1 | ρ|w3 , α w3 i = 1 h 22N i 1 + (±i) N . (3.107) This expression is nonreal if N is odd. Example 3, with the discussion after Example 1, shows how interference can elim  inate nonreality from a quasiprobability. In Example 3, = à ρ does not neces  sarily vanish. Hence the coarse-grained = A˜ρ does not obviously vanish. But   = A˜ρ = 0 according to the discussion after Example 1. Summing Example 3’s   nonzero = à ρ values must quench the quasiprobability’s nonreality. This quenching illustrates how interference can wash out quasiprobabilities’ nonclassicality. Yet interference does not always wash out nonclassicality. Section 3.3 depicts A˜ρ ’s that have nonzero imaginary components (Figures 3.10, 3.15, and 3.18). Example 3 resonates with a finding in [109, 110]. Solinas and Gasparinetti’s quasiprobability assumes nonclassical values when the initial state has coherences relative to the energy eigenbasis. Property 6. Marginalizing à ρ (.) over all its arguments except any one yields a probability distribution. Consider, as an example, summing Eq. (3.34) over every tuple except (w3 , α w3 ). P The outer products become resolutions of unity, e.g., (w2 ,α w2 ) |w2 , α w2 ihw2 , α w2 | = 1. A unitary cancels with its Hermitian conjugate: U †U = 1. The marginalization yields hw3 , α w3 |U ρU † |w3 , α w3 i. This expression equals the probability that preparing ρ, time-evolving, and measuring the W̃ eigenbasis yields the outcome (w3 , α w3 ). This marginalization property, with the structural and operational resemblances between à ρ and the KD quasiprobability, accounts for our calling à ρ an extended quasiprobability. The general K -extended Ã(ρK ) obeys Property 6. Property 7 (Symmetries of Ã(1/d) ). Let ρ be the infinite-temperature Gibbs state 1/d. The OTOC quasiprobability Ã(1/d) has the following symmetries. (A) Ã(1/d) (.) remains invariant under the simultaneous interchanges of (w2 , α w2 ) with (w3 , α w3 ) and (v1 , λ v1 ) with (v2 , λ v2 ): Ã(1/d) (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ) = Ã(1/d) (v2 , λ v2 ; w3 , α w3 ; v1 , λ v1 ; w2 , α w2 ). 87 (B) Let t = 0, such that |w` , α w` i = |v` , λ v` i (under the assumption that [W,V ] = 0). Ã(1/d) (.) remains invariant under every cyclic permutation of its arguments.   Equation (3.34) can be recast as a trace. Property 7 follows from the trace’s cyclicality. Subproperty (B) relies on the triviality of the t = 0 time-evolution operator: U = 1. The symmetries lead to degeneracies visible in numerical plots (Sec. 3.3). Analogous symmetries characterize a regulated quasiprobability. Maldacena et al. regulated F (t) to facilitate a proof [7]:14   Freg (t) := Tr ρ1/4 W (t) ρ1/4V ρ1/4 W (t) ρ1/4V . (3.108) Freg (t) is expected to behave roughly like F (t) [7, 43]. Just as F (t) equals a moment of a sum over à ρ , Freg (t) equals a moment of a sum over reg à ρ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ) (3.109) := hw3 , α w3 |U ρ1/4 |v2 , λ v2 ihv2 , λ v2 | ρ1/4U † |w2 , α w2 i × hw2 , α w2 |U ρ1/4 |v1 , λ v1 ihv1 , λ v1 | ρ1/4U † |w3 , α w3 i ≡ hw3 , α w3 |Ũ |v2 , λ v2 ihv2 , λ v2 |Ũ † |w2 , α w2 i (3.110) × hw2 , α w2 |Ũ |v1 , λ v1 ihv1 , λ v1 |Ũ † |w3 , α w3 i . The proof is analogous to the proof of Theorem 1 in [38]. Equation (3.110) depends on Ũ := Z1 e−iHτ , which propagates in the complex-time variable τ := t − 4Ti . The ∗ Hermitian conjugate Ũ † = Z1 eiHτ propagates along τ ∗ = t + 4Ti . reg à e −H /T /Z has the symmetries of Ã(1/d) (Property 7) for arbitrary T. One might ( ) reg expect à ρ to behave similarly to à ρ , as Freg (t) behaves similarly to F (t). Numerical simulations largely support this expectation. We compared A˜ρ (.) with P reg reg A˜ρ (.) := degeneracies à ρ (.) . The distributions vary significantly over similar reg time scales and have similar shapes. A˜ρ tends to have a smaller imaginary component and, as expected, more degeneracies. The properties of à ρ imply properties of P(W,W 0 ). We discuss these properties in Appendix B.1. 14 The name “regulated” derives from quantum field theory. F (t) contains operators W † (t) and W (t) defined at the same space-time point (and operators V † and V defined at the same space-time point). Products of such operators encode divergences. One can regulate divergences by shifting one operator to another space-time point. The inserted ρ1/4 = Z 11/4 e −H /4T shifts operators along an imaginary-time axis. 88 Bayes-type theorem and retrodiction with à ρ We reviewed, in Sec. 3.1, the KD quasiprobability’s role in retrodiction. The KD 0 0 0 quasiprobability Ã(1) ρ generalizes the nontrivial part <(h f |aiha| ρ | f i) of a conditional quasiprobability p̃(a| ρ, f ) used to retrodict about an observable A. Does à ρ play a role similar to Ã(1) ρ ? It does. To show so, we generalize Sec. 3.1 to composite observables. Let A, B, . . . , K denote K observables. K . . . BA might not be Hermitian but can be symmetrized. For example, Γ := K . . . A + A . . . K is an observable.15 Which value is most reasonably attributable to Γ retrodictively? A weak value Γweak given by Eq. (3.3). We derive an alternative expression for Γweak . In our expression, Γ eigenvalues are weighted by K -extended KD quasiprobabilities. Our expression reduces exponentially, in the system’s size, the memory required to calculate weak values, under certain conditions. We present general theorems about Ã(ρK ) , then specialize to the OTOC à ρ . Theorem 2 (Retrodiction about composite observables). Consider a system S associated with a Hilbert space H . For concreteness, we assume that H is discrete. P P Let A = a a|aiha| , . . . , K = k k |kihk | denote K observables defined on H . Let Ut denote the family of unitaries that propagates the state of S along time t. Suppose that S begins in the state ρ at time t = 0, then evolves under Ut 00 until P t = t 00. Let F = f f | f ih f | denote an observable measured at t = t 00. Let f denote the outcome. Let t 0 ∈ (0,t 00 ) denote an intermediate time. Define ρ0 := Ut 0 ρUt†0 and | f 0i := Ut†00 −t 0 | f i as time-evolved states. The value most reasonably attributable retrodictively to the time-t 0 Γ := K . . . A + A . . . K is the weak value X h Γweak ( ρ, f ) = (a . . . k) p̃→ (a, . . . , k | ρ, f ) a,...,k i + p̃← (k, . . . , a| ρ, f ) . (3.111) The weights are joint conditional quasiprobabilities. They obey analogs of Bayes’ 15 So is Γ̃ := i(K . . . A − A . . . K ). An operator can be symmetrized in multiple ways. Theo- rem 2 governs Γ. Appendix B.2 contains an analogous result about Γ̃. Theorem 2 extends trivially to Hermitian (already symmetrized) instances of K . . . A. Corollary 1 illustrates this extension. 89 Theorem: p̃→ (a, . . . , k, f | ρ) p( f | ρ) <(h f 0 |kihk | . . . |aiha| ρ0 | f 0i) ≡ , h f 0 | ρ0 | f 0i p̃→ (a, . . . , k | ρ, f ) = (3.112) (3.113) and p̃← (k, . . . , a, f | ρ) p( f | ρ) <(h f 0 |aiha| . . . |kihk | ρ0 | f 0i) ≡ . h f 0 | ρ0 | f 0i p̃← (k, . . . , a| ρ, f ) = (3.114) (3.115) Complex generalizations of the weights’ numerators, K) (a, . . . , k, f ) := h f 0 |kihk | . . . |aiha| ρ0 | f 0i Ã(ρ,→ (3.116) K) Ã(ρ,← (k, . . . , a, f ) := h f 0 |aiha| . . . |kihk | ρ0 | f 0i , (3.117) and are K -extended KD distributions. A rightward-pointing arrow → labels quantities in which the outer products, |kihk |, . . . , |aiha|, are ordered analogously to the first term K . . . A in Γ. A leftwardpointing arrow ← labels quantities in which reading the outer products |aiha|, . . . , |kihk | backward—from right to left—parallels reading K . . . A forward. Proof. The initial steps come from [15, Sec. II A], which recapitulates [74–76]. For every measurement outcome f , we assume, some number γ f is the guess most reasonably attributable to Γ. We combine these best guesses into the effective obn o P servable Γest := f γ f | f 0ih f 0 |. We must optimize our choice of γ f . We should quantify the distance between (1) the operator Γest we construct and (2) the operator Γ we wish to infer about. We use the weighted trace distance   D ρ0 (Γ, Γest ) = Tr ρ0[Γ − Γest ]2 . (3.118) ρ0 serves as a “positive prior bias” [15]. Let us substitute in for the form of Γest . Expanding the square, then invoking the trace’s linearity, yields Xh D ρ0 (Γ, Γest ) = Tr( ρ0 Γ2 ) + γ 2f h f 0 | ρ0 | f 0i f i − γ f (h f 0 | ρ0 Γ| f 0i + h f 0 |Γ ρ0 | f 0i) . (3.119) 90 The parenthesized factor equals 2<(h f 0 |Γ ρ0 | f 0i). Adding and subtracting X h f 0 | ρ0 | f 0i[<(h f 0 |Γ ρ0 | f 0i)]2 (3.120) f to and from Eq. (3.119), we complete the square: X D ρ0 (Γ, Γest ) = Tr( ρ0 Γ2 ) − h f 0 | ρ0 | f 0i[<(h f 0 |Γ ρ0 | f 0i)]2 f !2 X <(h f 0 |Γ ρ0 | f 0i) 0 0 0 + h f |ρ | f i γf − . 0 | ρ0 | f 0i h f f (3.121) n o Our choice of γ f should minimize the distance (3.121). We should set the square to zero: γf = <(h f 0 |Γ ρ0 | f 0i) . h f 0 | ρ0 | f 0i (3.122) Now, we deviate from [15, 74–76]. We substitute the definition of Γ into Eq. (3.122). Invoking the linearity of < yields γf = <(h f 0 |K . . . A ρ0 | f 0i) <(h f 0 |A . . . K ρ0 | f 0i) + . h f 0 | ρ0 | f 0i h f 0 | ρ0 | f 0i (3.123) We eigendecompose A, . . . , K . The eigenvalues, being real, can be factored out of the <’s. Defining the eigenvalues’ coefficients as in Eqs. (3.113) and (3.115), we reduce Eq. (3.123) to the form in Eq. (3.111).  Theorem 2 reduces exponentially, in system size, the space required to calculate Γweak , in certain cases.16 For concreteness, we focus on a multiqubit system and on P l-local operators A, . . . , K . An operator O is l-local if O = j O j , wherein each O j operates nontrivially on, at most, l qubits. Practicality motivates this focus: The lesser the l, the more easily l-local operators can be measured. We use asymptotic notation from computer science: Let f ≡ f (N ) and g ≡ g(N ) denote any functions of the system size. If g = O( f ), g grows no more quickly than (is upper-bounded by) a constant multiple of f in the asymptotic limit, as N → ∞. If g = Ω( f ), g grows at least as quickly as (is lower-bounded by) a constant multiple of f in the asymptotic limit. If g = Θ( f ), g is upper- and lowerbounded by f : g = O( f ), and g = Ω( f ). If g = o( f ), g shrinks strictly more quickly than f in the asymptotic limit. 16 “Space” means “memory,” or “number of bits,” here. 91 Theorem 3 (Weak-value space saver). Let S denote a system of N qubits. Let H denote the Hilbert space associated with S. Let | f 0i ∈ H denote a pure state and ρ0 ∈ D (H ) denote a density operator. Let B denote any fixed orthonormal basis for H in which each basis element equals a tensor product of N factors, each of which operates nontrivially on exactly one site. B may, for example, consist of tensor products of σ z eigenstates. Let K denote any polynomial function of N: K ≡ K (N ) = poly(N ). Let A, . . . , K denote K traceless l-local observables defined on H , for any constant l. Each observable may, for example, be a tensor product of ≤ l nontrivial Pauli operators and ≥ N − l identity operators. The composite observable P Γ := A . . . K + K . . . A is not necessarily l-local. Let A = a a|aiha| , . . . , K = P k k |kihk | denote eigenvalue decompositions of the local observables. Let OB denote the matrix that represents an operator O relative to B. Consider being given the matrices A B , . . . , KB , ρ0B , and | f 0iB . From this information, the weak value Γweak can be computed in two ways: (1) Conventional method (A) Multiply and sum given matrices to form ΓB = KB . . . A B + A B . . . KB . (B) Compute h f 0 | ρ0 | f 0i = h f 0 |B ρ0B | f 0iB .  0  h f |B ΓB ρ0B | f 0 iB (C) Substitute into Γweak = < . h f 0 | ρ0 | f 0 i (2) K -factored method (A) Compute h f 0 | ρ0 | f 0i. (B) For each nonzero term in Eq. (3.111), (i) calculate p̃→ (.) and p̃← (.) from Eqs. (3.113) and (3.115). (ii) substitute into Eq. (3.111). Let Σ(n) denote the space required to compute Γweak , aside from the space required to store Γweak , with constant precision, using method (n) = (1), (2), in the asymptotic limit. Method (1) requires a number of bits at least exponential in the number K of local observables:   Σ(1) = Ω 2K . (3.124) 92 Method (2) requires a number of bits linear in K : Σ(2) = O(K ) . (3.125) Method (2) requires exponentially—in K and so in N—less memory than Method (1). Proof. Using Method (1), one computes ΓB . ΓB is a 2 N × 2 N complex matrix. The matrix has Ω(2K ) nonzero elements: A, . . . , K are traceless, so each of A B , . . . , KB contains at least two nonzero elements. Each operator at least doubles the number of nonzero elements in ΓB . Specifying each complex number with constant preci  sion requires Θ(1) bits. Hence Method (1) requires Ω 2K bits. Let us turn to Method (2). We can store h f 0 | ρ0 | f 0i in a constant number of bits. Step (B) can be implemented with a counter variable CO for each local operator O, a running-total variable G, and a “current term” variable T. CO is used to iterate through the nonzero eigenvalues of O (arranged in some fiducial order). O has O(2l ) nonzero eigenvalues. Hence CO requires O(l) bits. Hence the set of K counters CO requires O(lK ) = O(K ) bits. The following algorithm implements Step (B): (i) If CK < its maximum possible value, proceed as follows: (a) For each O = A, . . . , K , compute the (2CO ) th nonzero eigenvalue (according to the fiducial ordering). (b) Multiply the eigenvalues to form a . . . k. Store the product in T. (c) For each O = A, . . . , K , calculate the (2CO ) th eigenvector column (according to some fiducial ordering). (d) Substitute the eigenvector columns into Eqs. (3.113) and (3.115), to compute p̃→ (.) and p̃← (.). h (e) Form (a . . . k) p̃→ (a, . . . , k | ρ, f ) + p̃← (k, . . . , a| ρ, f ). Update T to this value. (f) Add T to G. (g) Erase T. (h) Increment CK . 93 (ii) If CK equals its maximum possible value, increment the counter of the preceding variable, J , in the list; reset CK to one; and, if J has not attained its maximum possible value, return to Step (i). Proceed in this manner— incrementing counters; then resetting counters, incrementing preceding counters, and returning to Step (i)—until CA reaches its maximum possible value. Then, halt. The space needed to store G is the space needed to store Γweak . This space does not contribute to Σ(2) . How much space is needed to store T? We must calculate Γweak with constant precision. Γweak equals a sum of 2l K terms. Let ε j denote the error in term j.   P lK The sum 2j=1 ε j must be O(1). This requirement is satisfied if 2l K max j |ε j | =   o(1), which implies max j |ε j | = o 2−l K . We can specify each term, with a smallenough roundoff error, using O(lK ) = O(K ) bits. Altogether, the variables require O(K ) bits. As the set of variables does, so does the O-factored method.  Performing Method (2) requires slightly more time than performing Method (1). Yet Theorem 3 can benefit computations about quantum many-body systems. Consider measuring a weak value of a quantum many-body system. One might wish to predict the experiment’s outcome and to compare the outcome with the prediction. Alternatively, consider simulating quantum many-body systems independently of laboratory experiments, as in Sec. 3.3. One must compute weak values numerically, using large matrices. The memory required to store these matrices can limit computations. Theorem 3 can free up space. Two more aspects of retrodiction deserve exposition: related studies and the physical significance of K . . . A. Related studies: Sequential weak measurements have been proposed [12] and realized recently [33–35]. Lundeen and Bamber proposed a “direct measurement” of a density operator [12]. Let ρ denote a density operator defined on a dimension-d Hilbert space H . Let Ba := {|a` i} and Bb := {|b` i} denote orthonormal mutually unbiased bases (MUBs) for H . The interbasis inner products have constant magnitudes: |ha` |bm i| = √1 ∀`, m. Consider measuring Ba weakly, then Bb weakly, then d Ba strongly, in each of many trials. One can infer (1) a KD quasiprobability for ρ and (2) a matrix that represents ρ relative to Ba [12]. 94 KD quasiprobabilities are inferred from experimental measurements in [34, 35]. Two weak measurements are performed sequentially also in [33]. Single photons are used in [33, 34]. A beam of light is used in [35]. These experiments indicate the relevance of Theorem 2 to current experimental capabilities. Additionally, composite observables AB + BA accompany KD quasiprobabilities in e.g., [130]. Physical significance of K . . . A: Rearranging Eq. (3.111) offers insight into the result: X Γweak ( ρ, f ) = (k . . . a) p̃→ (k, . . . , a| ρ, f ) k,...,a + X (a . . . k) p̃← (a, . . . , k | ρ, f ) . (3.126) a,...,k Each sum parallels the sum in Eq. (3.6). Equation (3.126) suggests that we are retrodicting about K . . . A independently of A . . . K . But neither K . . . A nor A . . . K is Hermitian. Neither operator seems measurable. Ascribing a value to neither appears to have physical significance, prima facie. Yet non-Hermitian products BA have been measured weakly [33–35]. Weak measurements associate a value with the supposedly unphysical K . . . A, just as weak measurements enable us to infer supposedly unphysical probability amplitudes A ρ . The parallel between K . . . A and A ρ can be expanded. K . . . A and A . . . K , being non-Hermitian, appear to lack physical significance independently. Summing the operators forms an observable. Similarly, probability amplitudes A ρ and A∗ρ appear to lack physical significance independently. Multiplying the amplitudes forms a probability. But A ρ and K . . . A can be inferred individually from weak measurements. We have generalized Sec. 3.1. Specializing to k = 3, and choosing forms for A, . . . K , yields an application of à ρ to retrodiction. Corollary 1 (Retrodictive application of à ρ ). Let S, H , ρ, W (t), and V be defined as in Sec. 3.1. Suppose that S is in state ρ at time t = 0. Suppose that the P observable F = W = w3 ,α w3 w3 |w3 , α w3 ihw3 , α w3 | of S is measured at time t 00 = t. P Let (w3 , α w3 ) denote the outcome. Let A = V = v1 ,λ v1 v1 |v1 , λ v1 ihv1 , λ v1 |, B = P P W (t) = w2 ,α w2 w2 U † |w2 , α w2 ihw2 , α w2 |U, and C = V = v2 ,λ v2 v2 |v2 , λ v2 ihv2 , λ v2 | . Let the composite observable Γ = ABC = V W (t)V . The value most reasonably 95 attributable to Γ retrodictively is the weak value X Γweak ( ρ; w3 , α w3 ) = v1 w2 v2 (v1 ,λ v1 ),(v2 ,λ v2 ),(w2 ,α w2 ) × p̃↔ (v2 , λ v2 ; w2 , α w2 ; v1 , λ v1 | ρ; w3 , α w3 ) . (3.127) The weights are joint conditional quasiprobabilities that obey an analog of Bayes’ Theorem: p̃↔ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 | ρ; w3 , α w3 ) p̃↔ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 | ρ) = p(w3 , α w3 | ρ) (3.128) ≡ <(hw3 , α w3 |U |v2 , λ v2 ihv2 , λ v2 |U † |w2 , α w2 i × hw2 , α w2 |U |v1 , λ v1 ihv1 , λ v1 | ρU † |w3 , α w3 i) /hw3 , α w3 | ρ|w3 , α w3 i . (3.129) A complex generalization of the weight’s numerator is the OTOC quasiprobability: Ã(3) ρ,↔ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ) = à ρ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ) . (3.130) The OTOC quasiprobability, we have shown, assists with Bayesian-type inference, similarly to the KD distribution. The inferred-about operator is V W (t)V , rather than the W (t)V W (t)V in the OTOC. The missing W (t) plays the role of F. This structure parallels the weak-measurement scheme in the main text of [38]: V , W (t), and V are measured weakly. W (t) is, like F, then measured strongly. à ρ (.) values as coefficients in an operator decomposition Let B denote any orthonormal operator basis for H . Every state ρ ∈ D(H ) can be decomposed in terms of B, as in Sec. 3.1. The coefficients form a KD distribution. Does à ρ consist of the coefficients in a state decomposition? Summing à ρ (.) values yields a coefficient in a decomposition of an operator ρ0.17 ρ0 results from asymmetrically “decohering” ρ. This decoherence relates to timereversal asymmetry. We expect ρ0 to tend to converge to ρ after the scrambling time t ∗ . By measuring à ρ after t ∗ , one may infer how accurately one prepared the target initial state. 17 This ρ0 should not be confused with the ρ0 in Theorem 2. 96 Theorem 4. Let X ρ0 := ρ − |v2 , λ v2 ihw3 , α w3 |U (v2 ,λ v2 ),(w3 ,α w3 ) : hw3 ,α w3 |U |v2 ,λ v2 i,0 × hv2 , λ v2 | ρU † |w3 , α w3 i (3.131) denote the result of removing, from ρ, the terms that connect the “input state” U † |w3 , α w3 i to the “output state” |v2 , λ v2 i. We define the set ) ( |v2 , λ v2 ihw3 , α w3 |U (3.132) B := hw3 , α w3 |U |v2 , λ v2 i hw3 ,α w |U |v2 ,λ v i,0 3 2 of trace-one operators. ρ0 decomposes in terms of B as X (v2 ,λ v2 ),(w3 ,α w3 ) : hw3 ,α w3 |U |v2 ,λ v2 i,0 (w3 ,α w ) C(v2 ,λ v )3 2 |v2 , λ v2 ihw3 , α w3 |U . hw3 , α w3 |U |v2 , λ v2 i (3.133) The coefficients follow from summing values of the OTOC quasiprobability: X (w3 ,α w ) à ρ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ) . (3.134) C(v2 ,λ v )3 := 2 (w2 ,α w2 ), (v1 ,λ v1 )  Proof. We deform the argument in Sec. 3.1. Let the {|ai} in Sec. 3.1 be |v2 , λ v2 i . n o Let the {| f i} be U † |w3 , α w3 i . We sandwich ρ between resolutions of unity: ρ =  P
P a |aiha| ρ f | f ih f | . Rearranging yields ρ= X |v2 , λ v2 ihw3 , α w3 |U (v2 ,λ v2 ),(w3 ,α w3 ) × hv2 , λ v2 | ρU † |w3 , α w3 i . (3.135) We wish to normalize the outer product, by dividing by its trace. We assumed, in Sec. 3.1, that no interbasis inner product vanishes. But inner products could vanish here. Recall Example 1: When t = 0, W (t) and V share an eigenbasis. That eigenbasis can have orthogonal states |ψi and |φi. Hence hw3 , α w3 |U |v2 , λ v2 i can equal hψ|φi = 0. No such term in Eq. (3.135) can be normalized. We eliminate these terms from the sum with the condition hw3 , α w3 |U |v2 , λ v2 i , 0. The left-hand side of Eq. (3.135) is replaced with the ρ0 in Eq. (3.131). We divide 97 and multiply by the trace of each B element: X |v2 , λ v2 ihw3 , α w3 |U ρ0 = hw3 , α w3 |U |v2 , λ v2 i (v ,λ ),(w ,α ) : 2 v2 3 w3 hw3 ,α w3 |U |v2 ,λ v2 i,0 × hw3 , α w3 |U |v2 , λ v2 ihv2 , λ v2 | ρU † |w3 , α w3 i . (3.136) The coefficients are KD-quasiprobability values. Consider inserting, just leftward of the ρ, the resolution of unity   X 1 = U † |w2 , α w2 ihw2 , α w2 |U    w2 ,α w2  ×   X v1 ,λ v1 In the resulting ρ0 decomposition, the  |v1 , λ v1 ihv1 , λ v1 |  .  P (3.137) P v1 ,λ v1 is pulled leftward, to just after |v2 ,λ v2 ihw3 ,α w3 |U the hw3 ,α w |U |v2 ,λ v i . This double sum becomes a sum of à ρ ’s. The ρ0 weights have 3 2 the form in Eq. (3.134). w2 ,α w2  Theorem 4 would hold if ρ were replaced with any bounded operator O ∈ B(H ). Four more points merit discussion. We expect that, after the scrambling time t ∗ ,   there tend to exist parameterizations α w` and λ vm such that B forms a basis. Such a tendency could facilitate error estimates: Suppose that à ρ is measured after t ∗ . One can infer the form of the state ρ prepared at the trial’s start. The target initial state may be difficult to prepare, e.g., thermal. The preparation procedure’s accuracy can be assessed at a trivial cost. Third, the physical interpretation of ρ0 merits investigation. The asymmetric decoherence relates to time-reversal asymmetry. Fourth, the sum in Eq. (3.134) relates to a sum over trajectories, a marginalization over intermediate-measurement outcomes. n f|o Relationship between scrambling and completeness of B: The |aih h f |ai in Sec. 3.1 forms a basis for D (H ). But suppose that ρ0 , ρ. B fails to form a basis. What does this failure imply about W (t) and V ? The failure is equivalent to the existence of a vanishing ξ := |hw3 , α w3 |U |v2 , λ v2 i|. Some ξ vanishes if some degenerate eigensubspace H0 of W (t) is a degenerate eigensubspace of V : Every eigenspace of every Hermitian operator has an orthogonal basis. H0 therefore has an orthogonal basis. One basis element can be labeled U † |w3 , α w3 i; and the other, |v2 , λ v2 i. 98 The sharing of an eigensubspace is equivalent to the commutation of some component of W (t) with some component of V . The operators more likely commute before the scrambling time t ∗ than after. Scrambling is therefore expected to magnify the similarity between the OTOC quasiprobability à ρ and the conventional KD distribution. Let us illustrate with an extreme case. Suppose that all the ξ’s lie as far from zero as possible: 1 ξ=√ ∀ξ . (3.138) d Equation (3.138) implies that W (t) and V eigenbases are mutually unbiased biases (MUBs) [131]. MUBs are eigenbases of operators that maximize the lower bound in an uncertainty relation [132]. If you prepare any eigenstate of one operator (e.g., U † |w` , α w` i) and measure the other operator (e.g., V ), all the possible outcomes have equal likelihoods. You have no information with which to predict the outcome; your ignorance is maximal. W (t) and V are maximally incompatible, in the quantum-information (QI) sense of entropic uncertainty relations. Consistency between this QI sense of “mutually incompatible” and the OTOC sense might be expected: W (t) and V eigenbases might be expected to form MUBs after the scrambling time t ∗ . We elaborate on this possibility in Sec. 3.6. KD quasiprobabilities are typically evaluated on MUBs, such as position and momentum eigenbases [11, 12, 35]. One therefore might expect à ρ to relate more closely the KD quasiprobability after t ∗ than before. The OTOC motivates a generalization of KD studies beyond MUBs. Application: Evaluating a state preparation’s accuracy: Experimentalists wish to measure the OTOC F (t) at each of many times t. One may therefore wish to measure à ρ after t ∗ . Upon doing so, one may be able to infer not only F (t), but also the accuracy with which one prepared the target initial state. Suppose that, after t ∗ , some B that forms a basis for H . Consider summing latetime à ρ (.) values over (w2 , α w2 ) and (v1 , λ v1 ). The sum equals a KD quasiprobability for ρ. The quasiprobability encodes all the information in ρ [11, 12]. One can reconstruct the state that one prepared [33–35]. The prepared state ρ might differ from the desired, or target, state ρtarget . Thermal states e−H/T /Z are difficult to prepare, for example. How accurately was ρtarget prepared? One may answer by comparing ρtarget with the KD quasiprobability à ρ for ρ. 99 Reconstructing the KD quasiprobability requires a trivial sum over already-performed measurements [Eq. (3.134)]. One could reconstruct ρ independently via conventional quantum-state tomography [133]. The ρ reconstruction inferred from à ρ may have lower precision, due to the multiplicity of weak measurements and to the sum. But independent tomography would likely require extra measurements, exponentially many in the system size. Inferring à ρ requires exponentially many measurements, granted.18 But, from these measurements, one can infer à ρ , the OTOC, and ρ. Upon reconstructing the KD distribution for ρ, one can recover a matrix representation for ρ via an integral transform [12]. The asymmetrically decohered ρ0: What does the decomposed operator ρ0 signify? ρ0 has the following properties: The term subtracted off in Eq. (3.131) has trace zero. Hence ρ0 has trace one, like a density operator. But the subtracted-off term is not Hermitian. Hence ρ0 is not Hermitian, unlike a density operator. Nor is ρ0 anti-Hermitian, necessarily unitarity, or necessarily anti-unitary. ρ0 plays none of the familiar roles—of state, observable, or time-evolution operator— in quantum theory. The physical significance of ρ0 is not clear. Similar quantities appear in weak-measurement theory: First, non-Hermitian products BA of observables have been measured weakly (see Sec. 3.5 and [33–35]). Second, nonsymmetrized correlation functions characterize quantum detectors of photon absorptions and emissions [126]. Weak measurements imbue these examples with physical significance. We might therefore expect ρ0 to have physical significance. Additionally, since ρ0 is non-Hermitian, non-Hermitian quantum mechanics might offer insights [134]. The subtraction in Eq. (3.131) constitutes a removal of coherences. But the subtraction is not equivalent to a decohering channel [114], which outputs a density operator. Hence our description of the decoherence as asymmetric. The asymmetry relates to the breaking time-reversal invariance. Let U † |w3 , α w3 i =: | w̃3 i be fixed throughout the following argument (be represented, relative to any given basis, by a fixed list of numbers). Suppose that ρ = e−H/T /Z. The removal of hv2 , λ v2 | ρ| w̃3 i terms from ρ is equivalent to the removal of hv2 , λ v2 |H | w̃3 i terms from H: ρ 7→ ρ0 ⇔ H 7→ H 0. Imagine, temporarily, that H 0 could represent a 18 One could measure, instead of à ρ , the coarse-grained quasiprobability A ρ =: ˜ P degeneracies Ãρ ˜ ˜ (Sec. 3.2). From A ρ , one could infer the OTOC. Measuring A ρ would require exponentially fewer measurements. But from A˜ρ , one could not infer the KD distribution. One could infer a coarsegrained KD distribution, akin to a block-diagonal matrix representation for ρ. 100 Hamiltonian without being Hermitian. H 0 would generate a time evolution under which | w̃3 i could not evolve into |v2 , λ v2 i. But |v2 , λ v2 i could evolve into | w̃3 i. The forward process would be allowed; the reverse would be forbidden. Hence ρ 7→ ρ0 relates to a breaking of time-reversal symmetry. Interpretation of the sum in Eq. (3.134): Summing à ρ (.) values, in Eq. (3.134), yields a decomposition coefficient C of ρ0. Imagine introducing that sum into Eq. (3.130). The OTOC quasiprobability à ρ (.) would become a KD quasiprobability. Consider applying this summed Eq. (3.130) in Eq. (3.127). We would change from retrodicting about V W (t)V to retrodicting about the leftmost V . Relationship between out-of-time ordering and quasiprobabilities The OTOC has been shown to equal a moment of the complex distribution P(W,W 0 ) [38]. This equality echoes Jarzynski’s [39]. Jarzynski’s equality governs out-of-equilibrium statistical mechanics. Examples include a quantum oscillator whose potential is dragged quickly [135]. With such nonequilibrium systems, one can associate a difficult-to-measure, but useful, free-energy difference ∆F. Jarzynski cast ∆F in D E terms of the characteristic function e− βW of a probability distribution P(W ).19 Similarly, the difficult-to-measure, but useful, OTOC F (t) has been cast in terms of D E 0 0 the characteristic function e−( βW + β W ) of the summed quasiprobability P(W,W 0 ) [38]. Jarzynski’s classical probability must be replaced with a quasiprobability because [W (t),V ] = 0. This replacement appeals to intuition: Noncommutation and quasiprobabilities reflect nonclassicality as commuting operators and probabilities do not. The OTOC registers quantum-information scrambling unregistered by time-ordered correlators (TOCs). One might expect TOCs to equal moments of coarse-grained quasiprobabilities closer to probabilities than à ρ is. We prove this expectation. First, we review the TOC FTOC (t). Then, we introduce the TOC analog ATOC of the probability amplitude A ρ [Eq. (3.32)]. A ρ encodes no ρ time reversals, as expected. Multiplying a forward amplitude ATOC by a backward ρ  ∗ TOC requires amplitude ATOC yields the TOC quasiprobability ÃTOC ρ ρ . Inferring à ρ only two weak measurements per trial. ÃTOC reduces to a probability if ρ = ρV ρ [Eq. (3.35)]. In contrast, under no known condition on ρ do all à ρ (.) values reduce to probability values. Summing ÃTOC under constraints yields a complex distribuρ 19 Let P(W ) denote a probability distribution over a random variable W . The characteristic R function G(s) equals the Fourier transform: G(s) := dW ei sW . Defining s as an imaginary-time D E D E variable, is ≡ − β, yields e − βW . Jarzynski’s equality reads, e − βW = e − β∆F . 101 tion PTOC (W,W 0 ). The TOC FTOC (t) equals a moment of PTOC (W,W 0 ). Time-ordered correlator FTOC (t) The OTOC equals a term in the expectation value h.i of the squared magnitude |.| 2 of a commutator [. , .] [7, 31], D E C(t) := [W (t),V ]† [W (t),V ] (3.139) D E D E = − W † (t)V †V W (t) − V † W † (t)W (t)V + 2<(F (t)) . (3.140) The second term is a time-ordered correlator (TOC), D E FTOC (t) := V † W † (t)W (t)V . (3.141) D E The first term, W † (t)V †V W (t) , exhibits similar physics. Each term evaluates to one if W and V are unitary. If W and V are nonunitary Hermitian operators, the TOC reaches its equilibrium value by the dissipation time t d < t ∗ (Sec. 3.1). The TOC fails to reflect scrambling, which generates the OTOC’s Lyapunov-type behavior at t ∈ (t d , t ∗ ). TOC probability amplitude ATOC ρ We define ATOC ρ ( j; v1 , λ v1 ; w1 , α w1 ) √ := hw1 , α w1 |U |v1 λ v1 ihv1 λ v1 | ji p j (3.142) as the TOC probability amplitude. ATOC governs a quantum process P ATOC . Figρ ure 3.19a, analogous to Fig. 3.2a, depicts P ATOC , analogous to the P A in Sec. 3.1: (1) Prepare ρ. (2) Measure the ρ eigenbasis, {| jih j |}. (3) Measure Ṽ . (4) Evolve the system forward in time under U. (5) Measure W̃. 102 Equation (3.142) represents the probability amplitude associated with the measurements’ yielding the outcomes j, (v1 , λ v1 ), and (w1 , α w1 ), in that order. All the meaTOC surements are strong. P ATOC is not a protocol for measuring ATOC ρ . Rather, P A facilitates the physical interpretation of ATOC ρ . Measure W̃. (w1 , ↵w1 ) Measure W̃. -t -t U U Experiment time 0 Measure Ṽ . Prepare ⇢. (w1 , ↵w1 ) Measure {|jihj|}. (v1 , v1 ) Measure Ṽ . Prepare ⇢. j (a) Experiment time 0 Measure {|jihj|}. (v2 , v2 ) j (b) Figure 3.19: Quantum processes described by the probability amplitudes ATOC in the ρ time-ordered correlator (TOC) FTOC (t): FTOC (t), like F (t), equals a moment of a summed quasiprobability (Theorem 5). The quasiprobability, ÃTOC ρ , equals a sum of multiplied TOC probability amplitudes Aρ [Eq. (3.144)]. Each product contains two factors: ATOC ρ ( j; v1 , λ v 1 ; w1 , α w1 ) denotes the probability amplitude associated with the “forward” process in Fig. 3.19a. The system, S, is prepared in a state ρ. The ρ eigenbasis {| jih j |} is measured, yielding outcome j. Ṽ is measured, yielding outcome (v1 , λ v1 ). S is evolved forward in time under the unitary U. W̃ is measured, yielding outcome (w1 , α w1 ). Along the abscissa runs the time measured by a laboratory clock. Along the ordinate runs the t in ∗ U := e −i H t . The second factor in each ÃTOC product is ATOC ρ ρ ( j; v2 , λ v 2 ; w1 , α w1 ) . This factor relates to the process in Fig. 3.19b. The operations are those in Fig. 3.19a. The processes’ initial measurements yield the same outcome. So do the final measurements. The middle outcomes might differ. Complex-conjugating ATOC yields the probability ρ amplitude associated with the reverse process. Figures 3.19a and 3.19b depict no time reversals. Each analogous OTOC figure (Fig. 3.2a and Fig. 3.2b) depicts two. P ATOC results from eliminating, from P A, the initial U, W̃ measurement, and U † . A ρ encodes two time reversals. ATOC encodes none, as one might expect. ρ TOC quasiprobability ÃTOC ρ Consider a P ATOC implementation that yields the outcomes j, (v2 , λ v2 ), and (w1 , α w1 ). Such an implementation appears in Fig. 3.19b. The first and last outcomes [ j and (w1 , α w1 )] equal those in Fig. 3.19a, as in the OTOC case. The middle outcome can differ. This process corresponds to the probability amplitude ATOC ρ ( j; v2 , λ v2 ; w1 , α w1 ) √ = hw1 , α w1 |U |v2 , λ v2 ihv2 , λ v2 | ji p j . (3.143) 103 Complex conjugation reverses the inner products, yielding the reverse process’s amplitude. We multiply this reverse amplitude by the forward amplitude (3.142). Summing over j yields the TOC quasiprobability: ÃTOC ρ (v1 , λ v1 ; w1 , α w1 ; v2 , λ v2 ) X ∗ TOC := ATOC ρ ( j; v2 , λ v2 ; w1 , α w1 ) A ρ ( j; v1 , λ v1 ; w1 , α w1 ) (3.144) j = hv2 , λ v2 |U † |w1 , α w1 ihw1 , α w1 |U |v1 , λ v1 i × hv1 , λ v1 | ρ|v2 , λ v2 i . (3.145) Like à ρ , ÃTOC is an extended Kirkwood-Dirac quasiprobability. ÃTOC is 2-extended, ρ ρ whereas à ρ is 3-extended. ÃTOC can be inferred from a weak-measurement protoρ TOC col P : (1) Prepare ρ. (2) Measure Ṽ weakly. (3) Evolve the system forward under U. (4) Measure W̃ weakly. (5) Evolve the system backward under U † . (6) Measure Ṽ strongly. P TOC requires just two weak measurements. The weak-measurement protocol P for inferring à ρ requires three. P TOC requires one time reversal; P requires two. In a simple case, every ATOC ρ (.) value reduces to a probability value. Suppose that ρ shares the Ṽ eigenbasis, as in Eq. (3.35). The (v2 , λ v2 ) in Eq. (3.145) comes to equal (v1 , λ v1 ); Figures 3.19a and 3.19b become identical. Equation (3.145) reduces to ATOC ρV (v1 , λ v1 ; w1 , α w1 ; v2 , λ v2 ) (3.146) = |hw1 , α w1 |U |v1 , λ v1 i| 2 pv1 ,λ v1 δv1 v2 δ λ v1 λ v2 (3.147) = p(w1 , α w1 |v1 , λ v1 ) pv1 ,λ v1 δv1 v2 δ λ v1 λ v2 (3.148) = p(v1 , λ v1 ; w1 , α w1 ) δv1 v2 δ λ v1 λ v2 . (3.149) 104 The p(a|b) denotes the conditional probability that, if b has occurred, a will occur. p(a; b) denotes the joint probability that a and b will occur. All values ÃTOC ρV (.) of the TOC quasiprobability have reduced to probability values. Not all values of à ρV reduce: The values associated with (v2 , λ v2 ) = (v1 , λ v1 ) or (w3 , α w3 ) = (w2 , α w2 ) reduce to products of probabilities. [See the analysis around Eq. (3.36).] The OTOC quasiprobability encodes nonclassicality—violations of the axioms of probability—more resilient than the TOC quasiprobability’s. 0 Complex TOC distribution PTOC (WTOC ,WTOC ) 0 Let WTOC and WTOC denote random variables analogous to thermodynamic work. 0 We fix the constraints WTOC = w1 v2 and WTOC = w1 v1 . (w1 and v2 need not be complex-conjugated because they are real, as W and V are Hermitian.) Multiple outcome sextuples (v2 , λ v2 ; w1 , α w1 ; v1 , λ v1 ) satisfy these constraints. Each sextuple corresponds to a quasiprobability ÃTOC ρ (.). We sum the quasiprobabilities that satisfy the constraints: X 0 PTOC (WTOC ,WTOC ) := (v1 ,λ v1 ),(w1 ,α w1 ),(v2 ,λ v2 ) × ÃTOC ρ (v1 , λ v1 ; w1 , α w1 ; v2 , λ v2 ) δW (w1∗ v2∗ ) δW 0 (w1 v1 ) . (3.150) 0 PTOC forms a complex distribution. Let f denote any function of WTOC and WTOC . The PTOC average of f is D E 0 f (WTOC ,WTOC ) X 0 0 := f (WTOC ,WTOC )PTOC (WTOC ,WTOC ). (3.151) 0 WTOC ,WTOC TOC as a moment of the complex distribution The TOC obeys an equality analogous to Eq. (11) in [38]. Theorem 5 (Jarzynski-like theorem for the TOC). The time-ordered correlator (3.141) equals a moment of the complex distribution (3.150): FTOC (t) = wherein β, β0 ∈ R. ∂ 2 D −( βWTOC + β 0W 0 ) E TOC e , ∂ β ∂ β0 β,β 0 =0 (3.152) 105  Proof. The proof is analogous to the proof of Theorem 1 in [38]. D E 0 Equation (3.152) can be recast as FTOC (t) = WTOCWTOC , along the lines of Eq. (3.45). Higher-order OTOCs as moments of longer (summed) quasiprobabilities Differentiating a characteristic function again and again yields higher- and higher35 0 Proof. The proof is analogous to the proof of Theorem 1 Let usagain. evaluate Eq.But (91) on particular arguments: 35 point correlation functions. So does differentiating P(W,W ) again and in [31]. 1us, evaluate . . ; v on, particular ;w ,↵ ) to the proof of Theorem 1 ¯Ã= Let(v arguments: each resulting correlator encodes Proof. just The Kproof = is3analogous time reversals. Let K (K; w +, ↵Eq. 1); .(91) = in [31]. = hw 2 , ↵ |U |v , ihv , |U |w , ↵ i E. Higher-order OTOCs as moments of longer à (v , ;w ,↵ ;...;v , ;w ,↵ ) (summed) ⇥ . . . ⇥ hw , ↵ |U |v , ihv , |⇢U |w ,↵ i. 2, 3, . . ., for K = 3, 5, . . . A K¯ -fold OTOC hasquasiprobabilities been defined [123, 124]: .ihv .. , = hw ,↵ |U |v , |U |w , ↵ i 35 35 0 4. Complex TOC distribution PTOC (WTOC , WTOC ) 0 4. Complex TOC distribution PTOC (WTOC , WTOC ) illustrated in Fig. 19. We focus on Hermitian W and V , 34 35, illustrated in for Fig.simplicity. 19. We focus on Hermitian W and V as in [6, 100], (K ) 34 0 vK¯ wK¯+1 35 K¯ K¯+1 2 simplicity. Let WTOC and WTOC denote random variables⇢anal- 1 asv1in [6,2100],wfor 0 0 34 4. to TOC distribution P (Wthe , WTOCanal) illustrated in Fig. 19. We focus on Hermitian† W and V , ogous thermodynamic work. fix constraints Let WComplex and WTOC denote We random variables TOC TOC TOC W(t) W(t) W(t) 0 34 ⇤ ⇤ 0 ¯19. We ¯ Vw, K¯ vK¯ focusK¯ vK¯ TOC W distribution P1TOC , WTOCout) K¯+1as illustrated in for Fig. on Hermitian WK and inw[6, 100], simplicity. 34 W =Complex w1 v2 and = w v1fix . (Wthe Multiple K TOCconstraints ogous thermodynamic work. We K¯+1 TOC4. to TOC (K t 2 , ↵w ) W(t) basis, as in Eq.)(24). The (w in Eq. (134) comes is analogous to the proof of Theorem 1 W(t)to ¯Proof. 34 The proof W(t) ⇤ ⇤ (v , 0 0 ,↵ ;v , ¯ as in [6, 100], for simplicity. come ; w ) satisfy these 34 1 v 2 w v w WLet w v and W = w v . Multiple out¯ ¯ 2 v 1 w 1 v K K +1 ⇢ 1 2 W= and W denote random variables anal2 1 1 equal (w , ↵ ); Figures 18a and 18b become identical. in [30]. TOCsextuples 1 1 † K K +1 TOC 1 2 TOC TOC t basis, as1 (134) in wEq.reduces (24). The in Eq. (134)1comesvto Proof. to the proof Theorem 1 0 1 The v1proof is analogous wK¯of+1 Equation to (w22, ↵w )w constraints. Each corresponds avariables quasiprobaK¯+1 2 † 1 come sextuples (vW v1 , fix satisfy 34 34 ogous thermodynamic work. constraints Let to WTOC andsextuple denote random anal2 , TOC v2 ; w 1 , ↵w1 ;We vto 1 )the equal (w1these ,as ↵w ); Figures 18a(wand 18b become identical. in [30].The proof is analogous U U † basis, in Eq. (24). The , ↵ ) in Eq. (134) comes to Proof. to the proof of Theorem 1 2 w TOC ⇤ ⇤ W(t) W(t) TOC 0 corresponds ¯(135)vK¯ inE.[30].K ¯ W(t) ¯ bility à as (.). Wesextuple sumWthe quasiprobabilities that Aconstraints (wsat, ↵1 ,w↵K ;reduces w¯2+1 , Figures ↵w to ; v1 ,18a )and wv K vK¯ OTOCs asKmoments w Equation (134) 1 ¯+1 ¯ longer 34 constraints. Each a quasiproba⇢ W = thermodynamic wmoments = w . tothe Multiple outequal (w 18b become identical. Higher-order of ogous work. We † K K (145) TOC ⇢to 1 v1fix w ); E. Higher-order OTOCs of longer 1 v2 and TOC U t 2 ,(w basis, as in Eq. The 2The (w ↵w ) wW(t) in)Eq. (134) comes toto Proof. is analogous analogous theproof proofofof Theorem basis, as (134) in†(24). Eq.reduces (24). in U Eq. (134) comes Proof.The The proof proof totothe Theorem 1 1 W(t) 2, ↵ W(t) TOC ⇤ ⇤ (summed) quasiprobabilities Equation 0 , quasiprobabilities isfy theà constraints: = |hv |U ,);2↵,wFigures (136) bility the that come sextuples (vand , vW hese 34 1 , (w v sat1w ,↵v )and w w ↵become ↵ identical. ATOC , ↵1); ;Figures ↵wi| 18a ; pvtow (135) WTOC w1function v2We vv11. ) satisfy Multiple out2sum 1 ↵w= 1 ,1 1(w w|w 1 , and ⇢ 2; w 1 ; vw ⇢= (.). equal (w become inin[30]. † OTOCs equal ↵w 18b identical. [30].Higher-order E. as moments of longer Experiment Di↵erentiating a characteristic again and 1, ↵ TOC (summed) quasiprobabilities TOC t ; v18a basis, in (w Eq. (w in Eq. (134) Proof. The is analogous to he of Theorem 1 2 ,,↵ ⇥ .1,1(134) .(24). hw , wwv↵18b |U |v1comes ,(136) |⇢U |wquasiprobabilities , ↵proof A1as ,†,reduces ↵↵w.w ⇥ ,to ↵ ))w (135) ¯ X ) a satisfy 2w w2 to vto1 ihv 1 , Higher-order v1proof Equation ==p(v )wpThe (137) (summed) ⇢, (134) time Equation reduces ¯+1 i .34 constraints. Each0 sextuple quasiprobaE. OTOCs as+1 momentsw ofK longer isfy thesextuples constraints: v |w ↵ 2↵ † K come (v2 , v2 ; wcorresponds |hv |wFigures i|,↵ pw21wand 1 , ↵w1 ; v1 , vto 1 , 1these v↵|U 1 , ↵w 0 ,↵ w wbecome ↵ ↵ identical. Di↵erentiating ais analogous characteristic function again and 1 equal (w , ); 18a 18b in [30]. U U w (K ) Experiment basis, as in Eq. (24). The (w , ↵ ) in Eq. (134) comes to Proof. The proof to the proof of Theorem 1 † (summed) quasiprobabilities w again yields higher- and bility higher-point correlation funcTOC PTOCÃ(W , WWe ) :=the quasiprobabilities TOC TOC TOC = |hvsat|w ,ww↵ i|2;↵pv⇢vw12,↵),↵à (136) TOC = p(v ,(w w ,|U ↵↵ . w w ↵ ↵from (138) 1,;,↵ 1w (.). sum that again yields higher- and higher-point correlation func1A A=One w ,);↵ ;w (135) (w ,;,reduces ↵v,↵ (135) the Equation (134) 2 X to a quasiprobaconstraints. Each sextuple corresponds can interferometry scheme w w1 ,18a v ) 18b ⇢ ⇢ p(v ↵w); ) infer p2w,wto (137) time equal (w ,v1↵11w become identical. in [30]. E. Higher-order OTOCs as moments of longer † E. Higher-order OTOCs as moments of longer ⇢ (145) 1⇢, 1 v 1|w w Figures wand w ↵ ↵ V (134) comesVto tions. 0 U Di↵erentiating aisVanalogous characteristic again and 0 0 So does di↵erentiating P (W, Wfunction ) again and again. basis, in Eq. (24). in U Eq. Proof. The (summed) proof to the proof of Theorem 1 , v1 ),(w = p(v , (134) ↵w conditional )0 pThe (137) 2 ,w↵probability w 1 1 ,↵w1 ),(v2 ,The v2 )p(a|b) TOC quasiprobabilities 1as v† |w 1the wto ,↵2(w w )↵ ↵ Equation †,reduces (summed) quasiprobabilities TOC isfy the constraints: PTOC ,W ) :=(v tions. So does di↵erentiating PÃ(W, Wfunction )TOC again and again. denotes that, if = |hv ,w i| pvi| (136) Di↵erentiating acorrelators characteristic function againtime andfunc|hv |w ↵2w;w p⇢vw ↵ (136) . each .E.[30]. bility (.). We sum the quasiprobabilities that satp(v ,(w )Figures (138) 1= 1w w ,↵ A ,;↵w,wv1|w ;↵ ,w ↵ )and (135) 1|U 1 ,w ,↵ w . w w w↵become ↵↵ ↵ identical. again yields higherand Experiment higher-point 1,[31] w ↵ 1 ,18a ⇢ ⇢ TOC of these encodes justcorrelation three reequal (w ↵,|U );2↵ 18b in Higher-order as moments of longer Di↵erentiating a characteristic again and b has in from weak measurements. Upon implementw ill V the V But V OTOCs occurred, occur. joint TOC = p(v1 ,(w1va,;and w . (138) 0 th di↵erentiating again yields higherand higher-point funcTOC 1 , ;↵w w ,)↵ w ;wvp(a; ↵ denotes A= ↵↵ ,1↵st b) (135) tions. does P Wcorrelation ) again and again. X W(t)V 11 w W(t)V pair ,The K -fold OTOC has been 89]: )↵ p2wwconditional (137) (summed) quasiprobabilities time Equation (134) ⇤ 2W p(v ,a |ww ),↵ p2woccur. (137) 1 ,↵ v0= E. So OTOCs as(W, moments of[88, longer ⇥ Ãthe (v1 , v1 ; just w , ↵wK ; v(v21,,=vv12),(w .vdenotes isfy constraints: 1⇢,) 1|w v†, and w↵ ↵↵ pair ↵ 0defined 2 (ww1⇤1v),(v (w p(a|b) if versals. =)p(v |hv |U |w ,,reduces ptow,↵1w,↵ wv w probability (136) But each resulting correlator 3) Wfunctime ⇢encodes 1 b↵w will 1 11 , that 11the w w ↵ Di↵erentiating acorrelators characteristic again Di↵erentiating characteristic again andand 0 1 . each .AHigher-order tions. So does di↵erentiating P (W, Wfunction )function again and again. 2 ) probability (K 0i| conditional (K ) ↵ that, Experiment But of these encodes just) three time re† 2 ⇢ (summed) quasiprobabilities The p(a|b) denotes the probability that, if again yields higherand higher-point correlation PTOCTOC (WTOC , WTOC ) := TOC TOC . . . = |hv , ,(w |w ,occur. ↵ i| pvp(a; (136) b All has occurred, ill b) denotes joint =ing p(v , w ↵ ) . (138) values à (.) of the TOC quasiprobability have 11;⇢ v1v1,|U 1w w w ,↵ w . w ↵ the ↵ p(v ; w , ↵ ) (138) the interferometry scheme, one can infer à for st again yields higherand higher-point correlation functh 1 w w w ↵ ↵ (K ) But each of these correlators encodes just three time reagain yields higherand higher-point correlation func1 w w w ↵ A , ↵ ; , ↵ ; , (135) 1 ⇢ X (141) One can infer à from the interferometry scheme w 2 w 1 v , .v |w1 , ↵aw will ) pwoccur. ¯ = (K +1) 1p(a; W(t)V pairhas (t) hW(t)V .OTOC . W(t)V i time ⌘ . longer . again . W(t)V ). E. := OTOCs asTr(⇢ moments of[88, ⇢↵ denotes ,↵ w wW(t)V ↵Vpair the (137) AHigher-order K -fold been defined 89]: and b(wp(v has1⇢occurred, b) joint V Fversals. 0= ⇥ à ; v ,, v v3, ) 1W↵.(w 0W(t)V th time 1 , ↵w reversals. Let Kdi↵erentiating = 2,(vTOC 3,1 ,W . ,.vW .⇤.v2⇤ )2reduced W )that Di↵erentiating a.V characteristic 0.,; w 0 for 1 1 (v12= 2 5, probability and b will occur. 1= 1p(v |-1) {z }P |)again {z } tions. So di↵erentiating P(W, (W,defined again and again. versals. A K -fold OTOC has been [88,and 89]: 1st wtime (2does tions. So does di↵erentiating WW0 )function again. ),(w ,The |w ↵values. )b0 pwconditional w11),(v v2 ) to p(a|b) probability all 1 , that vdenotes w values ↵ ↵ of à †, and (summed) quasiprobabilities 2 1 ⇢ (137) PTOC )K := tions. So does P⇢ (W, )TOC again and again. probability a1the occur. p(a|b) denotes conditional probability that, if if The ⇢wOne Di↵erentiating acorrelators function again and |hv ,w ,the ↵wwill i|,↵2Not (136) . ).each . yields 2kcharacteristic 2k time = p(v= ,K ,|U ↵ )w1the (138) ,↵ . probability w w has ↵ ↵ that, again higherand higher-point correlation 1[31] 1and w w TOC ↵p ↵ all -values: measured all the inner prodAll values Ã1 ;TOC (.)w|w of quasiprobability have reversal But of these encodes just three re-) . (K each But ofreversal these correlators encodes just hree re-funcin weak measurements. Upon implementreduce: The associated with (wV3 , ↵ )joint = Fagain := hW(t)V ..and .W(t)V i Tr(⇢ ⌘ Tr(⇢ .time . .funcW(t)V All values the quasiprobability have V. W(t)V b has aTOC occur. p(a; b) denotes the wthe (K ) (t) boccurred, has occurred, occur. b). denotes joint V F =(141) p(v , ⇢ và ;values will , (.) ↵will )from (138) 0W(t)V th di↵erentiating (142) higherhigher-point ⇢ 1a w of w w TOC ↵ forms a complex distribution. Let ⇤f can denote any th time (t)yields := hW(t)V .OTOC . {z i }⌘been W(t)V . .[88, . W(t)V ). } tions. So does P (W,defined Wcorrelation ) again and again. 1↵1ststp(a; W(t)V pair A K¯each -fold resulting OTOC has beenP⇥TOC defined W(t)V pair ,The )p(a|b) versals. AA K -fold OTOC has been defined 89]: | | {z versals. K -fold has [88, 89]: time (2 -1) = p(v , |w , ↵ ) p (137) 1 ,↵ 2reduced v0↵ ÃTOC (v1 ,[112, w1113]: , ↵w0K ; v(v21,,=vv12),(w ) W . 1 v 1 w w ,↵ w w ↵ ↵ 0 2 v1 ; just to probability values. Not all values of à (ww1⇤1v),(v ) W (w v ) denotes the conditional probability that, if (w , ) or (v , ) = (v , ) reduce to products of But correlator 3 time | {z } | {z } ⇢encodes 1 probability b will occur. ⇢ 1 1 that 2 reduced 2 probability v and 1values. v occur. probability that a and b will Di↵erentiating acorrelators again and re. each . So . does tions. di↵erentiating P (W, Wfunction ) again and 2 (K to Not allprobability values of à 2kcharacteristic 2kagain. ucts ha|U|bi. Multiplying arbitrarily many in⇢ together But of these encodes just)2k three time The p(a|b) denotes the conditional that, if Each reversal reversal function of W and W . The P average of f is TOC 2k TOC TOC TOC . . . probabilities. [See the analysis around Eq. (25).] The such correlation function contains K HeisenbergTOC b has occurred, a ill occur. p(a; b) denotes the joint can reduce: The values associated with (w , ↵ ) = All values à (.) of the TOC quasiprobability have = p(v , ; w , ↵ ) . (138) ing the interferometry scheme, one can infer à for st All values à (.) of the TOC quasiprobability have th these correlators encodes just three reagain higherhigher-point correlation func33 , ↵ww ) = v ⇢ FIG. w w w 1K ↵ W(t)V (K)each ) yields 19: out-of-time-ordered correlator ⇢0W(t)V (142) (141) P⇥TOC forms distribution. Let denote any can reduce: associated (w pair W(t)V pair FBut (t) W(t)V Tr(⇢ W(t)V .(142) . W(t)V versals. AofhW(t)V K -fold has been defined 89]: F(K (t):= := hW(t)V .OTOC . ..and W(t)V ii⌘⌘ Tr(⇢ .time-0 .[88, .time W(t)V . ). has occurred, a1values will occur. p(a;↵-fold b)with denotes the joint picture 0b à ; w 1 , ↵w ; v2= , v3, ) W .(w .1⇢2 , The st time more th W(t) reversals. Let K¯ = 12 (K +1) =TOC 3,10,a. .vcomplex K .1⇤.v2⇤f)OTOC W ))that quasiprobability has nonclassicality resilient interleaved with K[88, op⇢ 2,(v 1., for 2 5, probability a2v,and b= will occur. 1) vor ||-1) {z again. (w ,(w ↵(w (v ) v= (v ,yields )v1reduce to of versals. A -fold OTOC has}}P been defined 89]: (2 K time {zencodes |)|again {z } } tions.operators So does di↵erentiating (W, W and 2ner w orTOC (vdenotes )values. (v ,v Not )The reduce toproducts products 01 an arbitrarily quasiprobabilreduced tow1products probability all all values of Ã⇢à reduced to probability Not values of 2, ↵ 1 ¯) (OTOC): conventional [Eq. (22)], just probability that and b1values. will occur. ⇢ of ifOTOChigh-K The the conditional probability that, (K function of, W WTOC of fp(a|b) is 2k function 2k 2k 2k .each the TOC quasiprobability’s. erators V. . correlation Fcorrelation (t) correlators encodes 2K 1 just time reversals, as TOC)iand WTOC . The PTOC average all K One has measured all the inner prodAll values à (.)FIG. of the TOC quasiprobability have reversal hf (WiTOC (142) probabilities. [See the analysis around Eq. Each K Heisenberg(K.such )such ofreversal these encodes three reTOC ⇢ -values: probabilities. analysis around Eq. (25).] The Each contains KW(t)V Heisenberg(141) 19: K -fold out-of-time-ordered correlator can reduce: The associated with (w(25).] ,↵ )The = (K ) .function FA(K :=OTOC hW(t)Vhas . . . W(t)V ⌘ Tr(⇢ W(t)V . . . W(t)VLet) f.than FBut . . W(t)V icontains ⌘ Tr(⇢ .time .(142) . W(t)V ). values Ã[See (.) of the TOC quasiprobability have can reduce: The values associated (w )K =-fold 3 (K ) (t) := hW(t)V bAll has occurred, athe will occur. p(a; b)with denotes the joint 3 ,w↵ w (142) (K )OTOC ⇢values PTOC forms a complex distribution. denote any st time moreThe th W(t) three time F-1) encodes Fversals. := .(t) . W(t)V i ⌘onTr(⇢ W(t)V . .W .time-0 W(t)V ., } illustrated in Fig. 19. We focus Hermitian and K¯(t) -fold defined [112, 113]: {z OTOC quasiprobability has nonclassicality resilient A K -fold OTOC has defined [88, 89]: 1reversals. (2hW(t)V time picture operators interleaved with OTOC quasiprobability has nonclassicality more resilient picture W(t) interleaved with time-0 op-}V)opreduced to allallvalues of (w ↵0reduced orw (v (v= to products ||(K {z{z } }been | K| K {zin(w )to or (v2), (OTOC): )values. )The reduce to products of ⇢OTOC Having inferred some à ,(t)Voperators one need perform 2 ,ity. wprobability 2 , that 1values. v1 ,)Not | {z been } X |and } average a= b, (v will occur. 31not 2), ↵ v and vreduce 0 0 {z conventional [Eq. (22)], just Not values of à Ã⇢⇢ of 0 2kencodes 2k ucts ha|U|bi. Multiplying together arbitrarily many (K ) ) (t) aserators in that [6, 97], for implicity. reversal reversal function of W W . The P of f is than the TOC quasiprobability’s. erators V . F encodes 2K 1 time reversals, as than the TOC quasiprobability’s. 2K 1 time reversals. The time passes in a laboratory . F (t) encodes 2K 1 time reversals, as 2k 2k := f (W , W )P (W , W ) . TOC hf (W , W )i (142) TOC TOC 03 [See the analysis Eq.Eq. (25).] TOC TOC TOC Each uch functioncontains containsKKHeisenbergHeisenbergTOC probabilities. the analysis around (25).] Each correlation can reduce: The associated with (w , ,↵↵wwThe ))The = TOC¯ TOC Let All values Ã[See (.) of the TOC have )such correlation (K ).function FIG. 19: K -fold out-of-time-ordered 4. TOC Complex TOC distribution Paround (Wquasiprobability ,W ⇢values PTOC formsTOC a complex distribution. fprobabilities. denote any can reduce: The values associated with (w = TOC TOC TOC 3The (K +2) (K )W(t)V F (KOTOC (t) := hW(t)V . W(t)V i⌘ Tr(⇢ .time-0 . . W(t)V ). (142) reversals. K -fold Fcorrelator (t) encodes illustrated in 19. focus on Hermitian and Vop,(142) 2K¯ 2K illustrated inFig. Fig. 19.We focus Hermitian OTOC quasiprobability has(valong nonclassicality more silient picture operators W(t) interleaved with KW opabscissa. The represents the time 0 OTOC quasiprobability has nonclassicality resilient picture operators W(t) interleaved with time-0 (wnew ,(w ↵reduced )three = ,time ) the reduce to products of ordinate |(22)], {zWe } on | K {zW and }V , 2ner WTOCX ,WTOC , )↵wor ) (v or2 ,(v2v,runs (v reduce tomore products 0 0 products yields an arbitrarily high-K quasiprobabil31 31 à experiments to infer à . To infer from to Not all values of à v ) = 1values. 1 ,v v )The (OTOC): conventional OTOC [Eq. encodes just ⇢ of 0w2of 0 ⇢ ⇢ (K ) implicity. aserators in [6, 97], for simplicity. (K )(t) in in [6, 97], for (K¯) function of W and W . The P average f is 2k 2k than the TOC quasiprobability’s. erators V . F encodes 2K 1 time reversals, as TOC TOC than the TOC quasiprobability’s. 2K 1 time reversals. The time that passes a laboratory V .experiments. F encodes 2K 1W(t) time reversals, as W(t) W(t) :=(WiTOC f (W WTOC (WTOC ,W )and[See hf WTOC )i (t)V the analysis around Eq. The Each such uch correlation function contains Heisenberg0 parameter 0 0 analt, which may inverted in The TOC , TOC TOC the analysis Eq. (25).] The Each correlation contains K K Heisenberg(144) FIG. 19: K -fold out-of-time-ordered correlator TOC 4.(142) Complex TOC distribution Paround (W ,(25).] W ) function F (t) := hW(t)V . . . W(t)V ⌘, Tr(⇢ . . . )P W(t)V ) . probabilities. Let W W[See denote random variables can reduce: The values associated with (w TOC TOC 4.probabilities. Complex TOC distribution PTOC (W ,W ))be TOC TOC 3, ↵ w )K=-fold TOC (K TOC TOC (142) three time reversals. The F (K (t) encodes illustrated in 19. We focus on Hermitian and , illustrated in Fig. Fig. We focus on Hermitian WW and V , Vopt )OTOC OTOC has nonclassicality more resilient picture operators W(t) interleaved with time-0 OTOC quasiprobability has nonclassicality more resilient picture operators W(t) interleaved with K K time-0 opweak one first prepares ⇢. One performs runs along abscissa. The represents the time 0 to thermodynamic fix he constraints (wquasiprobability )measurements, or (vorange, ) = (vleftmost , v the )The reduce products dot represents preparation ⇢. ity. inferred some Ãoftime ,state one need perform | {z } X |)i , W 0 {z )P (W} ogous 31not 2 , ↵wHaving 2 , (OTOC): v work. 1We W ,WTOC 31 conventional OTOC [Eq. (22)], just ⇢ordinate 0the 0 (K ) (K ) implicity. asthe [6, for implicity. asininthat [6,W(t) for than TOC ⇤ .TOC ⇤ quasiprobability’s. erators V97], F (t) encodes 1 time reversals, W(t) than the quasiprobability’s. 2K 1 time reversals. The in aencodes laboratory erators V97], . .passes Fcorrelation (t) encodes 2K2K 1 W(t) time reversals, as as W(t) := f (W ,W 00 the analysis hf (WTOC (142) 0 (25).] TOC TOC W(t) W(t) WTOC =Complex vand W = w vdot .(W Multiple 0 outTOC , WTOC probabilities. [See Eq. Each such function contains Heisenberg0W TOC¯ TOC Let W)2TOC and denote variables anal1random † Each parameter t,P1around which inverted in (K 1 TOC TOC 4. TOC distribution P , Wmay Each represents aThe W(t) ainterspersed Vinexperiments. purple TOC 4.w Complex TOC distribution (W ,TOC WTOC )be TOC TOC Let W W¯ denote random variables analTOC TOC UOTOC U. Fig. TOC (K +2) (K ) KKuni.focus .The .online tt or TOC three time reversals. The K -fold F19. (t) encodes illustrated focus Hermitian W and Vop, V, 2K¯ 2K for the TOC illustrated in Fig. 19.We We on Hermitian W and = 2(wand K 1green weak with 5. equality X abscissa. The ordinate represents the time comeK sextuples ↵ ; v1We ,the )measurements satisfy these 0Jarzynski-like OTOC quasiprobability has more resilient picture operators W(t) interleaved with time-0 ogous to thermodynamic work. fix the constraints 1 , ↵runs w ; w2 ,along w nonclassicality vto ¯ WTOC ,WTOC 31 new experiments infer à . To infer from 31 à ogous to thermodynamic work. We fix he constraints orange, leftmost dot represents the state preparation ⇢. 0 0 ⇢ ⇢ represents a unitary time volution. The diagram, scanned as in [6, 97], for simplicity. (K ) Each such correlation function contains K Heisenberg⇤ ⇤ 0 in [6, 97], for implicity. constraints. Each sextuple corresponds to a quasiprobathan the TOC quasiprobability’s. 2K 1= time Theinverted time that passes in a. laboratory erators V .experiments. F (t) encodes 2K W(t) reversals, as WTOC =)⇤ .w w1 vwhich Multiple outW(t) W(t) := f (WTOC , WTOC )PTOC (WTOC ,taries. W 0 1 .reversals. W(t) W(t) 0and 1 TOC 00W TOC= parameter t, may (144) TOC (K ) U in 4. w distribution (W ,W )be WLet = v1⇤2 vTOC and Wistribution wTOC vTOC Multiple outU† † W denote random variables analExperiment TOC . .The Let WComplex and W denote random variables 4.W omplex P ,W )analTOC 1P 1 .(W TOC (One measures Ṽ0TOC weakly, evolves with U1W(t) ,timemeaTOC TOC 2and 2 1 2 1 F (K¯ ) 1 W(t) (t) := hW (t)V . . . W (t)V i ≡ Tr( ρ W (t)V . . . W (t)V ) . | {z } | {z } 2 1 1 1 1 1 1 1 11 2K¯ 11 1 W(t) W(t) 1 ... ... ... w2 w2 (3.153) 2 w1 w1 w2 w2 w1 w2 21 2 w1 w1 w2 w2 w2 w1 w2 1 2 w1 11 12 21 w1 2 1 w2 w1 1 w1 1 12 w1 w2 1 w1 1 12 1 1 W(t) 1 2 2 w1 1 1 2 w1 w2 2 1 w1 w2 w2 w11 2 1 2w1 w1 1 w1 1 2 w1 11 1 1 1 W(t) 2 w1 w2 1 2 w1 1 w1 1 2 2 1 21 12 1 ww1 11 ww1 1 2 1 11 1 1 1 2 w1 1 1 11 1 1 1 1 1 W(t) 1 2 1 2 1 12 1 w1 2 1 1 w1 21 2 2 w1 1 w2 1 w1 1 12 11 1 1 11 W(t) 2 w1 1 1 2 1 1 1 1 W(t) 1 1 1 1 11 W(t) 1 W(t) 2K¯ 2 1 W(t) 1 1 1 1 2 2 w1 1 1 w1 w2 w1 w2 1 2 w1 w11 ww1 2 W(t) w2 3 w2 1 2 w1 w2 W(t) W(t) Each such correlation function contains K¯ Heisenberg-picture operators W (t) in¯ terleaved with K¯ time-0 operators V . F (K ) (t) encodes 2K¯ − 1 = K time rever. . . performs weak one first prepares ⇢. One sals, illustrated in Fig. 3.20. Wepicture focus on W(t) Hermitian W and operV , assures in measurements, [7, 29], for operators interleaved with K¯ time-0 evolves with U , etc.) Finally, ¯ 1 weak ...... one K =W̃ 2Kweakly, measurements interspersed withmeauni¯ Heisenbergators . F correlation (t) encodes 2K¯ contains 1=K K time reversals, Each Vsuch function sures W̃ strongly. The strong measurement corresponds . .... . U , meataries. (One measures Ṽ weakly, evolves with ¯ illustrated in Fig. 19. We focus on Hermitian W and V , picture operators W(t) interleaved with K time-0 opersimplicity. ¯ + 1 in (w to the anomalous index K , ↵ ). sures W̃ weakly, evolves with U , etc.) Finally, . . . one meaas in [6, 125], for simplicity. 1 W(t) W(t) 2 2 2 1 1 2 1 2 w1 3 w2 3 1 W(t) W(t) W(t) 3 W(t) 2 2 2 1 2 W(t) 1 W(t) 3 W(t) 2 2 2 3 3 1 2 1 W(t) 3 2 2 1 1 5. 2 1 TOC TOC TOC Each green represents a FW(t) orU ascanned V Each purple line from to left toHermitian Wtime bility come Ã⇢ sextuples (.).TOC We sum that sat(w , the ↵TOC ;quasiprobabilities w vdot ) represents atisfy these illustrated inU. Fig. 19.from We focus on and V , t t (t), 1runs w left 2, ↵ w ;right, 1 , v abscissa. Jarzynski-like equality for the TOC along the The ordinate represents the time come ogous sextuples (w1 , ↵ ↵wleftmost ; v1We ,fixv fixhe ) dot satisfy these ogous to thermodynamic We constraints orange, represents to thermodynamic constraints w ; work. 2 , work. 0the state preparation 31 ⇢. TOC ,W TOC a Jarzynski-like equality analogous constraints. sextuple to the a quasiprobaThe W TOC obeys to isfy the constraints: as 97], for simplicity. represents avt, unitary time evolution. diagram, scanned † in⇢ [6,The ⇤ ⇤Each W(t) W(t) ⇤sextuple ⇤right. 0 corresponds W(t) 00 W W(t) W(t) W = à vand W =quasiprobabilities w Multiple 0 outconstraints. Each corresponds o a variables quasiprobaWLet = wand vWe = w Multiple out0and V W W denote analTOC TOC 1random parameter which may inverted in The TOC 1. v 1 .represents 2 1(.). TOC TOC 2and 1 Each TOC green dot a W(t) Vexperiments. .V†† Each purple line W(t)V Experiment TOC 4.w TOC distribution P1TOC (W WTOC )be X Let W denote random variables analbility sum the satTOC ,that UU a UU (Kt )or TOC Experiment time ⇢ Complex TOC TOC Jarzynski-like equality for the TOC 0(w1 sum Eq. (11) in5.[31]. . . to t (t), scanned from . ft from represents come sextuples ↵ ,;quasiprobabilities ↵w vw1We , right, )v satisfy hese bility à (.). We hat satcome sextuples ↵wwwork. ↵;to )dot atisfy these F 0the ogous to thermodynamic work. the constraints w1 ,;the 2 left 2 , leftmost 1v,fix Pogous (W ,W ), (w = time ⇢the TOC TOC isfy constraints: TOC to thermodynamic We fix he constraints ⇢ state orange, represents preparation ⇢. k W(t)V a; vunitary time evolution. scanned ⇤ ⇤ represents 0 0 1 The W(t)V Vdiagram, pair pair constraints. Each corresponds a, quasiprobaW =to v1 and = ,↵wX Multiple outconstraints. sextuple corresponds a quasiprobaThe TOC obeys a Jarzynski-like equality analogous isfy the constraints: ⇤ w ⇤Each ⇢ TOC 1 vto 1. o W(t)U † W(t)V W(t)V 2sextuple 00W TOC (w ,↵ ),(w ),(v ) right. (K ) U WTOC = w v and W = w v . Multiple outExperiment Let W and W denote random variables anal1 1 TOC TOC 2TOC 1 TOC 0(w V . .left . (K .P Each green represents W(t)t (t), orthe Vdistribution purple line from F from to ) V Experiment U ascanned U.V† Each greater K , these he bilityTOC (.). We the quasiprobabilities that atcome sextuples ↵TOC ;the w ↵to ;right, vdot atisfy hese PÃTOC (W , Wsum ),sum := bility à (.). We that sat-alonger Theorem 45.[31]. (Jarzynski-like forthe theTOC TOC). 1The w left 2 ,quasiprobabilities wX 1 , he v ) represents ⇢ The timetime TOC ⇢TOC Jarzynski-like theorem equality for come sextuples (w , ↵ ; w , ↵ ; v , ) satisfy 0 Eq. (11) in . . . ogous to thermodynamic work. We fix the constraints 1 time (2k-1) 1 W(t)V pair k W(t)V 1 w 2 w 1 v 0 ⇥PTOC Ãthe (w ↵,wWEach ;TOC w2 ,)↵sextuple , corresponds 1 ,to w ; v1(w v ) ),(w (W = Wa (w,↵ v) ),(v ) W (w)time v ). (K (K ) timepair ⇢constraints. to a, quasiprobaThe TOC obeys a Jarzynski-like equality analogous isfy constraints: TOC isfy the constraints: ⇢ The We represents unitary evolution. diagram, scanned †0 ⇢reversal ⇤sextuple ⇤right. 0 ,↵ F reversal of which (t) equals a moment. define P ¯ 1 W(t)V pair k W(t)V pair Experiment time-ordered correlator (132) obeys the Jarzynski-like constraints. Each corresponds to a quasiproba. . . W = w v and W = v . Multiple outw V V V TOC ¯ V V V TOC 1 1 † K +1 1 TOC (139)satK +1 ¯ bility Ã⇢ (.).2 We sum quasiprobabilities ) U (w the ,↵ ),(wX ,↵X ),(v ) that U Experiment time TOC (K 0↵w sum ¯ in 4[31]. Eq. (11) 1 time scanned from. .left . .(2k-1) (K )) 0(w right, F (K (t), totime ⇥ ÃTOC (w wfrom ,2 , v↵to ) ;W . bility ÃTOC (.). We that come sextuples ↵:= ;vquasiprobabilities w v1(w , the these 1 ,TOC 21,The ww ;left 1greater v )) represents Watisfy (w v, )satK the longer the distribution w PTOC (W , TOC W );TOC := ⇢TOC time ⇢the P (W , Win ),↵the ators V . F (K ) (t) encodes 2The K 1(Jarzynski-like = K Jarzynski-like timetheorem reversals, Theorem for the TOC). The isfy constraints: equality ⇢reversal three steps: We the00 19:quasiprobability ÃP⇢reversal FIG. -foldpair out-of-time-ordered correlator TOC ¯respect 111KW(t)V W(t)V (2k-1) time W(t)V k W(t)V Functions be to . recall Let ⇥ à (wcan ↵define ;averaged w2strongly. ,right. ↵sextuple ;2 vwith , corresponds ) ),(w Vpair V V pairpair TOC to a, quasiprobaTOC obeys a to analogous isfy the constraints: 1 ,to w Each w(w 1(w v),(w ⇢ time W (wThe v) ),(v )P ). e K random variables (K ⇢constraints. X ,↵ ,↵(K ),(v , W )(wtrong ,↵ ) v(139) sures W̃ measurement )) (20)], -OTOC): conventional [Eq. encodes reversalTheV reversal The conventional OTOC corresponds 3 equality and 0 ,↵ (t) which FW equals a(K moment. We define P 0of V corresponds V Experiment . (K TOC .OTOC ..P time-ordered correlator K (132) the fTOC). denote any of W and .introduce TheK Pthat The greater , heat-measurable longer the distribution (91)]. We random variables X TOC TOC PTOC (W , W[Eq. )sum := bility Ãfunction We the quasiprobabilities (Jarzynski-like theorem for The TOCthe (139) (K ) time illustrated in Fig.(2) 19. WeTheorem focus Hermitian W=obeys and VEthe , Jarzynski-like TOC ⇢TOC(.). FIG. 19:11time KW(t)V -fold out-of-time-ordered correlator TOC 0wbe The K -fold OTOC (t) Eq. (11) on in4 [31]. . . .(2k-1) Functions averaged respect PTOC Let (2k-1) timeF pair W(t)V pair 1 time time reversals. time (K ⇥PTOC ÃTOC (w ↵(w ,; vwith .) . just three 0 ⇥(W à ↵w2 ,);↵ w= )(K w2 , ;↵vw1(w v1 , ) v),(w ,wcan W (w v) ),(v )to (w)(w v. )v¯ (2K¯)D ( W (K ))(20)], average of fthe is1 ,constraints: 0W 1 ,;TOC ⇢isfy W (w vW ), W ⇢TOC ¯ = 2: F (t) the 0 0 ,↵ ,↵These ⇢reversal @ (K -OTOC): The conventional OTOC [Eq. encodes equality 0 reversal reversal to anomalous index K + 1 in (w , ↵ ). of which F (t) equals a moment. We define P reversal W and W . variables participate in constraints in three teps: We recall he quasiprobability à encodes 2K 1 time reversals. The time that passes in a ¯ FIG. 19: K -fold out-of-time-ordered correlator 1 W(t)V pair 0 + W ) k W(t)V pair K = F (t). If K < 3, F (t) is not OTO. time-ordered ⇢ ` of Wwith .K . . ¯+1 f, denote any be function and The (139) P (139) TOC obeys V ) TOC respect ` ,↵W TOCthe Jarzynski-like KV +1The VKw-fold Functions can averaged to .P Let TOC (K FTOC (t) = correlator e (132) (143) X (w ,↵ ),(w ),(v , TOC ) . TOC 0 just three¯ OTOC (t) as in [6, 125], for simplicity. TOC (K runs along the abscissa..OTOC ordinate represents (K The conventional [Eq. encodes 1time time reversals. time (K ) )F (20)], (K .(2k-1) (W ,W )i (140) . .The ⇥PTOC Ãany , ↵,wW ;0of w2 ,W ↵:= ; v1greater , v )W average of(W f(w is12 TOC w W0 (w) he v= )The W2, (w v, The K longer the distribution P TOC ⇢W denote function and PTOC )` @ 0 [Eq. We variables Theorem 4@ (Jarzynski-like theorem for the f0hfTOC). The {w }(91)]. 8` 3,) .he . measurable .laboratory .FIG. ,-OTOC): K + 1random and (146) TOC equality TOC reversal encodes 2K 1 time reversals. The time that passes TOC .introduce in three recall the quasiprobability à (Kin a on sums values. FIG. 19: K -fold correlator `↵TOC 19: K -fold out-of-time-ordered ⇢reversal X ¯steps: ⇢ time parameter t,out-of-time-ordered whichThe may inverted in Fpair experiments. 11time time (2k-1) time W(t)V pair k correlator W(t)V just three reversals. Kbe -fold OTOC (t) Functions be PWe Let ⇥ ÃTOC w vwith , of ) à . the Functions can be averaged with respect P . )Let E , average TOC =0 w 2, ↵ w0 ;2 1(w v respect W (w,↵ to v) ),(v )to W (w.) v(139) define K random (K (K )(20)], 0 ;averaged of e f(wcan is1,,fW 0(W ,↵ ),(w ,0 TOC laboratory runs along the abscissa. The ordinate represents 0 3 0 and (K -OTOC): The conventional OTOC [Eq. (20)], encodes (W ,W )Pand ,W ) .variables reversal reversal hf⇢(W )i (140) @ 2 Dto( K The conventional OTOC corresponds 0TOC (K -OTOC): The conventional OTOC [Eq. encodes of which F equals a moment. We define P TOC TOC The orange, leftmost dot represents the state preparation TOCfunction 0 .(t) TOC TOC TOC W and W . These variables participate in constraints time-ordered correlator (132) Jarzynski-like . . . encodes 2K 1 time reversals. The time that passes in a⇢. 0 e WTOC= +obeys WTOCthe ) f:=denote any of W W The P [Eq. (91)]. introduce measurable random variables ` f denote any function of W and W . The P Let us evaluate Eq. (91) on particular arguments: TOC TOC TOC TOC ` TOC (139) (K ) TOC X00 can be averaged with respect (K ) the parameter t,out-of-time-ordered whichThe may be in Fexperiments. FIG. -fold correlator 0 PTOC FTOC (t) =0 e , Functions TOC just three time reversals. K -fold OTOC Fpurple (t)line Each green dot aThe W(t) orinverted aThe V(K . ordinate Each ¯Kruns ,W . vLet just three eversals. K -fold OTOC (t) E average 1time timerepresents (2k-1) time 0 ; v1 , v0 ) (K 0 (140) )(147) laboratory along the abscissa. represents ⇥(143) à (w12 ; w2 ,,↵0Ww} of f, ⇢W is )to hf W (W )i,f↵(W w{v W (w v ,)= (w ) . .The average ofTOC W 8` 1, .the .time ,19: K .The := )PTOC WWTOC ) . 2, TOC 0 @(@2K¯0 )D(t)( is TOC orange, leftmost dot represents theà state preparation ⇢. (K -OTOC): conventional OTOC (20)], encodes 0TOC `ofand reversal W W . (W variables participate in constraints in three steps: quasiprobability encodes 2K 11 time reversals. The time that passes FIG. 19: K -fold correlator `f0 isbe + 0W areversal unitary time evolution. diagram, scanned K¯ = 2: W(t) F (t) = F (2) (t). wherein Ifequality <, 3,2@FR. not OTO. encodes 2K time reversals. The time that passes in in a a ⇢[Eq. ` sums f, denote any function WTOC and WThese The on ofrespect X TOC TOC `0à TOC ) ⇢ the time parameter t,out-of-time-ordered which may beThe inverted in Functions can averaged with to .values. PWe .Precall Let represents TOC 0 =0 (K experiments. ) TOC (139) FK e WTOC (143) , hf Each green dot represents W(t) . ordinate Each line (Kor ) a VOTOC W ,W 0 W(t) TOC (t) = just three reversals. -fold Fpurple (t) 0 0 (140) 0 laboratory runs along theaThe abscissa. The ordinate represents (K -OTOC): The conventional OTOC [Eq. (20)], encodes from left to¯time right, epresents FK (t), scanned from left to ⇢. (K laboratory runs along the abscissa. The represents 0TOC (K ).introduce (W ,W W )i average ofTOC f,fW is(W , W)WTOC )P (We ,W )TOC hf (W )i TOC TOC TOC;W The orange, leftmost dot represents the state preparation TOC TOC TOC f:=denote any function of and The P.(140) @ W(t) @ 0 [Eq. (91)]. measurable random variables 2 {w } 8` = 2, 3, . . . , K + 1 and (146) à (v , w , ↵ ; . ; v , ; w , ↵ ) TOC represents unitary time volution. The diagram, scanned encodes 2K 1 +1 time reversals. The time that passes TOC (Kin a Let us evaluate Eq. (91) on particular arguments: 1 v 2 w K v K w on sums of à values. ` ` FIG. 19: K -fold out-of-time-ordered correlator X ⇢ 1 2 K K +1 ⇢ the time parameter , which may be inverted in experiments. X right. 0 the time parameter which may be inverted in experiments. just three time reversals. The K -fold OTOC F (t) 5. equality for therespect TOC to P (K ) can be averaged . Let from D Eof, Theorem Proof. The proof to the0 proof 1is,fW(W =0 Functions Each green dot represents aF W(t) orscanned a ordinate V . Each purple 0 0 TOC 0 0)Pwith0(W 0 (140) left to right, represents (t), from left to line⇢. average of,WTOC fJarzynski-like laboratory runs along the abscissa. The 0 t 0 := W hf WTOC , W, TOC ) .) . (W )i ,TOC @is2 analogous (Korange, -OTOC): The conventional OTOC [Eq. (20)], encodes f (W , Wus )Pand 0represents TOC TOC TOC The leftmost dot represents the state preparation 0TOC TOC TOC The orange, leftmost dot represents the state preparation ⇢. a and W . (W These variables participate in constraints TOC encodes 2K 1’s time reversals. The time passes in ` of f,:=denote function WTOC W . W The PTOC Let evaluate (91) on particular arguments: represents unitary time evolution. The iagram, TOC right. TOC Consider fixing values of the W and the W ’s. X ) scanned parameter t,† which may be inverted 5. any equality TOC 0 Eq. 0purple FTOC (t), = 2 R. e ( WTOC + WTOC ) (143) inwherein [31]. = hw ,the ↵for`w0 the |U |v ihv ,to |U |w ,F↵ iOTOC .inthat .Fexperiments. .(K 00 Jarzynski-like green dot aabscissa. W(t) V.⇥ .Each Each line) ¯dot ,W just three time eversals. K -fold (t) +1 v Each K vK K w Each green represents aThe W(t) or a aThe V line (K )or W , W,W 0 0K `purple K(W +1 ) ,= K laboratory runs along ordinate represents (K hf W (W )i (140) average ofTOC from left right, represents (t), scanned from ft to (K )K 8` 1, .the ., time ,(K K .`represents (147) The greater the Kdot , the the longer the distribution P (K := f (W{v , 0W} )PTOC TOCW The TOC obeys Jarzynski-like equality analogous to) . 2,Krepresents TOC` TOC WTOC The orange, leftmost represents the state preparation TOC wherein , 0 2@ R.@ 0 `f0ais2 a;(K unitary time evolution. The diagram, scanned encodes 2K 1 +1 time The time that passes in ⇢. a ) Ãon (v ;w w ↵ ) be represents unitary time evolution. The diagram, sums ofv1the à X 1 , for 2 , ↵values. w 0 ; . . ; vof K v w ⇢TOC the time t,, reversals. which may inscanned experiments. ))K 0 =0 ⇢ equality K† K +1 right. which Fparameter (t) equals alonger moment. We purple define P (K (K ) , Theorem Each green dot represents aabscissa. W(t) aThe Vinverted . ordinate Each (K)or W 5. Jarzynski-like W(t) † W(t) Proof. The proofW(t) (K (11)The in [30]. 0 is analogous to the proof ofEq. 1,W,fW(W 0 a Jarzynski-like The greater the K,,satisfy the distribution Pline left to right, epresents (t), scanned from left to ⇢. (K ) TOC laboratory runs along the represents ⇥, Whw ↵TOC |U |v ,, Wv2TOC ihv |w ↵ i)the from left to right, epresents FF scanned from ft to(K TOC analogous := (W )to . 1; ,v rom hf (W )i (140) Certain quasiprobability values à (.) the con2 ,)P wequality 1↵ vrepresents K +1 wvolution. TOC TOC The orange, leftmost dot represents the state preparation TOCobeys 2; 1 1, |⇢U TOC K +1 )(t), ) ⇢ à (v , w , ; . . . ; w , ↵ a unitary time The diagram, scanned (K ) (K Let us evaluate Eq. (91) on particular arguments: 1 v 2 w K v K +1 w † U ⇢ equality U Ksteps: +1 inright. three We the ofEach which F,parameter (t)represents equals aK define P Ã⇢) line the time t, recall which may bequasiprobability inverted right. 5. for the TOC Eq. inJarzynski-like [30].Jarzynski-like (K 5. equality the TOC Proof. is analogous to the proof ofTheorem Theorem 1X 4,W (Jarzynski-like theorem the TOC). green dot aFmoment. W(t) aiWe V⇥ . Each W (11) in [31].The proof = hw ,for ↵1for |U0,2W |vThe ihv |U |w , ↵) (t), .inpreparation . experiments. .0 purple 0 0K , left to right, epresents scanned from left(K to) (K ) K +1 w v[Eq. K vleftmost K wor (145) 0 0 t K(W +1 Krom K K := f (W , W )P ) . (89)]. We introduce measurable random variables TOC TOC TOC The orange, dot represents the state ⇢. The greater the K , the longer the distribution P TOC TOC , 2 R. OTOCs as moments of The TOC correlator obeys analogous to v †time 0right. E. longer in three steps: We recall Ãscanned straints WJarzynski-like = and W = for all `)`(t) and `the,.moment. Summing represents a(K unitary evolution. The diagram, ⇢ time-ordered (130) obeys the Jarzynski-like Consider values the ’s and the W `fixing ` equality 5. Jarzynski-like equality for Theorem 4 ,W (Jarzynski-like theorem for theTOC TOC). The 0 ’s. 0 W inwherein [31].Higher-order = hw)Kw ,the ↵ |U 0|vK , of ihv ,`†to |U |w ↵(Kquasiprobability irandom .in. `purple .define (K )) +1 wthe v` K vThese K w Each green dot represents a aW(t) aV .⇥ Each W and W variables constraints K +1 ` K K of` The which F.We equals We P (K (K ) to Eq. (11) in W[30]. [Eq. (89)]. introduce measurable variables left right, represents F participate (t), scanned fromPline left (K greater the the longer the distribution P The greater the K longer distribution The TOC obeyscorrelator a Jarzynski-like analogous to.to ⇥(130) hw ,,equality ↵wvequality |U |v , w2v;1 ihv |w ,,,↵↵the i)or)the .K (summed) quasiprobabilities equality The TOC obeys a Jarzynski-like analogous time-ordered Jarzynski-like W(t) W(t) 21obeys 1↵ 1; ,vKrom vrepresents K w W(t) (K ) +1 à (v ; w , . . w 2the 1, |⇢U K +1 (K unitary time evolution. The diagram, scanned 0 ;a (K ) (K )) 2 v K +1 w (K ) (K ) (K ) ⇢ 1 K K +1 W and W . ⇢ These variables participate in define constraints †F on of values. these values yields of which (t) equals moment. We define P inright. hree steps: We thei) evoà ` ums ) Asa Jarzynski-like in Corollary 1, the The subscript tleft denotes the time 5. Jarzynski-like for the TOC which F|w (t) equals aalonger We P (K (K `à (11) in Proof. The proof is analogous to the proof ofEq. Theorem 1[30].Dquasiprobability Eq. (11) inquasiprobability equality The greater the K the the distribution P Theorem 4[30]. (Jarzynski-like theorem for the ⇥ equality hw |vTOC). ihv ,,satisfy ↵recall .quasiprobability E|U to right, represents Fmoment. scanned from left to⇢ ) The TOC equality analogous to1 , of Certain à the con2 , ↵w 1 , v1values vrom K +1 wK (K )(.) (145) Experiment 1 |⇢U +1 (t), ⇢ @ 2 obeys (K (K ) introduce )) [Eq. (89)]. We random variables W + W † )2 on sums ofsteps: Ã(K values. †recall U ⇢ (K U† in three We recall the quasiprobability quasiprobability moments of longer of which F (t) equals a measurable moment. define PÃ(K inright. three steps: the FTOC (t) (11) =4 (Jarzynski-like e (D= ,i.(141) ⇢Ã⇢ time-ordered correlator (130) obeys the Jarzynski-like Eq. in [30].Jarzynski-like 5. equality for TOC Theorem theorem TOC). The inE. [31].Higher-order OTOCs as hw ↵ |U |vOne , can ihv ,0à |U |w iWe ⇥ .in. .constraints in U |w ,Ethe ↵ infer via the interTheorem theorem for the TOC). The ⇢ K +1W,for w K v[Eq. K vThese K wKthe (145) `the w 0, ↵ @ 2lution K` The K ` 0 @ obeys @4 (Jarzynski-like time (K ) (K ) W and Wsteps: .We participate ( W + equality ) K +1 (89)]. introduce measurable random variables greater the Kvariables , the longer distribution (89)]. introduce measurable random variables ` We Di↵erentiating a characteristic function of again and The TOC a Jarzynski-like analogous to v` 0[Eq. E. Higher-order OTOCs as moments longer equality in three We recall the quasiprobability Ã⇢ P FTOC (t) correlator =4 (Jarzynski-like e(130) , 0(141) straints W = w and W = for all ` and ` . Summing ,the =0 time-ordered obeys the Jarzynski-like (summed) quasiprobabilities time-ordered correlator (130) obeys the Jarzynski-like ` ` Theorem theorem for TOC). The 0 (K ) 0 (K ) (K 0 (K ) 0 0 0 0 ` ferometry scheme in [31] and from weak measurements. @ @ As in Corollary 1, the subscript and W .We variables participate constraints of which F (t) equals alonger moment. We define P) ) W and W .to These variables participate ininconstraints ` `sums on of values. Eq.P(11) in [30]. [Eq. (89)]. introduce variables t,want the time `† 31, `.à equality ,¯analogous =0 greater the K the distribution P (K (W ,W ,hw+. .2Wobeys W .denotes .⇢(KThese ,(K W )measurable (148) ⇢ equality [Brian, add any ⇥3(130) ,.↵,wW |UK|v , ,v1W ihv ,W |w ,,references?] ↵¯the i evo. random (summed) The obeys a Jarzynski-like equality to11 again yields higherand quasiprobabilities higher-point correlation functime-ordered correlator the Jarzynski-like 1 v1The K +1 w wherein , 0 2TOC R. (K 2E K +1 (K ) +1 2|⇢U 0 (K K @0 2 2D ))) values. W and W .to variables participate in define constraints on of these values yields Time not be reversed in any trial. )equals 31 ) in hree steps: We recall the quasiprobability Ã⇢) `sums As in need Corollary the subscript t want denotes the time evo⇢⇢These on sums ofinterferometry values. of which Fà (t) a moment. P (K `à [Brian, add any references?] Eq. inquasiprobability [30]. equality V V V FTOC (t) (11) =4, (Jarzynski-like e (D W , (141) Theorem theorem The E the wherein R. 22 X X 2 lution in +U ††Wfor |w ,E1, ↵TOC). can infer à the inter(K⇢) (145)We tions. So does di↵erentiating P (W, W 0 ) again and again. Experiment 0@D @@ w` i. One (K ) @ correlator ) via [Eq. (89)]. introduce measurable random variables ( W ( W+ W ) `) on of steps: ÃWe values. ⇢ (K E. . .Higher-order OTOCs as moments of longer insums three We recall theinterquasiprobability Ã⇢ Di↵erentiating a characteristic function again and The weak-measurement scheme begins with preparaF (t) = e , (141) time-ordered (130) obeys the Jarzynski-like F (t) = e , (141) , =0 TOC D E 2 TOC U lution in U |w , ↵ i. One can infer à via the Theorem 4 (Jarzynski-like theorem for the TOC). The U† 0 0 ⇢These `) win 0 ` [31] and @ @ @ @ @ferometry time function W`from and Wweak variables participate in variables constraints ButDi↵erentiating each .resulting correlator encodes just three time remeasurements. := ( W + scheme W (89)]. introduce measurable random ` .We a characteristic again and equality 31 [Eq. FTOC (t) = correlator e , (141) , Jarzynski-like =0 (summed) quasiprobabilities , =0 0 time-ordered (130) obeys the [Brian, want to add any references?] tion of ⇢. One performs 2K 1 weak measurements inagain yields higherand higher-point correlation func0 0 031on 0t of 0 time 0 (K ) measurements. 1 wherein (K , )2 R.@ @ferometry 0 th st scheme [31] from weak and W variables participate values. `sums 1,in=0 the evo- in constraints versals. Let K =W(t)V (2K pair + 1), for K = 1, 2,P .wherein . ..funcneed not be reversed inW any interferometry trial. `.Ã.to 0A(W ,As Win3K ,Corollary .Wwith .W ,W , subscript Wand W .denotes .⇢(KThese ,(K W ⇢1 W(t)V pair equality [Brian, add anythe references?] 0 again yields higher-point [Brian, want to add any references?] 0E unitaries. 0(One DTime 2and ¯) 2,...,W ,W 2 R. K,¯20+1 1 ,3131in 2 want ) Ṽ terspersed measures weakly, im- (148) ,...,W ¯++1 @ 22 3R. tions. SoOTOC doeshigherdi↵erentiating P (W, W 00 ) correlation againwherein and again. 2, ,W Time need not be reversed any trial. ) K )1 ,W on[Brian, sums ofinterferometry Ãto⇢ add values. want any references?] K¯ can V V 0 -fold.So FTOC (t) = , 0 2 R. e (D Wweak-measurement , (141)scheme K has beenV defined [91, wherein The begins preparaX 2 lution in +U, †Wmeasures |w Ã⇢with via the intertions. di↵erentiating Pencodes (W, W 92]: ) again and again. † `),E↵X w` i. One . .does @ @ 0@The ( W U ButDi↵erentiating each resulting just three time reweakly,infer implements , etc.) acorrelator characteristic function again and weak-measurement scheme with UpreparaFTOC (t) = plements e of (141) , =0 , W̃ . . . 0 ⇢. One performs 2K 1 begins weak measurements in(K )reBut correlator encodes just three time ferometry scheme in [31] and from weak measurements. @ @tion := Experiment 1 0 each resulting 0 st th 31 (K ) 1 sttime -1) à (v ,One ↵with ;unitaries. . ., .=0; vW̃K , v[Brian, wThe ,weakly, ↵ ) versals. Let(2 K = .time (2K + 1), for K = 1, 2, .⇢. ..func0A to any references?] ¯strongly. ¯add tion 2K 1Kweak measurements inmeasures again(t)yields higherhigher-point correlation v1 ;ofw⇢. 2one w wKmea¯ ;want ¯+1 12.and wherein 2)R. 0 th Fversals. := W(t)V ⌘ Tr(⇢ W(t)V . .2, . W(t)V .1 ,Finally, K +1 2performs terspersed (One measures Ṽstrong imLethW(t)V =.time +}i1), for K .wherein . .. , A 1 W(t)V pair Time need not be0 reversed any interferometry pair 0 0 0| = 1,{z want to add any references?] trial. 0(One |Kreversal {z 2 (2Kdefined reversal 2surement ⇣ ⌘31in⇣[Brian, ⌘ K -foldSoOTOC hasW(t)V been [91, W}2, ,W ,...,WK¯+1 terspersed with unitaries. measures Ṽ weakly, corresponds to the index K †+ 1 imin W10 ,W tions. does di↵erentiating P (W, W 92]: ) again and again. 3R. ¯anomalous 2 ,...,W 0 K K 0 -fold. .OTOC has2K been defined [91, 92]: 2K 0 The weak-measurement begins with aUpreparaplements U , measures W̃scheme weakly, implements † , etc.) (w , ↵ ). But 0 each. resulting correlator encodes just three⇥ time replements U , measures W̃ weakly, implements U , etc.) K +1 w ⇢ 0v ⇥ . . . ⇥ W 0 v K +1W ⇥ . of. .⇢.⇥ . (144) W2 w2Finally, W ¯+1 wKW̃ ¯+1 ¯ (K 0 ) 1The tion One performs 2K 1 weak measurements in(K ) K one measures strongly. strong mea1 1 0 K st th F (K ) (t) := hW(t)V . .time . W(t)V ⌘ Tr(⇢ W(t)V .2, . W(t)V )1 . ,Finally, ¯ time (2 à (v , ↵with ; . . . ; vW̃K¯(One , vKmeasures ,weakly, ↵wKmea)¯ K LethW(t)V K -1) = +}ii1), for K = 1,..{z .⇢. .. A VFIG. 19:1 K V-fold out-of-time-ordered V ¯+1strong measures strongly. correlator v1 ; w2one w ¯ ; wThe ¯+1 Fversals. (t) := . . W(t)V ⌘ Tr(⇢ W(t)V . . W(t)V ) . K 2 unitaries. 2. (2K | {z | } terspersed Ṽ imsurement corresponds to the anomalous anomalous index K + 1 in 0 | has2K {z } contains |K 0 Heisenberg{z } reversal 0 ⇣ ⌘ KOTOC -fold been defined [91, 92]: Each such OTOC correlation function surement corresponds to the ⌘ index K †+ 1 in 2K 0 ⇣ (OTOC): The conventional [Eq.reversal (22)], 2K 0 encodes just 0 2K 0 plements U , measures W̃ weakly, implements U , etc.) (w , ↵ ). K +1 w K +1 (K ) picture operators W(t) interleaved with K time-0⇥oper(wK +1 ). (144) 0 v K +1 function . ., .↵w⇥ . . ⇥ mea. 0 K¯ ) The P W10 v1The is⇥ .strong th (144) Wcharacteristic w ⇥ WK ¯+1 wKW̃ ¯+1of ¯F-fold (KW(t)V ) OTOC ¯ K¯ one W measures strongly. 1st W(t)V pair three time The Kators F (correlator (t) (K 0hW(t)V ) 0 i ⌘ Tr(⇢ W(t)V . . . W(t)V 2 pair [Brian, should weK add any references?] . . . W(t)V ) .342Finally, V .(t) F := 2Kencodes 1 = K |time reversals, FIG. 19: reversals. K¯-fold out-of-time-ordered |(t) encodes {z } contains {z } surement corresponds to the anomalousX index K + 1X in Each such correlation correlation function K 00 HeisenbergHeisenberg2(OTOC): K¯ 1 = KThe = conventional 3, 5, . . . time reversals. The time measured 0 0 Each such function contains K 2K 2K(K ) 0). 0 OTOC [Eq. (22)], encodes justK 0 (w , ↵ 0 time-0 picture operators W(t) interleaved with operK +1 w G (s , . . . , s , s , . . . , s ) := K +1 picturethe operators W(t) interleaved with K time-0 oper(144)2characteristic K¯+1 1function by a laboratory clock runs along abscissa. The ordinate K¯ of P (K ) is (K¯ ) The 34 [Brian, should we th time(K 00 ) ¯-1) add any any references?] references?] 1st time (2 Kators three time reversals. The -fold 00 ators F (K ) (t) (t) F encodes(t) 2Kencodes =K time reversals, reversals, 34 [Brian, should we add .. OTOC F encodes 2K 11 = W2 ,...,WK¯+1 W10 ,...,W 0 ¯ represents the time parameter t,VVwhich may be inverted inK time reversal X XK Each such correlation function contains K 0 Heisenbergreversal 2K¯ 1 = K = 3, 5, . . . time reversals. The time measured (K ) 0 0 experiments. The orange, leftmost dot represents the state (K ) 0 0 picturethe operators W(t) interleaved with K time-0 G Poper(s2(W , .34.2., ,W sK ,,sW . ¯. . , s, 0K ¯+1 ¯ )0 ,:= 1,K by a laboratory clock runs along abscissa. The ordinate , . . . W W 0 3 1 2 , . . . , WK¯ ) [Brian, should we add+1 any references?] preparation ⇢. Each green dot represents W(t) or ators V . F (K ) (t)a encodes 2Ka0 V1. = K time reversals, 0 0⌘ ⇣ W ,...,WK¯+1 W1 ,...,W represents the time parameter t, which may be inverted in 0 2 0 0 0 K¯ Each purple line represents a unitary time evolution. The is2 W2 ⇥ e(K ⇥ . . . ⇥ eisK¯+1 WK¯+1 0 eis10W1 ⇥ . . .0 ⇥ eisk WK¯ . experiments. The orange, leftmost dot represents the state ¯ ) diagram, scanned from left to right, represents F (K ) (t), P (W , W , . . . , W , W , W , . . . , W ) ¯ 2 3 K +1 1 2 K¯ preparation ⇢. Each green dot represents a W(t) or a V . 0 1 2 1 (K¯) 1 1 2 1 1 2 1 st 2 stst 0 1 w2 1 ⇤ ⇤ 2 1 ⇤ ⇤ 11 w2 2 2 w2 1 2 1 w1 1 w1 2 1 1 v1 v1 0 2 1 1 1 0 1 1 v1 1 v1 0 TOC TOC TOC TOC TOC ... TOC 2 2 2 1 v1 ⇤ ⇤ 2 1 ⇤ ⇤ 1 2 2 w2 1 w1 1 1 w2 1 0 1 ⇤ ⇤ 2 1 1 0 1 1 v1 0 th th st th ststst th th th ststst th th th st th st th stst th th 1 1 st th 1 1 0 TOC 0 TOC 0 TOC TOC 0 TOC TOC 0 TOC 0 TOC TOC t w1 2 1 ⇤ ⇤⇤ ⇤ 0 0 1 2 1 1 v1 1 11 1 2 2 w2 2 1 1w1 1 1 1 th st The conventional OTOC corresponds to K = 3 and K¯ = 2: F (t) = F (2) (t). If ¯ K < 3, F (K ) (t) is not OTO. 1 th v1 1 ⇤ 1 ⇤ 1 2 2 w2 w1 21 1 1 1 2 1 2 w2 2 1 w1 1 1 1 1 2 w1 1 2 1 1 2 0 TOC 0 ... 0 0 TOC 0 TOC 0 TOC 0 TOC 0 0 00 0 ... 0 TOC 0 TOC 0 TOC TOC 0 00 0 TOCTOC 0 0 TOC TOC 0 0 0 0 0 0 0 TOC TOC 0 TOC 0 0 TOC 0 0 0 ... ... Figure 3.20: K¯ -fold out-of-time-ordered correlator (OTOC): The conventional OTOC (149)⌘ ⇣ from left to right. (K¯ ) (t) encodes Each purple line represents evolution.F The W [Eq. (3.28)], encodes just three timescanned reversals. The Ka ¯unitary -foldtimeOTOC ⇥ eis W ⇥ . . . ⇥ eis eis W ⇥ . . . ⇥ eis W . diagram, scanned from left to right, represents F (t), 0 The s and s variables are regarded as imaginary` (K ) ` (149) The greater thetoK , measured the longer the distribution P scanned from left right. 2K¯ − 1 = K = 3, 5, . . . time reversals. The time by a laboratory clock runs temperature variables, to parallel the fluctuation-relation ¯ of which F (K ) (t) equals a moment. We define P (K ) in 0 literature (e.g.,s0 [126]): is` ⌘are `regarded , and is0` as ⌘ imaginary` . DifThebe s` and three steps: Wethe recall the Klonger -extended quasiprobability (K ) may ` ) variables along the abscissa. The ordinate represents the time parameter t, which inverted in (K The greater K , the the distribution P ferentiating Gvariables, yieldstothe K¯-fold OTOC. (K ) temperature parallel the fluctuation-relation ¯ ( K ) (K ) à [Eq.F(91)].(t)We introduce measurable random of⇢ which equals a moment. We define P variin 0 0 literature (e.g., [126]): ` , and is` ⌘ experiments. The orange, leftmost dot represents the state preparation green dot is` ⌘ equality ` ¯. Difables W W`0recall . These variables participate in con-ρ. Each ` andWe three steps: the K -extended quasiprobability Theorem 5 (Jarzynski-like for the K -fold ferentiating G (K ¯) yields the K¯-fold OTOC. (K ) (K ) straints on sums of Ã⇢introduce (.) values. OTOC). The K -fold OTOC obeys the Jarzynski-like Ã⇢ line [Eq. (91)]. We measurable random varirepresents a W (t) or a V . Each purple represents a unitary time evolution. The 0 ables W` and W` . These variables participate in conTheorem 5 (Jarzynski-like equality for the K¯-fold ¯ (K (K )) straints on sums F of Ã⇢ (.) values. The K¯-fold OTOC obeys the Jarzynski-like diagram, scanned from left to right, represents (t), scanned from left OTOC). to right. 2 (K¯) K¯+1 2 K¯+1 0 1 0 1 0 k 0 0 0 0 0 0 0 ¯ The greater the K , the longer the distribution P (K ) of which F (K ) (t) equals a moment. We define P (K ) in three steps: We recall the K -extended quasiprobability Ã(ρK ) [Eq. (3.99)]. We introduce measurable random variables W` and W`00 . These variables participate in constraints on sums of Ã(ρK ) (.) values. 0 0 K¯ 106 Let us evaluate Eq. (3.99) on particular arguments: Ã(ρK ) (v1 , λ v1 ; w2 , α w2 ; . . . ; vK¯ , λ vK¯ ; wK¯ +1 , α wK¯ +1 ) = hwK¯ +1 , α wK¯ +1 |U |vK¯ , λ vK¯ ihvK¯ , λ vK¯ |U † |wK¯ , α wK¯ i × . . . × hw2 , α w2 |U |v1 , λ v1 ihv1 , λ v1 | ρU † |wK¯ +1 , α wK¯ +1 i . (3.154) One can infer Ã(ρK ) from the interferometry scheme in [38] and from weak measurements. Upon implementing one batch of the interferometry trials, one can infer Ã(ρK ) for all K -values: One has measured all the inner products ha|U |bi. Multiplying together arbitrarily many inner products yields an arbitrarily high-K quasiprobability. Having inferred some Ã(ρK ) , one need not perform new experiments to infer Ã(ρK +2) . To infer Ã(ρK ) from weak measurements, one first prepares ρ. One performs K = 2K¯ − 1 weak measurements interspersed with unitaries. (One measures Ṽ weakly, evolves with U, measures W̃ weakly, evolves with U † , etc.) Finally, one measures W̃ strongly. The strong measurement corresponds to the anomalous index K¯ + 1 in (wK¯ +1 , α wK¯ +1 ). We define 2K¯ random variables W` ∈ {w` } W`00 ∈ {v`0 } ∀` = 2, 3, . . . , K¯ + 1 ∀`0 = 1, 2, . . . , K¯ . and (3.155) (3.156) Consider fixing the values of the W` ’s and the W`00 ’s. Certain quasiprobability values Ã(ρK ) (.) satisfy the constraints W` = w` and W`00 = v`0 for all ` and `0. Summing these quasiprobability values yields 0 P (K ) (W2 ,W3 , . . . ,WK¯ +1 ,W10 ,W20 , . . . ,WK ¯) X X := (3.157) W2 ,W3 ,...,WK¯ +1 W10 ,W20 ,...,W 0 ¯ K Ã(ρK ) (v1 , λ v1 ; w2 , α w2 ; . . . ; vK¯ , λ vK¯ ; wK¯ +1 , α wK¯ +1 )    × δW2 w2 × . . . × δWK¯ +1 wK¯ +1 δW10 v1 × . . . × δW 0 ¯ vK¯ . K Theorem 6 (The K¯ -fold OTOC as a moment). The K¯ -fold OTOC equals a 2K¯ th moment of the complex distribution (3.157): ¯ ∂ 2K F (t) = ∂ β2 . . . ∂ βK¯ +1 ∂ β10 . . . ∂ β0 ¯  + K ¯ ¯ * +1 K KX  X exp −  β` W` + β`0 0 W`00   ,    β ,β 0 =0 ∀`,`0  0 ` `0 ` =1  `=2  0 wherein β` , β` ∈ R. (K¯ ) (3.158) 107 Proof. The proof proceeds in analogy with the proof of Theorem 1 in [38].  The greater the K , the “longer” the quasiprobability Ã(ρK ) . The more weak measurements are required to infer Ã(ρK ) . Differentiating Ã(ρK ) more does not raise the number of time reversals encoded in the correlator. DQ ¯  Q ¯ E ¯ K +1 K 0 Equation (3.158) can be recast as F (K ) (t) = W W , along ` `=2 ` 0 =1 ` 0 the lines of Eq. (3.45). 3.6 Outlook We have characterized the quasiprobability à ρ that “lies behind” the OTOC F (t). à ρ , we have argued, is an extension of the Kirkwood-Dirac distribution used in quantum optics. We have analyzed and simplified measurement protocols for à ρ , calculated à ρ numerically and on average over Brownian circuits, and investigated mathematical properties. This work redounds upon quantum chaos, quasiprobability theory, and weak-measurement physics. As the OTOC equals a combination of à ρ (.) values, à ρ provides more-fundamental information about scrambling. The OTOC motivates generalizations of, and fundamental questions about, KD theory. The OTOC also suggests a new application of sequential weak measurements. At this intersection of fields lie many opportunities. We classify the opportunities by the tools that inspired them: experiments, calculations, and abstract theory. Experimental opportunities We expect the weak-measurement scheme for à ρ and F (t) to be realizable in the immediate future. Candidate platforms include superconducting qubits, trapped ions, ultracold atoms, and perhaps NMR. Experimentalists have developed key tools required to implement the protocol [11–14, 32–36, 51, 64]. Achievable control and dissipation must be compared with the conditions needed to infer the OTOC. Errors might be mitigated with tools under investigation [117]. Opportunities motivated by calculations Numerical simulations and analytical calculations point to three opportunities. Physical models’ OTOC quasiprobabilities may be evaluated. The Sachdev-YeKitaev model, for example, scrambles quickly [30, 31]. The quasiprobability’s functional form may suggest new insights into chaos. Our Brownian-circuit calculation (Sec. 3.4), while a first step, involves averages over unitaries. Summing 108 quasiprobabilities can cause interference to dampen nonclassical behaviors [15]. Additionally, while unitary averages model chaotic evolution, explicit Hamiltonian evolution might provide different insights. Explicit Hamiltonian evolution would also preclude the need to calculate higher moments of the quasiprobability. In some numerical plots, the real part <(A˜ρ ) bifurcates. These bifurcations resemble classical-chaos pitchforks [120]. Classical-chaos plots bifurcate when a differential equation’s equilibrium point branches into three. The OTOC quasiprobability à ρ might be recast in terms of equilibria. Such a recasting would strengthen the parallel between classical chaos and the OTOC. Finally, the Brownian-circuit calculation has untied threads. We calculated only the first moment of A˜ρ . Higher moments may encode physics less visible in F (t). Also, evaluating certain components of A˜ρ requires new calculational tools. These tools merit development, then application to A˜ρ . An example opportunity is discussed after Eq. (3.90). Fundamental-theory opportunities Seven opportunities concern the mathematical properties and physical interpretations of à ρ . The KD quasiprobability prompts the question, “Is the OTOC definition of ‘maximal noncommutation’ consistent with the mutually-unbiased-bases definition?” n f|o Recall Sec. 3.5: We decomposed an operator ρ0 in terms of a set B = |aih h f |ai h f |ai,0 of operators. In the KD-quasiprobability literature, the bases Ba = {|ai} and B f = {| f i} tend to be mutually unbiased (MU): |h f |ai| = √1 ∀a, f . Let A d and B denote operators that have MU eigenbases. Substituting A and B into an uncertainty relation maximizes the lower bound on an uncertainty [132]. In this quantum-information (QI) sense, A and B noncommute maximally. n o  In Sec. 3.5, Ba = |v2 , λ v2 i , and B f = U † |w3 , α w3 i . These B’s are eigenbases of V and W (t). When do we expect these eigenbases to be MU, as in the KDquasiprobability literature? After the scrambling time t ∗ —after F (t) decays to zero—when W (t) and V noncommute maximally in the OTOC sense. The OTOC provides one definition of “maximal noncommutation.” MUBs provide a QI definition. To what extent do these definitions overlap? Initial results show that, in some cases, the distribution over possible values of |hv2 , λ v2 |U |w3 , α w3 i| peaks at √1 . But the distribution approaches this form before t ∗ . Also, the disd tribution’s width seems constant in d. Further study is required. The overlap be- 109 tween OTOC and two QI definitions of scrambling have been explored already: (1) When the OTOC is small, a tripartite information is negative [29]. (2) An OTOC-like function is proportional to a frame potential that quantifies pseudorandomness [123]. The relationship between the OTOC and a third QI sense of incompatibility—MUBs and entropic uncertainty relations—merits investigation. Second, à ρ effectively has four arguments, apart from ρ (Sec. 3.5). The KD quasiprobability has two. This doubling of indices parallels the Choi-Jamiolkowski (CJ) representation of quantum channels [119]. Hosur et al. have, using the CJ representation, linked F (t) to the tripartite information [29]. The extended KD distribution might be linked to information-theoretic quantities similarly. Third, our P(W,W 0 ) and weak-measurement protocol resemble analogs in [109, 110]. {See [111–113] for frameworks similar to Solinas and Gasparinetti’s (S&G’s).} Yet [109, 110] concern quantum thermodynamics, not the OTOC. The similarity between the quasiprobabilities in [109, 110] and those in [38], their weakmeasurement protocol and ours, and the thermodynamic agendas in [109, 110] and [38] suggest a connection between the projects [107, 108]. The connection merits investigation and might yield new insights. For instance, S&G calculate the heat dissipated by an open quantum system that absorbs work [109, Sec. IV]. OTOC theory focuses on closed systems. Yet experimental systems are open. Dissipation endangers measurements of F (t). Solinas and Gasparinetti’s toolkit might facilitate predictions about, and expose interesting physics in, open-system OTOCs. Fourth, W and W 0 suggest understudies for work in quantum thermodynamics. Thermodynamics sprouted during the 1800s, alongside steam engines and factories. How much work a system could output—how much “orderly” energy one could reliably draw—held practical importance. Today’s experimentalists draw energy from power plants. Quantifying work may be less critical than it was 150 years ago. What can replace work in the today’s growing incarnation of thermodynamics, quantum thermodynamics? Coherence relative to the energy eigenbasis is being quantified [136, 137]. The OTOC suggests alternatives: W and W 0 are random variables, analogous to work, natural to quantum-information scrambling. The potential roles of W and W 0 within quantum thermodynamics merit exploration. Fifth, relationships amongst three ideas were identified recently: (1) We have linked quasiprobabilities with the OTOC, following [38]. 110 (2) Aleiner et al. [138] and Haehl et al. [139, 140] have linked the OTOC with Schwinger-Keldysh path integrals. (3) Hofer has linked Schwinger-Keldysh path integrals with quasiprobabilities [128]. The three ideas—quasiprobabilities, the OTOC, and Schwinger-Keldysh path integrals— form the nodes of the triangle in Fig. 3.21. The triangle’s legs were discovered recently; their joinings can be probed further. For example, Hofer focuses on singletimefold path integrals. OTOC path integrals contain multiple timefolds [138– 140]. Just as Hofer’s quasiprobabilities involve fewer timefolds than the OTOC quasiprobability à ρ , the TOC quasiprobability ÃTOC (3.144) can be inferred from ρ fewer weak measurements than à ρ can. One might expect Hofer’s quasiprobabilities to relate to ÃTOC ρ . Kindred works, linking quasiprobabilities with out-of-time ordering, include [125–129]. OTOC • Aleiner et al. (2016) • Haehl et al. (2016) SchwingerKeldysh path integral Yunger Halpern (2017) Hofer (2017) Quasiprobability Figure 3.21: Three interrelated ideas: Relationships amongst the out-of-time-ordered correlator, quasiprobabilities, and Schwinger-Keldysh path integrals were articulated recently. Sixth, the OTOC equals a moment of the complex distribution P(W,W 0 ) [38]. The OTOC has been bounded with general-relativity and Lieb-Robinson tools [7, 72]. A more information-theoretic bound might follow from the Jarzynski-like equality in [38]. Finally, the KD distribution consists of the coefficients in a decomposition of a quantum state ρ ∈ D(H ) [11, 12] (Sec. 3.1). ρ is decomposed in terms of a n f|o set B := |aih h f |ai of operators. B forms a basis for H only if h f |ai , 0 ∀a, f . The inner product has been nonzero in experiments, because {|ai} and {| f i} are chosen to be mutually unbiased bases (MUBs): They are eigenbases of “maximally noncommuting” observables. The OTOC, evaluated before the scrambling time t = t ∗ , motivates a generalization beyond MUBs. What if, F (t) prompts us to ask, h f |ai = 0 for some a, f (Sec. 3.5)? The decomposition comes to be of an 111 “asymmetrically decohered” ρ0. This decoherence’s physical significance merits investigation. The asymmetry appears related to time irreversibility. Tools from non-Hermitian quantum mechanics might offer insight [134]. 112 BIBLIOGRAPHY [1] N. Yunger Halpern, B. Swingle, and J. Dressel, Phys. Rev. A 97, 042105 (2018). [2] S. H. Shenker and D. Stanford, Journal of High Energy Physics 3, 67 (2014). [3] S. H. Shenker and D. Stanford, Journal of High Energy Physics 12, 46 (2014). [4] S. H. Shenker and D. Stanford, Journal of High Energy Physics 5, 132 (2015). [5] D. A. Roberts, D. Stanford, and L. Susskind, Journal of High Energy Physics 3, 51 (2015). [6] D. A. Roberts and D. Stanford, Physical Review Letters 115, 131603 (2015). [7] J. Maldacena, S. H. Shenker, and D. Stanford, 1503.01409. [8] H. J. Carmichael, Statistical Methods in Quantum Optics I: Master Equations and Fokker-Planck Equations (Springer-Verlag, 2002). [9] J. G. Kirkwood, Physical Review 44, 31 (1933). ArXiv e-prints (2015), [10] P. A. M. Dirac, Rev. Mod. Phys. 17, 195 (1945). [11] J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011). [12] J. S. Lundeen and C. Bamber, Phys. Rev. Lett. 108, 070402 (2012). [13] C. Bamber and J. S. Lundeen, Phys. Rev. Lett. 112, 070405 (2014). [14] M. Mirhosseini, O. S. Magaña Loaiza, S. M. Hashemi Rafsanjani, and R. W. Boyd, Phys. Rev. Lett. 113, 090402 (2014). [15] J. Dressel, Phys. Rev. A 91, 032116 (2015). [16] R. W. Spekkens, Phys. Rev. Lett. 101, 020401 (2008). [17] C. Ferrie, Reports on Progress in Physics 74, 116001 (2011). [18] A. G. Kofman, S. Ashhab, and F. Nori, Physics Reports 520, 43 (2012). [19] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, Rev. Mod. Phys. 86, 307 (2014). 113 [20] M. Howard, J. Wallman, V. Veitch, and J. Emerson, Nature 510, 351 (2014). [21] N. Delfosse, P. Allard Guerin, J. Bian, and R. Raussendorf, Phys. Rev. X 5, 021003 (2015). [22] J. Maldacena, International Journal of Theoretical Physics 38, 1113 (1999). [23] E. Witten, Advances in Theoretical and Mathematical Physics 2, 253 (1998). [24] S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Physics Letters B 428, 105 (1998), hep-th/9802109. [25] D. Stanford, Journal of High Energy Physics 10, 9 (2016), 1512.07687. [26] A. A. Patel and S. Sachdev, ArXiv e-prints (2016), 1611.00003. [27] D. Chowdhury and B. Swingle, ArXiv e-prints (2017), 1703.02545. [28] A. A. Patel, D. Chowdhury, S. Sachdev, and B. Swingle, ArXiv e-prints (2017), 1703.07353. [29] P. Hosur, X.-L. Qi, D. A. Roberts, and B. Yoshida, Journal of High Energy Physics 2, 4 (2016), 1511.04021. [30] S. Sachdev and J. Ye, Phys. Rev. Lett. 70, 3339 (1993). [31] A. Kitaev, A simple model of quantum holography, KITP strings seminar and Entanglement 2015 program, 2015. [32] V. Bollen, Y. M. Sua, and K. F. Lee, Phys. Rev. A 81, 063826 (2010). [33] Y. Suzuki, M. Iinuma, and H. F. Hofmann, New Journal of Physics 18, 103045 (2016). [34] F. Piacentini et al., Phys. Rev. Lett. 117, 170402 (2016). [35] G. S. Thekkadath et al., Phys. Rev. Lett. 117, 120401 (2016). [36] T. C. White et al., npj Quantum Information 2, 15022 (2016). [37] J. P. Groen et al., Phys. Rev. Lett. 111, 090506 (2013). [38] N. Yunger Halpern, Phys. Rev. A 95, 012120 (2017). [39] C. Jarzynski, Physical Review Letters 78, 2690 (1997). [40] M. Campisi and J. Goold, ArXiv e-prints (2016), 1609.05848. [41] N. Tsuji, T. Shitara, and M. Ueda, ArXiv e-prints (2016), 1612.08781. [42] B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hayden, Phys. Rev. A 94, 040302 (2016). 114 [43] N. Y. Yao et al., ArXiv e-prints (2016), 1607.01801. [44] A. Bohrdt, C. B. Mendl, M. Endres, and M. Knap, ArXiv e-prints (2016), 1612.02434. [45] G. Zhu, M. Hafezi, and T. Grover, ArXiv e-prints (2016), 1607.00079. [46] I. Danshita, M. Hanada, and M. Tezuka, ArXiv e-prints (2016), 1606.02454. [47] N. Tsuji, P. Werner, and M. Ueda, 1610.01251. Phys. Rev. A 95, 011601 (2017), [48] J. Li et al., ArXiv e-prints (2016), 1609.01246. [49] M. Gärttner et al., ArXiv e-prints (2016), 1608.08938. [50] K. X. Wei, C. Ramanathan, and P. Cappellaro, 1612.05249. ArXiv e-prints (2016), [51] S. Hacohen-Gourgy et al., Nature 538, 491 (2016). [52] R. P. Rundle, T. Tilma, J. H. Samson, and M. J. Everitt, ArXiv e-prints (2016), 1605.08922. [53] M. Takita et al., Phys. Rev. Lett. 117, 210505 (2016). [54] J. Kelly et al., Nature 519, 66 (2015). [55] R. W. Heeres et al., ArXiv e-prints (2016), 1608.02430. [56] D. Ristè et al., Nature Communications 6, 6983 (2015). [57] S. A. Gardiner, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 79, 4790 (1997). [58] S. K. Choudhary, T. Konrad, and H. Uys, Phys. Rev. A 87, 012131 (2013). [59] L. G. Lutterbach and L. Davidovich, Phys. Rev. Lett. 78, 2547 (1997). [60] S. Debnath et al., Nature 536, 63 (2016). [61] T. Monz et al., Science 351, 1068 (2016). [62] N. M. Linke et al., ArXiv e-prints (2016), 1611.06946. [63] N. M. Linke et al., ArXiv e-prints (2017), 1702.01852. [64] A. Browaeys, D. Barredo, and T. Lahaye, Journal of Physics B: Atomic, Molecular and Optical Physics 49, 152001 (2016). [65] C. Guerlin et al., Nature 448, 889 (2007), 0707.3880. [66] K. W. Murch, S. J. Weber, C. Macklin, and I. Siddiqi, Nature 502, 211 (2013). 115 [67] L. Xiao and J. A. Jones, Physics Letters A 359, 424 (2006). [68] D. Lu, A. Brodutch, J. Li, H. Li, and R. Laflamme, New Journal of Physics 16, 053015 (2014). [69] W. Brown and O. Fawzi, ArXiv e-prints (2012), 1210.6644. [70] P. Hayden and J. Preskill, Journal of High Energy Physics 2007, 120 (2007). [71] Y. Sekino and L. Susskind, Journal of High Energy Physics 2008, 065 (2008). [72] N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden, Journal of High Energy Physics 2013, 22 (2013). [73] Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). [74] L. M. Johansen, Phys. Lett. A 329, 184 (2004). [75] M. J. W. Hall, Phys. Rev. A 64, 052103 (2001). [76] M. J. W. Hall, Phys. Rev. A 69, 052113 (2004). [77] L. D. Landau and E. M. Lifshitz, Statistical Physics (Pergamon Press, Oxford, England, 1980). [78] J. Banerji, Contemporary Physics 48, 157 (2007). [79] A. Rihaczek, IEEE Transactions on Information Theory 14, 369 (1968). [80] L. Cohen, Proceedings of the IEEE 77, 941 (1989). [81] N. W. M. Ritchie, J. G. Story, and R. G. Hulet, Phys. Rev. Lett. 66, 1107 (1991). [82] G. J. Pryde, J. L. O’Brien, A. G. White, T. C. Ralph, and H. M. Wiseman, Phys. Rev. Lett. 94, 220405 (2005). [83] J. Dressel, C. J. Broadbent, J. C. Howell, and A. N. Jordan, Phys. Rev. Lett. 106, 040402 (2011). [84] M. F. Pusey, Phys. Rev. Lett. 113, 200401 (2014). [85] M. Waegell et al., ArXiv e-prints (2016), 1609.06046. [86] C. Ferrie and J. Combes, Phys. Rev. Lett. 113, 120404 (2014). [87] L. Vaidman, ArXiv e-prints (2014), 1409.5386. [88] E. Cohen, ArXiv e-prints (2014), 1409.8555. [89] Y. Aharonov and D. Rohrlich, ArXiv e-prints (2014), 1410.0381. 116 [90] D. Sokolovski, ArXiv e-prints (2014), 1410.0570. [91] A. Brodutch, Phys. Rev. Lett. 114, 118901 (2015). [92] C. Ferrie and J. Combes, Phys. Rev. Lett. 114, 118902 (2015). [93] Y. P. Terletsky, JETP 7, 1290 (1937). [94] H. Margenau and R. N. Hill, Prog. Theor. Phys. 26, 722 (1961). [95] L. M. Johansen and A. Luis, Phys. Rev. A 70, 052115 (2004). [96] H. M. Wiseman, Phys. Rev. A 65, 032111 (2002). [97] A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985). [98] C. Emary, N. Lambert, and F. Nori, Reports on Progress in Physics 77, 016001 (2014). [99] H. F. Hofmann, New Journal of Physics 14, 043031 (2012). [100] J. Dressel and A. N. Jordan, Phys. Rev. A 85, 012107 (2012). [101] H. F. Hofmann, Phys. Rev. A 89, 042115 (2014). [102] H. F. Hofmann, New Journal of Physics 16, 063056 (2014). [103] J. Z. Salvail et al., Nat Photon 7, 316 (2013). [104] M. Malik et al., Nat Commun 5 (2014), Article. [105] G. A. Howland, D. J. Lum, and J. C. Howell, Opt. Express 22, 18870 (2014). [106] H. Tasaki, arXiv e-print (2000), cond-mat/0009244. [107] J. Cotler, private communication, 2016. [108] P. Solinas, private communication, 2016. [109] P. Solinas and S. Gasparinetti, Phys. Rev. E 92, 042150 (2015). [110] P. Solinas and S. Gasparinetti, Phys. Rev. A 94, 052103 (2016). [111] J. J. Alonso, E. Lutz, and A. Romito, Phys. Rev. Lett. 116, 080403 (2016). [112] H. J. D. Miller and J. Anders, ArXiv e-prints (2016), 1610.04285. [113] C. Elouard, D. A. Herrera-Martí, M. Clusel, and A. Auffèves, npj Quantum Information 3, 9 (2017). [114] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010). 117 [115] J. von Neumann, Mathematische Grundlagen der Quantenmechanik (Springer, Berlin, 1932). [116] IBM, The quantum experience, 2017. [117] B. Swingle and N. Yunger Halpern, Resilience of scrambling measurements, in press, 1802.01587, accepted by Phys. Rev. E. [118] J. Dressel, T. A. Brun, and A. N. Korotkov, Phys. Rev. A 90, 032302 (2014). [119] J. Preskill, Quantum computation: Ch. 3: Foundations of quantum theory ii: Measurement and evolution, Lecture notes, 2015. [120] S. Strogatz, Non-linear Dynamics and Chaos: With applications to Physics, Biology, Chemistry and Engineering (Perseus Books, 2000). [121] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Advances in Physics 65, 239 (2016), http://dx.doi.org/10.1080/00018732.2016.1198134. [122] C. Gogolin and J. Eisert, Reports on Progress in Physics 79, 056001 (2016). [123] D. A. Roberts and B. Yoshida, ArXiv e-prints (2016), 1610.04903. [124] F. M. Haehl, R. Loganayagam, P. Narayan, and M. Rangamani, ArXiv eprints (2017), 1701.02820. [125] V. Man’ko and R. V. Mendes, Physica D: Nonlinear Phenomena 145, 330 (2000). [126] A. Bednorz, C. Bruder, B. Reulet, and W. Belzig, Phys. Rev. Lett. 110, 250404 (2013). [127] D. Oehri, A. V. Lebedev, G. B. Lesovik, and G. Blatter, Phys. Rev. B 93, 045308 (2016). [128] P. P. Hofer, ArXiv e-prints (2017), 1702.00998. [129] J. Lee and I. Tsutsui, ArXiv e-prints (2017), 1703.06068. [130] J. J. Halliwell, Phys. Rev. A 93, 022123 (2016). [131] T. Durt, B.-G. Englert, I. Bengtsson, and K. Życzkowski, ArXiv e-prints (2010), 1004.3348. [132] P. J. Coles, M. Berta, M. Tomamichel, and S. Wehner, ArXiv e-prints (2015), 1511.04857. [133] M. Paris and J. Rehacek, editors, Quantum State Estimation, Lecture Notes in Physics Vol. 649 (Springer, Berlin, Heidelberg, 2004). [134] N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge UP, 2011). 118 [135] S. An et al., Nature Physics 11, 193 (2015). [136] M. Lostaglio, D. Jennings, and T. Rudolph, Nature Communications 6, 6383 (2015), Article. [137] V. Narasimhachar and G. Gour, Nature Communications 6, 7689 EP (2015). [138] I. L. Aleiner, L. Faoro, and L. B. Ioffe, Annals of Physics 375, 378 (2016). [139] F. M. Haehl, R. Loganayagam, and M. Rangamani, ArXiv e-prints (2016), 1610.01940. [140] F. M. Haehl, R. Loganayagam, and M. Rangamani, ArXiv e-prints (2016), 1610.01941. 119 Chapter 4 MBL-MOBILE: MANY-BODY-LOCALIZED ENGINE This chapter appeared, in an earlier form, in [1]. Many-body localization (MBL) has emerged as a unique phase in which an isolated interacting quantum system does not thermalize internally. MBL systems are integrable and have local integrals of motion [2], which retain information about initial conditions for long times, or even indefinitely [3]. This and other aspects of MBL were recently observed experimentally [4–11]. In contrast, in thermalizing isolated quantum systems, information and energy can easily diffuse. Such systems obey the eigenstate thermalization hypothesis (ETH) [12–14]. A tantalizing question is whether the unique properties of MBL phases could be utilized. So far, MBL was proposed to be used for robust quantum memories [15]. We believe, however, that the potential of MBL is much greater. MBL systems behave athermally, and athermality (lack of thermal equilibrium) facilitates thermodynamic tasks. When a cold bath is put in contact with a hot environment, for instance, work can be extracted from the heat flow. More generally, athermal systems serve as thermodynamic resources [16–27]. Could MBL’s athermality have thermodynamic applications? We present a thermodynamic application of MBL: We formulate, analyze, and numerically simulate an Otto engine cycle for a quantum many-body system that has an MBL phase. The engine contacts a hot bath and a narrow-band cold bath, as sketched in Fig. 4.1. This application unites the growing fields of quantum thermal machines [28–39] and MBL [2, 15, 40–43]. Our proposal could conceivably be explored in cold-atom [4, 5, 7, 8, 11]; nitrogen-vacancy-center [9]; trapped-ion [10]; and possibly doped-semiconductor [44] experiments. Our engine relies on the spectral-correlation properties that distinguish MBL from thermal systems [43, 45]. Take an interacting finite spin chain as an example. Consider the statistics of gaps between consecutive energy eigenvalues far from the energy band’s edges. A gap distribution P(δ) encodes the probability that any given gap has size δ. The MBL gap distribution enables small (and large) gaps to appear much more often than in ETH spectra [46]. This difference enables MBL to enhance our quantum many-body Otto cycle. 120 Wb Figure 4.1: Schematic of many-body-localized (MBL) engine: We formulate an Otto engine cycle for a many-body quantum system that exhibits an MBL phase. The system is exemplified by the spin chain illustrated by the green dots and black arrows. A random disorder potential (the jagged red line) localizes the particles. Particles interact and hop between sites (as suggested by the horizontal red arrows). Consider strengthening the interactions and the hopping frequency. The system transitions from strong localization to a thermal phase (which obeys the eigenstate thermalization hypothesis), or at least to weak localization. The engine thermalizes with a hot bath (represented by the flames) and with a cold bath (represented by the ice cube). The cold bath has a small bandwidth Wb , to take advantage of small energy gaps’ greater prevalence in the highly localized regime. Let us introduce the MBL and ETH distributions in greater detail. Let hδiE denote the average gap at the energy E. MBL gaps approximately obey Poisson statistics [41, 46]: (E) PMBL (δ) ≈ 1 −δ/hδiE e . hδiE (4.1) (E) Any given gap has a decent chance of being small: As δ → 0, PMBL (δ) → hδi1 E > 0. Neighboring energies have finite probabilities of lying close together: MBL systems’ energies do not repel each other, unlike thermal systems’ energies. Thermalizing systems governed by real Hamiltonians obey the level statistics of random matrices drawn from the Gaussian orthogonal ensemble (GOE) [41]: (E) PGOE (δ) ≈ π δ − π δ2 /hδi2E e 4 . 2 hδi2E (4.2) (E) Unlike in MBL spectra, small gaps rarely appear: As δ → 0, PGOE (δ) → 0. MBL’s athermal gap statistics should be construed as a thermodynamic resource as athermal quantum states are [16–27]. In particular, MBL’s athermal gap statistics improve our engine’s reliability: The amount W of work extracted by our engine fluctuates relatively little from successful trial to successful trial. Athermal statistics also lower the probability of worst-case trials, in which the engine outputs net negative work, Wtot < 0. Furthermore, MBL’s localization enables the engine to scale 121 robustly: Mesoscale “subengines” can run in parallel without disturbing each other much, due to the localization inherent in MBL. Even in the thermodynamic limit, an MBL system behaves like an ensemble of finite, mesoscale quantum systems, due to its local level correlations [45, 47, 48]. Any local operator can probe only a discrete set of sharp energy levels, which emerge from its direct environment. This paper is organized as follows. Section 4.1 contains background about the Otto cycle and quantum work and heat. We present the MBL Otto engine in three steps in Sec. 4.2. In Sec. 4.2, we introduce the basic idea using a single qubit (two-level quantum system). In Sec. 4.2, we scale the engine up to a mesoscopic chain tuned between MBL and ETH. In Sec. 4.2, we show that the mesoscopic segments could be combined into a macroscopic MBL system, while operating in parallel. Our analytic calculations are tested in Sec. 4.3, with numerical simulations of disordered spin chains. In Sec. 4.4, we provide order-of-magnitude estimates for a localized semiconductor engine’s power and power density. We compare the localized engine with more traditional alternatives in Sec. 4.5. Background information, intuitive examples, and extensive calculations appear in [1]. 4.1 Thermodynamic background The classical Otto engine (see e.g., [49]) consists of a gas that expands, cools, contracts, and heats. During the two isentropic (constant-entropy) strokes, the gas’s volume is tuned between values V1 and V2 < V1 . The compression ratio is defined as r := VV21 . The heating and cooling are isochoric (constant-volume). The engine outputs a net amount Wtot of work per cycle, absorbing heat Qin > 0 during the heating isochore. A general engine’s thermodynamic efficiency is η := Wtot . Qin (4.3) The Otto engine operates at the efficiency η Otto = 1 − 1 r γ−1 < η Carnot . (4.4) γ := CCPv denotes a ratio of the gas’s constant-pressure and constant-volume specific heats. The Carnot efficiency η Carnot upper-bounds the efficiency of every thermodynamic engine that involves just two heat baths. A quantum Otto cycle [37] for harmonic oscillators has been formulated [29, 37, 50–55]. The quantum harmonic oscillator’s (QHO’s) gap plays the role of the clas- 122 sical Otto engine’s volume. Let ω1 and ω2 > ω1 denote the values between which the angular frequency is tuned. The ideal QHO Otto cycle operates at the efficiency η QHO = 1 − ω1 . ω2 (4.5) This oscillator model resembles the qubit toy model that informs our MBL Otto cycle (Sec. 4.2). The heat and work exchanged by slowly tuned systems are defined as ! Z τ dH W := dt Tr ρ , and dt 0 ! Z τ dρ H Q := dt Tr dt 0 (4.6) (4.7) in quantum thermodynamics [55]. This Q definition is narrower than the definition prevalent in the MBL literature [46, 56–58]: Here, all energy exchanged during unitary evolution counts as work. 4.2 The MBL Otto cycle During the MBL Otto cycle, a quantum many-body system is cycled between two disorder strengths and so between two level-repulsion strengths and two localization lengths. The system begins in the less localized regime, in thermal equilibrium with a hot bath at a temperature TH ≡ 1/ βH . (We set Boltzmann’s constant to one: k B = 1.) Next, disorder is effectively increased, suppressing level suppression. The system then thermalizes with a finite-size cold bath that has a narrow bandwidth at a temperature TC ≡ 1/ βC < βH . Finally, the disorder is decreased. The system then returns to its initial state by thermalizing with the hot bath. Below, we introduce the MBL Otto cycle in three steps: (1) A qubit toy model illustrates the basic physics. (2) A mesoscale engine (Fig. 4.2) is tuned between MBL and ETH phases. (3) Mesoscale subengines operate in parallel in a macroscopic MBL engine. Table 4.3 summarizes parameters of the mesoscale and macroscopic MBL engines. Qubit toy model At the MBL Otto engine’s basis lies a qubit Otto engine whose energy eigenbasis transforms during the cycle [59–62]. Consider a 2-level system evolving under the time-varying Hamiltonian Hqubit (t) := (1 − αt )hσ x + αt h0 σ z . (4.8) 123 Energy (Et) • Stroke 1 • Isentrope • W1 > 0 • Stroke 4 • Hot isochore • Q4 > 0 • Stroke 2 • Cold isochore • Heat -Q2 > 0 leaves the engine. • Stroke 3 • Isentrope • W3 > 0 ETH (“thermal”) MBL Hamiltonian [Hmeso(t)] Figure 4.2: Otto engine cycle for a mesoscale many-body-localized (MBL) system: Two energies in the many-body spectrum capture the cycle’s basic physics. The engine can be regarded as beginning each trial in an energy eigenstate drawn from a Gibbs distribution. Let the red dot denote the engine’s starting state in some trial of interest. The cycle consists of four strokes: During stroke 1, the Hamiltonian Hmeso (t) is tuned from “thermal” (obeying the eigenstate thermalization hypothesis, or ETH) to MBL. During stroke 2, the engine thermalizes with a cold bath. Hmeso (t) returns from MBL to thermal during stroke 3. Stroke 4 resets the engine, which thermalizes with a hot bath. The tunings (strokes 1 and 3) map onto the thermodynamic Otto cycle’s isentropes. The thermalizations (strokes 2 and 4) map onto isochores. The engine outputs work W1 and W3 during the tunings and absorbs heat Q2 and Q4 during thermalizations. The engine benefits from the discrepancy between MBL and thermal gap statistics: Energies have a greater probability of lying close together in the MBL phase than in the thermal phase. This discrepancy leads the engine to “slide down” the lines that represent tunings. During downward slides, the engine loses energy outputted as work. σ x and σ z denote the Pauli x- and z-operators. αt denotes a parameter tuned between 0 and 1. The engine begins in thermal equilibrium at the temperature TH . During stroke 1, the engine is thermally isolated, and αt is tuned from 0 to 1. During stroke 2, the engine thermalizes to the temperature TC . During stroke 3, the engine is thermally isolated, and αt returns from 1 to 0. During stroke 4, the engine resets by thermalizing with a hot bath. Let us make two simplifying assumptions (see [1, App. C] for a generalization): First, let TH = ∞ and TC = 0. Second, assume that the engine is tuned slowly enough to satisfy the quantum adiabatic theorem. We also choose1 h= δGOE 0 δMBL , h = 2 2 1 The gaps’ labels are suggestive: A qubit, having only one gap, obeys neither GOE nor MBL gap statistics. But, when large, the qubit gap apes a typical GOE gap; and, when small, the qubit gap apes a useful MBL gap. This mimicry illustrates how the mesoscopic engine benefits from the greater prevalence of small gaps in MBL spectra than in GOE spectra. 124 Symbol 5 Significance N Number of sites per mesoscale engine (in Sec. II B) or per mesoscale subengine (in the macroscopic engine, in Sec. II C). Chosen, in the latter case, to equal ⇠> . N Dimensionality of one mesoscale (sub)engine’s Hilbert space. E Unit of energy, average energy density per site. Hamiltonian parameter tuned from 0 (at which point the engine obeys the ETH, if mesoscopic, ↵t or is shallowly localized, if macroscopic) to 1 (at which point the engine is MBL, if mesoscopic, or deeply localized, if macroscopic). h i Average gap in the energy spectrum of a length-N MBL system. Wb Bandwidth of the cold bath. Is small: Wb ⌧ h i. Inverse temperature of the hot bath. H = 1/TH = 1/T Inverse temperature of the cold bath. C C Level-repulsion scale of a length-N MBL system. Minimal size reasonably attributable to any energy gap. Smallest gap size at which a Poissonian approximates the MBL gap distribution well. v Speed at which the Hamiltonian is tuned. Has dimensions of 1/Time2 . ⇠> Localization length of the macroscopic MBL engine in its shallowly localized phase. ⇠< Localization length of the macroscopic MBL engine in its deeply localized phase. Satisfies ⇠< < ⇠> . Xmacro Characteristic X of the macroscopic MBL engine (e.g., X = N, h i). g Strength of coupling between engine and cold bath. ⌧cycle Time required to implement one cycle. h i(L) Average energy gap of a length-L MBL system. TABLE I: Often-used parameters that characterize the mesoscopic and macroscopic MBL engines: Introduced Figure of constant the mesoscopic and MBL engines: Introduced in in Sections II B4.3: and IIParameters C. Boltzmann’s is set to one: kB macroscopic = 1. Sections 4.2 and 4.2. Boltzmann’s constant is set to one: kB = 1. We take advantage of the phases’ distinct level statisTr e H HGOE normalizes the state. ⇢(0) admits of an and δ  δ . MBL cold bath. Recall the tics using aGOE small-bandwidth ensemble interpretation: We can regard the engine as qubit engine of Sec. II A: The qubit engine outputs net beginning each trial in one energy level. E The engine’s D positive gap down which cold The probability beginning any given trial Letwork us because analyzeMBL the, the cycle’s energetics. systemH Eofjbegins with Hqubit (t)with=energy 0. Ej /Z. Averaging over trials involves an averthermalization drops the engine, is smaller than GOE , equals e the gap up which hot thermalization the engine: age with2 respect ⇢(0). Stroke 1 preserves the T =raises ∞ state /2. Stroke dropstothe energy to − δMBL 2 . The MBL < GOE . Like the qubit engine, the mesoscopic enδGOE During stroke4,1,the Hmeso (t) is resets tuned from HGOE to energy drops to across − 2 justduring stroke 3. During stroke engine to zero gine must cold-thermalize small gaps. The avWe approximate the tuning as quantumerage gap must upper-bound the cold bath’s bandwidth,δGOEHMBL . adiabatic. (Later, we calculate diabatic corrections average energy, absorbing heat hQ i = , on average. Wb : W b ⌧ h i. This bath can thermalize only 4energy 2 to adiabatic results.) Stroke 2—cold thermalization— levels separated by anomalously small gaps [56–58]. Such depends on the gap j0 between the j th and (j 1)th gaps appear more inexchanged MBL spectra during than in GOE spectra. (strokes The energy the tunings 1 and 3) gap constitutes work [Eq. MBL levels. This typically exceeds Wb . (4.6)], If it does, Applying level statistics requires one more subtlety: the cold bath does not change the engine’s energy. Zero Suppose that the Hamiltonian H (t) conserves parwhile the energy exchanged during the thermalizations (strokes 2 and 4) is heat meso net work is extracted during the cycle: Wtot = 0. With ticle number. The engine state ⇢(t) must occupy one Wb 2 some small probability ⇠ work gap is small enough h i , the [Eq. (4.7)]. The outputs the sector per-cycle power, or average outputted per particle-number sector andengine must remain in that 0 to thermalize: < W . In these cases, the cold bath throughout the cycle. The j hWtotbi δMBL 1 reason is a lack of repulsion cycle, (δ − δ ). The efficiency is η = = 1 − . This redrops the engine to level j 1. Stroke 3 brings the entot =not GOE MBL qubit between GOEhW energies in the same particle-number 2 δGOE hQ4 i gine to level j 1 of HGOE . The gap j between the sector (App. F). ⇢(t) occupies a Hilbert space of dimensult is equivalent to the efficiency η placeof a (jthermodynamic Otto engine [Eq.h (4.4)]. 1)th and j th HGOE levels is typically i Wb . The sionality N ⇠ 2N . Our numerical simulations takeOtto engine therefore outputs net positive work: Wtot > 0. δ γ−1 MBL at half-filling. The gap ratio δGOE plays the role of r .Hot η qubit also equals the efficiency η QHO thermalization (stroke 4) returns the engine to ⇢(0). 1 [Eq. (4.5)], if the frequency ratio ω/Ω is chosen to equal the gap ratio δMBL /δGOE . Qualitative analysis of the mesoscale engine As shown in Sections 4.2-4.2, however, the qubit engine can scale to a large comFigure 2 illustrates or positive-workposite engine aof“successful,” densely packed qubit subengines operating in parallel. The dense outputting, trial. The engine begins in the thermal 2 This expression is derived in Sec. II B 3. if partition the qubits are encoded in the MBL system’s localized degrees state packing ⇢(0) = e isHpossible /Z. The function Z := of freedom (`-bits, roughly speaking [2]). II B 2. H GOE 125 Level-statistics engine for a mesoscale system The next step is an interacting finite-size system tuned between MBL and ETH phases. Envision a mesoscale engine as a one-dimensional (1D) system of N ≈ 10 sites. This engine will ultimately model one region in a thermodynamically large MBL engine. We will analyze the mesoscopic engine’s per-trial power hWtot i, the efficiency η MBL , and work costs hWdiab i of undesirable diabatic transitions. Set-up for the mesoscale MBL engine The mesoscopic engine evolves under the Hamiltonian Hmeso (t) := E [(1 − αt )HGOE + αt HMBL ] . Q(αt ) (4.9) The unit of energy, or average energy density per site, is denoted by E. The tuning parameter αt ∈ [0, 1]. When αt = 0, the system evolves under a random Hamilto(E) nian HGOE whose gaps δ are distributed according to PGOE (δ) [Eq. (4.2)]. When αt = 1, Hmeso (t) = HMBL , a Hamiltonian whose gaps are distributed according to (E) PMBL (δ) [Eq. (4.1)]. We simulate HGOE and HMBL using a disordered Heisenberg model in Sec. 4.3. There, HGOE and HMBL differ only in their ratios of hopping frequency to disorder strength. The mesoscale engine’s cycle is analogous to the qubit cycle, including initialization at αt = 0, tuning of αt to one, thermalization with a temperature-TC bath, tuning of αt to zero, and thermalization [64–67] with a temperature-TH bath. To highlight the role of level statistics in the cycle, we hold the average energy gap, hδi, constant.2 We do so using renormalization factor Q(αt ).3 Section 4.3 details how we define Q(αt ) in numerical simulations. 2 hδi is defined as follows. Let µ(E) =≈ √ N 2π N E e −E /2N E denote the density of states at 2 2 1 energy E. Inverting µ(E) yields the local average gap: hδi E := µ(E) . Inverting the average of µ(E) yields the average gap: √ 1 N 2 πN =R∞ = E. (4.10) hδi := 2 (E) N hµ(E)ienergies dE µ −∞ 3 Imagine removing Q(α ) from Eq. (4.9). One could increase α —could tune the Hamiltonian t t from ETH to MBL [43]—by strengthening a disorder potential. This strengthening would expand the energy band. Tuning from MBL to ETH would compress the band. Expanding and compressing would generate an accordion-like motion. By interspersing the accordion motion with thermalizations, one could extract work. Such an engine would benefit little from properties of MBL, whose thermodynamic benefits we wish to highlight. Hence we “zero out” the accordion-like motion, by fixing hδi through Q(α t ). 126 The key distinction between GOE level statistics (4.2) and Poisson (MBL) statistics (4.1) is that small gaps (and large gaps) appear more often in Poisson spectra. A toy model illuminates these level statistics’ physical origin: An MBL system can be modeled as a set of noninteracting quasilocal qubits [2]. Let g j denote the j th qubit’s gap. Two qubits, j and j 0, may have nearly equal gaps: g j ≈ g j 0 . The difference |g j − g j 0 | equals a gap in the many-body energy spectrum. Tuning the Hamiltonian from MBL to ETH couples the qubits together, producing matrix elements between the nearly degenerate states. These matrix elements force energies apart. To take advantage of the phases’ distinct level statistics, we use a cold bath that has a small bandwidth Wb . According to Sec. 4.2, net positive work is extracted from the qubit engine because δMBL < δGOE . The mesoscale analog of δGOE is ∼ hδi, the typical gap ascended during hot thermalization. During cold thermalization, the system must not emit energy on the scale of the energy gained during cold thermalization. Limiting Wb ensures that cold thermalization relaxes the engine only across gaps δ ≤ Wb  hδi. Such anomalously small gaps appear more often in MBL energy spectra than in ETH spectra [68–70]. This level-statistics argument holds only within superselection sectors. Suppose, for example, that Hmeso (t) conserves particle number. The level statistics arguments apply only if the particle number remains constant throughout the cycle [1, App. F]. Our numerical simulations (Sec. 4.3) take place at half-filling, in a subspace of dimensionality N of the order of magnitude of the whole space’s dimensionality: N N ∼ √2 . N We are now ready to begin analyzing the mesoscopic engine Otto cycle. The engine   begins in the thermal state ρ(0) = e− βH HGOE /Z, wherein Z := Tr e− βH HGOE . The engine can be regarded as starting each trial in some energy eigenstate j drawn according to the Gibbs distribution (Fig. 4.2). During stroke 1, Hmeso (t) is tuned from HGOE to HMBL . We approximate the tuning as quantum-adiabatic. (Diabatic corrections are modeled in Sec. 4.2.) Stroke 2, cold thermalization, depends on the gap δ0j between the j th and ( j − 1) th MBL levels. This gap typically exceeds Wb . If it does, cold thermalization preserves the engine’s energy, and the cycle outputs 0 b Wtot = 0. With probability ∼ W hδi , the gap is small enough to thermalize: δ j < Wb . In this case, cold thermalization drops the engine to level j − 1. Stroke 3 brings the engine to level j − 1 of HGOE . The gap δ j between the ( j − 1) th and j th HGOE levels is hδi  Wb , with the high probability ∼ 1 − (Wb / hδi) 2 . Hence the engine likely outputs Wtot > 0. Hot thermalization (stroke 4) returns the engine to ρ(0). 127 Quantitative analysis of the mesoscale engine How well does the mesoscale Otto engine perform? We calculate average work hWtot i outputted per cycle and the efficiency η MBL . Details appear in Suppl. Mat. C.1. We focus on the parameter regime in which the cold bath is very cold, the cold-bath bandwidth Wb is very small, and the hot bath is very hot: TC  Wb  hδi, and √ N βH E  1. The mesoscale engine resembles a qubit engine whose state and (E) (δ j ) and gaps are averaged over. The gaps, δ j and δ0j , obey the distributions PGOE (E) PMBL (δ0j ) [Eqs. (4.2) and (4.1)]. Correlations between the HGOE and HMBL spectra can be neglected. We make three simplifying assumptions, generalizing later: (i) The engine is assumed to be tuned quantum-adiabatically. Diabatic corrections are calculated in Sec. 4.2. (ii) The hot bath is at TH = ∞. We neglect finite-temperature corrections,  2 b which scale as N ( βH E) 2 W hδi. (iii) The gap distributions vary negligibly with hδi (E) (E) energy: PGOE (δ j ) ≈ PGOE (δ j ), and PMBL (δ0j ) ≈ PMBL (δ0j ), while hδiE ≈ hδi. Average work hWtot i per cycle: The crucial question is whether the cold bath manages to relax the engine across the MBL-side gap δ0 ≡ δ0j . This gap obeys the Poisson distribution PMBL (δ0 ). If δ0 < Wb , the engine has a probability 1/(1+e−δ βC ) of thermalizing. Hence the overall probability of relaxation by the cold bath is ZWb 0 1 e−δ /hδi . hδi 1 + e− βC δ0 0 dδ0 pcold ≈ (4.11) − βC δ : b In a simple, illustrative approximation, we Taylor-expand to order W hδi and e 1 b pcold ≈ W hδi − βC hδi . A more sophisticated analysis tweaks the multiplicative con2 ln 2 b stants (Suppl. Mat. C.1): pcold ≈ W hδi − βC hδi . 0 Upon thermalizing with the cold bath, the engine gains heat hQi4 ≈ hδi, on average, during stroke 4. Hence the cycle outputs work ! 2 ln 2 , (4.12) hWtot i ≈ pcold hδi + hQ2 i ≈ Wb 1 − βC on average. hQ2 i denotes the average heat absorbed by the engine during cold thermalization: ZWb δ0 e−δ /hδi (Wb ) 2 ≈ − . dδ 0 2 hδi hδi 1 + e− βC δ 0 0 hQ2 i ≈ − 0 (4.13) 128 In hWtot i, hQ2 i cancels with terms, in hQ4 i, that come from high-order processes. We have excluded the processes from Eq. (4.11) for simplicity. See App. (C.1) for details. bandwidth This per-cycle power scales with the system size N as4 Wb  hδi ∼ #effective energy eigenstates ∼ √ E N N . Efficiency η MBL : The efficiency is η MBL = Wb hWtot i hQ4 i + hQ2 i = ≈1− . 2 hδi hQ4 i hQ4 i (4.14) Wb  1, because the cold bath has a small bandwidth. The imperfection is small, 2hδi This result mirrors the qubit-engine efficiency η qubit .5 But our engine is a manybody system of N interacting sites. MBL will allow us to employ segments of the system as independent qubit-like subengines despite interactions. In the absence of MBL, each subengine’s effective hδi = 0. With hδi vanishes the ability to extract hWtot i > 0 using a local cold bath. Diabatic corrections to the per-cycle power We have modeled the Hamiltonian tuning as quantum-adiabatic. Realistic tuning t speeds v := E dα dt are finite, inducing diabatic hops: Suppose that the engine starts some trial in the j th energy eigenstate, with energy E j . Suppose that Hmeso (t) is measured at the end of stroke 1, e.g., by the cold bath. The measurement’s outcome may be the energy E`0 of some MBL level other than the j th . The engine will be said to have undergone a diabatic transition. Transitions of three types can occur during stroke 1 and during stroke 3 (Fig. 4.4). If the engine jumps diabatically, its energy changes. Heat is not entering, as the engine is not interacting with any bath. The energy comes from the battery used to tune the Hamiltonian, e.g., to strengthen a magnetic field. Hence the energy change 4 The effective bandwidth is defined as follows. The many-body system has a Gaussian density √ 2 2 e −E /2N E . The states within a standard deviation E N of the mean obey √ Eqs. (4.1) and (4.2). These states form the effective band, whose width scales as E N. 5 η MBL is comparable also to η QHO [Eq. (4.5)]. Imagine operating an ensemble of independent QHO engines. Let the j th QHO frequency be tuned between Ω j and ω j , distributed according D E to PGOE (Ω j ) and PMBL (ω j ). The average MBL-like gap ω j , conditioned on ω j ∈ [0,Wb ], is ω j ∼ R Wb RW 1 dω j ω j PMBL (ω j ) ≈ W1b 0 b dω j ω j = W2b . Averaging the efficiency over the QHO Wb /hδi 0 of states: µ(E) ≈ √ N 2π N E Wb ensemble yields η QHO := 1 − hωi hΩi ≈ 1 − 2hδi ≈ η MBL . The mesoscale MBL engine operates at the ideal average efficiency of an ensemble of QHO engines. But MBL enables qubit-like engines to pack together densely in a large composite engine. 129 Energies (Et) APT APT HETH LandauZener LandauZener Frac-LandauZener Frac-LandauZener HMBL Hamiltonian [Hmeso(t)] Figure 4.4: Three (times two) classes of diabatic transitions: Hops to arbitrary energy levels, modeled with general adiabatic perturbation theory (APT), plague the ETH regime. Landau-Zener transitions and fractional-Landau-Zener transitions plague the many-body-localized regime. consists of work. In addition to depleting the battery, diabatic transitions can derail trials that would otherwise have outputted Wtot > 0. We estimate, to lowest order in small parameters, the average per-cycle work costs hWdiab i of diabatic jumps. Supplementary Materials C.1 contain detailed derivations. Numerics in Sec. 4.3 support the analytics: 1. Thermal-regime transitions modeled by general adiabatic perturbation theory (APT transitions): Tuning Hmeso (t) within the ETH phase ramps a perturbation. A matrix M represents the perturbation relative to the original energy eigenbasis. Off-diagonal elements of M may couple the engine’s state to arbitrary eigenstates of the original Hamiltonian. We model such couplings with general adiabatic perturbation theory (APT) [63], calling the induced transitions APT transitions (Suppl. Mat. C.1). APT transitions mimic thermalization with an infinite-temperature bath: The probability of transitioning across a size-δ gap does not depend on whether the gap lies above or below the engine’s initial state. More levels exist above the initial state than below, if the initial state is selected according to a Gibbs distribution at TH < ∞. Hence APT transitions tend to hop the engine upward, costing an amount ! 1 v 2 βH hδi2 −N ( βH E)2 /4 log e (4.15) hWAPT i ∼ √ v N E hδi of work per trial, on average. 130 Suppose that the engine starts at TH = ∞. APT transitions have no work to do during stroke 1, on average, by the argument above. As expected, the right-hand side of Eq. (4.15) vanishes. The logarithm in Eq. (4.15) is a regulated divergence. Let PAPT (n|m) denote the probability of the engine’s hopping from level m to level n. The probability diverges as the difference |En − Em | between the levels’ energies shrinks: PAPT (n|m) → ∞ as |En − Em | → 0. The consequent divergence in hWAPT i is logarithmic. We cut off the hWAPT i integral at the greatest energy difference that contributes significantly to the integral, |En − Em | ∼ hδi. The logarithm diverges in the adiabatic limit, as v → 0. Yet the v 2 in Eq. (4.15) vanishes more quickly, sending hWAPT i to zero, as expected. The exponential in Eq. (4.15) results from averaging over the thermal initial √ state, e− βH HGOE /Z. Since the hot bath is hot, N βH E  1, the exponential ∼ 1. The √1 and the logarithm scale subdominantly in the system size. N Let us recast the dominant factors in terms of small dimensionless parameters:  √ 4 √  2 v hWAPT i ∼ hδi ( N βH E) hδi hδi. The average work cost is suppressed E √ √ v fourfold in hδi  1, is suppressed linearly in N βH E  1, and is twofold large in hδi E  1. 2. Landau-Zener transitions: Landau-Zener-type transitions overshadow APT transitions in the MBL phase. Consider tuning the Hamiltonian parameter αt within the MBL regime but at some distance from the deep-localization value 1. Energies drift close together and separate. When the energies are close together, the engine can undergo a Landau-Zener transition [71] (Suppl. Mat. C.1). LandauZener transitions hop the engine from one energy level to a nearby level. (General APT transitions hop the engine to arbitrary levels.) Landau-Zener transitions cost zero average work, due to symmetries: hWLZ i = 0. The j th level as likely wiggles upward, toward the ( j + 1) th level, as it wiggles downward, toward the ( j − 1) th level. The engine as likely consumes work W > 0, during a Landau-Zener transition, as it outputs work W > 0. The consumption cancels the output, on average. 3. Fractional-Landau-Zener transitions: At the beginning of stroke 3, nonequilbrium effects could excite the system back across the small gap to energy level j. The transition would cost work and would prevent the trial from outputting Wtot > 0. We dub this excitation a fractional-Landau-Zener (frac-LZ) transition. 131 It could be suppressed by a sufficiently slow drive [63]. The effects, and the resultant bound on v, are simple to derive (see Suppl. Mat. C.1 for details). Let the gap start stroke 3 at size δ and grow to a size ∆ > δ. The probability of a frac-LZ transition between a small gap and a large gap δ < ∆ is [63] ! v 2 (δ − ) 2 1 1 v 2 (δ − ) 2 pfrac-LZ (δ) ≈ + ≈ . (4.16) 16 δ 6 ∆6 16δ6 δ − denotes the MBL level-repulsion scale, the characteristic matrix element introduced, by a perturbation, between eigenstates of an unperturbed Hamiltonian. This mode of failure must be factored into the success probability pcold of stroke2 cooling. To suppress the probability of a frac-LZ transition, the gap must satisfy δ > δmin := (vδ − /4) 1/3 . Neglecting TC > 0 and Wb / hδi corrections, we modify Eq. (4.11): ZWb pcold ≈ δmin   Wb − δmin dδ PMBL (δ) 1 − pfrac-LZ (δ) ≈ . hδi (4.17) To avoid frac-LZ costs, we must have Wb must  δ m , and v 4(Wb ) 3 δ− (4.18) Since Wb /δ −  1, v < (Wb ) 2 . MBL engine in the thermodynamic limit The mesoscale engine has two drawbacks. Consider increasing the system size √ E N N. The average gap declines exponentially: hδi ∼ 2 N . Hence the average work extracted per trial, hWtot i ∼ Wb  hδi, declines exponentially. Additionally, the tuning speed v must shrink exponentially: Hmeso (t) is ideally tuned quantumadiabatically. The time per tuning stroke must far exceed hδi−1 . The mesoscale engine scales poorly, but properties of MBL offer a solution. We introduce a thermodynamically large, or macroscopic, MBL Otto engine. The engine consists of mesoscale subengines that operate mostly independently. This independence hinges on local level correlations of the MBL phase, detailed in Sec. 4.2: Energy eigenstates localized near each other spatially tend to correspond to far-apart energies and vice versa. Local level correlations inform the engine introduced in Sec. 4.2. The engine cycle lasts for a time τcycle that obeys three constraints, introduced in Sec. 4.2. We focus on exponential scaling behaviors. 132 Local level correlations Consider subsystems, separated by a distance L, of an MBL system. The subsystems evolve roughly independently until times exponential in L, due to the localization [15]. We apply this independence to parallelize mesoscale engines in different regions of a large MBL system. This application requires us to shift focus from whole-system energy-level statistics to local level correlations [45, 47, 48]. We review local level correlations here. An MBL system has a complete set of quasilocal integrals of motion [15].6 Thus, each integral of motion can be associated with a lattice site. This association is unique, other than for a small fraction of the integrals of motion. Let |ψ1 i and |ψ2 i denote many-body energy eigenstates associated with the eigenvalues E1 and E2 . |ψ1 i and |ψ2 i are eigenstates of every integral of motion [15]. Let O denote a generic strictly local operator. O is represented, relative to the energy eigenbasis, by matrix elements O21 := hψ2 |O|ψ1 i. Local level correlations interrelate (1) the matrix-element size |O21 | and (2) the difference |E1 − E2 | between the states’ energies. Suppose that |ψ1 i and |ψ2 i correspond to the same configurations of the integrals of motion, of energy, and of particle density everywhere except in a size-L region. Such eigenstates are said to be “close together,” or “a distance L apart.” Let ξ denote the system’s localization length. If the eigenfunctions lie close together (L  ξ), the matrix-element size scales as |O21 | ∼ 2−L . (4.19) All lengths appear in units of the lattice spacing, set to one. If the states are far apart (L  ξ), |O21 | ∼ e−L/ξ 2−L . (4.20) Having related the matrix-element size |O21 | to the spatial separation L, we relate L to the energy difference |E1 − E2 |. Spatially close-together wave functions (L ≤ ξ) hybridize. Hybridization prevents E1 and E2 from having an appreciable probability of lying within Ee−L/ξ 2−L of one another (see [15, 40, 45, 72] and Suppl. 6 “Local” refers to spatial locality here. “Quasilocal” means that each integral of motion can be related to a local operator via a finite-depth unitary transformation that consists only of local unitaries, up to exponentially small corrections. 133 Mat. C.2). Hence small energy differences correlate with rearrangements of particles across large distances, which correlate with small matrix elements:7 |E1 − E2 |  Ee−L/ξ 2−L ↔ Lξ ↔ |O21 | ∼ e−L/ξ 2−L . (4.21) Conversely, large energy differences correlate with rearrangements of particles across small distances, which correlate with large matrix elements: |E1 − E2 |  E2−L ↔ Lξ ↔ |O21 | ∼ 2−L . (4.22) Application of local level correlations in the macroscopic MBL engine We apply local level correlations in constructing a scalable generalization of the mesoscale Otto engine. We denote properties of the macroscopic, composite engine with the subscript “macro.” (For example, as N denoted the number of sites in a mesoscale engine, Nmacro denotes the number of sites in the macroscopic engine.) Strokes 1 and 3 require modification: The Hamiltonian Hmacro (t) is tuned within the MBL phase, between a point analogous to HGOE and a point analogous to HMBL . The HGOE -like Hamiltonian has a localization length ξ > ; and HMBL -like Hamiltonian, ξ <  ξ > . We illustrate with ξ > = 1 and ξ < = 12 in Suppl. Mat. C.4. Particles mostly remain in regions of, at most, length ξ > . Such regions function as “subengines,” instances of the mesoscale engine. What happens in a subengine stays in a subengine. This subdivision boosts the engine’s power. A length-N mesoscale engine operates √ E N at the per-cycle power hWtot i ∼ Wb  hδi ∼ 2 N (Sec. 4.2). Suppose that the whole system consisted of one length-Nmacro engine. The power would scale as √ E Nmacro ∼ 2 Nmacro . This quantity → 0 in the thermodynamic limit, as Nmacro → ∞. But our engine consists of length-ξ > subengines. Local level correlations give each subengine an effective average gap √ E ξ> E (4.23) hδi ∼ ξ ∼ ξ 2> 2> 7 These features are consistent with globally Poisson level statistics: Suppose that E and E 1 2 denote large nearest-neighbor energies. |ψ1 i and |ψ2 i typically represent configurations that differ at extensively many sites. Hence |O21 | ∼ e − L/ξ 2 − L . This matrix element is exponentially smaller, in L, than the average gap 2 − L implied by Poisson statistics. 134 The composite-engine power hWtot imacro is suppressed not in Nmacro , but in the subengine length ξ > : √ ξ> (4.24) hWtot imacro ∼ Nmacro ξ E . 2> Time scales of the macroscopic MBL engine Three requirements constrain the time for which a cycle is implemented: (1) Subengines must operate mostly independently. Information propagates between subengines, albeit slowly due to localization. Hmacro (t) must be tuned too quickly for much information to cross-pollinate subengines (Suppl. Mat. C.4). (2) Tuning at a finite speed v > 0 induces diabatic transitions between energy levels (Sec. 4.2). v must be small enough to suppress the average work cost, hWdiab i, of undesirable diabatic transitions: hWdiab i  hWtot i (Suppl. Mat. C.4). (3) The cold bath has a small bandwidth, Wb  hδi; couples to the engine with a small strength g; and interacts locally. Stroke 2 must last long enough to thermalize each subengine nonetheless. We detail these requirements and bound the cycle time, τcycle . τcycle may be optimized via, e.g., shortcuts to adiabaticity [29, 52, 61, 62, 73–75]. Lower bound on the tuning speed v from the subengines’ (near) independence: The price paid for scalability is the impossibility of adiabaticity. Suppose that Hmacro (t) were tuned infinitely slowly. Information would have time to propagate from one subengine to every other. The slow spread of information through MBL [76] lower-bounds the tuning speed. We introduce notation, then sketch the derivation, detailed in Suppl. Mat. C.4. Let JL denote the level-repulsion scale—the least width reasonably attributable to any gap—of a length-L MBL system. (The δ − introduced earlier equals JN = Jξ > .) The time-t localization length is denoted by ξ (t). A length-L MBL system’s average gap is denoted by hδi(L) . (The average subengine gap hδi, introduced earlier, equals hδi(ξ > ) .) The engine must not lose too much work to undesirable adiabatic transitions. During tuning, energy levels approach each other. Typically, if such a “close encounter” results in an adiabatic transition, many particles shift across the engine. Subengines D E cost of work, on avereffectively interact, consuming a total amount ∼ Nmacro Wadiab age. Undesirable adiabatic transitions must cost less than the average work (4.24) 135 outputted by ideal (independent) subengines: D E cost Wadiab  hWtot i . (4.25) We approximate the left-hand side with ! Work cost 1 undesirable adiab. transition ! Prob. of undesirable adiab. transition × 1 close encounter ! # close encounters × 1 tuning stroke ! Avg. # strokes during which can lose work × . 1 cycle D E cost Wadiab ≈ (4.26) The first factor ∼ hδi. The second factor follows from the Landau-Zener proba2 2 bility PLZ = e−2πJ /v ∼ 1 − Jv that any given close encounter induces a diabatic transition. The Hamiltonian-matrix element that couples the approaching states has the size J ∼ J1.5ξ > . The 1.5ξ > encodes nearest-neighbor subengines’ isolation: Information should not propagate from the left-hand side of one subengine rightward, across a distance 1.5ξ > , to the neighbor’s center. We estimate the third factor in Eq. (4.26) as hδihδi (1.5) . This 1.5 has the same origin as the 1.5 in the J1.5ξ > . The b final factor in Eq. (4.26) ∼ W hδi , the fraction of the cycles that would, in the absence of undesirable transitions, output Wtot > 0. Upon substituting into Eq. (4.26), we substitute into Ineq. (4.25). The right-hand side ∼ Wb [Eq. (4.12)]. Solving for v yields v  (J1.5ξ > ) 2 hδi hδi(1.5ξ > ) ∼ E 2 e−3ξ > /ξ (t) 2−2.5ξ > (4.27) (4.28) Upper bound on v from the work cost hWdiab i of undesirable diabatic transitions: Tuning at a finite speed v > 0 induces diabatic transitions, (Sec. 4.2). Diabatic hops cost a subengine an amount hWdiab i of work per cycle, on average. For D E adiab the average work outputted by one ideal subengine, clarity, we relabel as Wtot D E adiab upper-bounds tuned adiabatically, per cycle. The requirement hWdiab i  Wtot v (Suppl. Mat. C.4). 136 When the engine is shallowly localized, APT transitions dominate hWdiab i [Eq. (4.15)]. They pose little risk if the speed is small, compared to the typical gap: v  hδi2 ∼ E2 ∼ E 2 2−2ξ > . 2 N (4.29) The third expression follows from (i) the text below Eq. (4.12) and (ii) the sub√ dominance of N in our scaling analysis. The final expression approximates hδi2 because the tuning rearranges particles across each subengine, across a distance L ∼ ξ. Such rearrangements induce the energy changes in (4.22). When the engine is very localized, fractional-Landau-Zener transitions dominate 2 2 hWdiab i. Equation (??) approximates, under  ≈ 13 , to hWfrac-LZ i ∼ v(W(δ −)5) + b 1 3 Wb . This work cost must be far less than the work hWtot i extracted adiabatically: hWfrac-LZ i  hWtot i. Solving for v yields v 1 (Wb ) 3 ∼ 3 e ξ > /ξ < 2−2ξ > E 2 . δ− 10 (4.30) The final expression follows if Wb ∼ hδi 10 . Both upper bounds, (4.29) and (4.30), lie above the lower bound (4.28), in an illustrative example in which ξ > = 12, ξ < = 1, and ξ (t) ∼ ξ > . Lower bound on the cycle time τcycle from cold thermalization: Thermalization with the cold bath (stroke 2) bounds τcycle more stringently than the Hamiltonian tunings do. The reasons are (1) the slowness with which MBL thermalizes and (2) the restriction Wb  hδi on the cold-bath bandwidth. We elaborate after introducing our cold-thermalization model (see [1, App. I] for details). We envision the cold bath as a bosonic system that couples to the engine locally, as via the Hamiltonian Hint = g Z Wb /ξ > −Wb /ξ > dω NX macro  c†j c j+1 + h.c.  bω + bω†  j=1 × δ(h0|c j Hmacro (τ)c†j+1 |0i − ω) . (4.31) The coupling strength is denoted by g. c j and c†j denote the annihilation and creation of a fermion at site j. Hmacro (t) denotes the Hamiltonian that would govern the engine at time t in the bath’s absence. Cold thermalization lasts from t = τ to t = τ0 (Fig. ??). bω and bω† represent the annihilation and creation of a frequency-ω boson in the bath. The Dirac delta function is denoted by δ(.). 137 The bath couples locally, e.g., to pairs of nearest-neighbor spins. This locality prevents subengines from interacting with each other much through the bath. The bath can, e.g., flip spin j upward while flipping spin j + 1 downward. These flips likely change a subengine’s energy by an amount E. The bath can effectively absorb only energy quanta of size ≤ Wb from any subengine. The cap is set by the bath’s speed of sound [77], which follows from microscopic parameters in the bath’s Hamiltonian [78]. The rest of the energy emitted during the spin flips, |E−Wb |, is distributed across the subengine as the intrinsic subengine Hamiltonian flips more spins. Let τth denote the time required for stroke 2. We estimate τth from Fermi’s Golden Rule, Γf i = 2π |h f |V |ii| 2 µbath . ~ (4.32) Cold thermalization transitions the engine from an energy level |ii to a level | f i. The bath has a density of states µbath ∼ 1/Wb . We estimate the matrix-element size |h f |V |ii| as follows. Cold thermalization transfers energy Ei f ∼ Wb from the subengine to the bath. Wb is very small. Hence the energy change rearranges particles across a large distance L  ξ = ξ < , due to local level correlations (4.21). V nontrivially transforms just a few subengine sites. Such a local operator rearranges particles across a large distance L at a rate that scales as (4.21), Ee−L/ξ 2−L ∼ δ − . Whereas E sets the scale of the level repulsion δ − , g sets the scale of |h f |V |ii|. The correlation length ξ = ξ < during cold thermalization. We approximate L with the subengine length ξ > . Hence |h f |V |ii| ∼ gδE− . We substitute into Eq. (4.32). The transition rate Γf i = τ1th . Inverting yields τcycle ∼ τth ∼ Wb E gδ − !2 . (4.33) To bound τcycle , we must bound the coupling g. The interaction is assumed to be Markovian: Information leaked from the engine dissipates throughout the bath quickly. Bath correlation functions must decay much more quickly than the coupling transfers energy. If τbath denotes the correlation-decay time, τbath < g1 . The small-bandwidth bath’s τbath ∼ 1/Wb . Hence g < Wb . This inequality, with Ineq. (4.33), implies τcycle = τth > E2 10 2ξ > /ξ < 3ξ > ∼ e 2 . E Wb (δ − ) 2 (4.34) 138 The final expression follows if Wb ∼ hδi 10 . Like Markovianity, higher-order processes bound τth . Higher-order processes occur at rates set by g a , wherein a > 1. Such processes transfer energy E > Wb between the engine and the cold bath. These transfers must be suppressed. The resulting bound on τth is less stringent than Ineq. (4.34) (Suppl. Mat. C.3). 4.3 Numerical simulations The engine can be implemented with a disordered Heisenberg model. A similar model’s MBL phase has been realized with cold atoms [4]. We numerically simulated a 1D mesoscale chain of N = 12 spin- 12 degrees of freedom, neglecting dynamical effects during strokes 1 and 3 (the Hamiltonian tunings). The chain evolves under the Hamiltonian # " NX −1 N X E z σ j · σ j+1 + h(αt ) hjσj . Hsim (t) = Q(h(αt )) j=1 j=1 (4.35) Equation (4.35) describes spins equivalent to interacting spinless fermions. Energies are expressed in units of E, the average per-site energy density. For γ = x, y, z, γ the γ th Pauli operator that operates nontrivially on the j th site is denoted by σ j . The Heisenberg interaction σ j · σ j+1 encodes nearest-neighbor hopping and repulsion. The tuning parameter αt ∈ [0, 1] determines the phase occupied by Hsim (t). The site- j disorder potential depends on a random variable h j distributed uniformly across [−1, 1]. The disorder strength h(αt ) varies as h(αt ) = αt hGOE +(1−αt )hMBL . When αt = 0, the disorder is weak, h = hGOE , and the engine occupies the ETH phase. When αt = 1, the disorder is strong, h = hMBL  hGOE , and the engine occupies the MBL phase. The normalization factor Q(h(αt )) preserves the width of the density of states (DOS) and so hδi. Q(h(αt )) prevents the work extractable via change of bandwidth from polluting the work extracted with help from level statistics, (Sec. 4.2). Q(h(αt )) is defined and calculated in Suppl. Mat. C.5. We simulated the spin chain using exact diagonalization, detailed in Suppl. Mat. C.5. The ETH-side field had a magnitude h(0) = 2.0, and the MBL-side field had a magnitude h(1) = 20.0. These h(αt ) values fall squarely on opposite sides of the MBL transition at h ≈ 7. 139 1.4 1.0 1.2 0.9 1.0 hWtoti/hδi 0.8 ηMBL 0.8 0.6 0.6 0.4 0.5 0.2 0.0 0.0 0.7 0.2 0.4 0.6 0.8 1.0 1.2 0.4 0.0 (a) 0.2 0.4 0.6 0.8 1.0 1.2 Wb/hδi Wb/hδi (b) Figure 4.5: Average per-cycle power hWtot i (top) and efficiency η MBL (bottom) as functions of the cold-bath bandwidth Wb : Each red dot represents an average over 1,000 disorder realizations of the random-field Heisenberg Hamiltonian (4.35). The slanted blue lines represent the analytical predictions (4.12) and (4.14). When Wb  hδi (in the gray shaded region), hWtot i and η MBL vary linearly with Wb , as predicted. Adiabatic engine performance We first simulated the evolution of each state in strokes 1 and 3 as though the Hamiltonian were tuned adiabatically. We index the energies E j (αt ) from least to greatest at each instant: E j (αt ) < Ek (αt ) ∀j < k. Let ρ j denote the state’s weight on eigenstate j of the pre-tuning Hamiltonian H (αt = 0). The engine ends the stroke with weight ρ j on eigenstate j of the post-tuning Hamiltonian H (1). The main results appear in Fig. 4.5. Figure 4.5a shows the average work extracted per cycle, hWtot i; and Fig. 4.5b shows the efficiency, η MBL . In these simulations, the baths had the extreme temperatures TH = ∞ and TC = 0. This limiting case elucidates the Wb -dependence of hWtot i and of η MBL : Disregarding finite-temperature corrections, on a first pass, builds intuition. Finitetemperature numerics appear alongside finite-temperature analytical calculations in Suppl. Mat. C.1. Figure 4.5 shows how the per-cycle power and the efficiency depend on the coldbath bandwidth Wb . As expected, hWtot i ≈ Wb . The dependence’s linearity, and the unit proportionality factor, agree with Eq. (4.12). Also as expected, the efficiency Wb . The linear dependence declines as the cold-bath bandwidth rises: η MBL ≈ 1 − 2hδi and the proportionality factor agree with Eq. (4.14). The gray columns in Fig. 4.5 highlight the regime in which the analytics were b performed, where W hδi  1. If the cold-bath bandwidth is small, Wb . hδi, the analytics-numerics agreement is close. But the numerics agree with the analytics 140 even outside this regime. If Wb & hδi, the analytics slightly underestimate η MBL : The simulated engine operates more efficiently than predicted. To predict the numerics’ overachievement, one would calculate higher-order corrections in Suppl. Mat. C.1: One would Taylor-approximate to higher powers, modeling subleading physical processes. Such processes include the engine’s dropping across a chain of three small gaps δ01 , δ02 , δ03 < Wb during cold thermalization. The error bars are smaller than the numerical-data points. Each error bar represents tot i the error in the estimate of a mean (of hWtot i or of η MBL := 1 − hW hQin i ) over 1,000 dis√ order realizations. Each error bar extends a distance (sample standard deviation)/ # realizations above and below that mean. Diabatic engine performance We then simulated the evolution of each state in strokes 1 and 3 as though the Hamiltonian were tuned at finite speed for 8 sites. (We do not simulate larger diabatic engines: That our upper bounds on tuning speed for a mesoscopic engine go as powers of the level spacing hδi ∼ 2−L means that these simulations quickly become slow to run.) We simulate a stepwise tuning, taking α(t) = (δt) bvt/(δt)c . (4.36) This protocol is considerably more violent than the protocols we treat analytically: In our estimates, we leave v general, but we always assume that it is finite. In the numerics, we tune by a series of sudden jumps. (We do this for reasons of numerical convenience.) We work at βC = ∞ and βH = 0, to capture the essential physics without the added confusion of finite-temperature corrections. In this case, we expect the engine to work well enough—to output a finite fraction of its adiabatic work output—for (Wb ) 3 (4.37) v δ− [c.f. Eq. (4.30)]. In Fig. 4.6, we show work output as a function of speed. Despite the simulated 3 protocol’s violence, Wtot is a finite fraction of its adiabatic value for v . (Wδ −b ) and 3 even for v > (Wδ −b ) : Our engine is much less sensitive to tuning speed than our crude diabatic-corrections bounds suggest. These numerics not only confirm the validity of our analytics, but also indicate the robustness of the MBL Otto engine to changes in the tuning protocol. 141 0.007 0.006 WT OT 0.005 0.004 0.003 0.002 0.001 0.000 10−3 10−2 10−1 100 vδ−/Wb3 101 102 Figure 4.6: Average per-cycle work as a function of tuning speed for 995 disorder realizations of the random-field Heisenberg Hamiltonian (4.35) at system size L = 8 (red dots), compared to the analytical estimate (4.12) for the adiabatic work output (blue line). Each error bar represents the √ error in the estimate of the mean, computed as (sample standard deviation)/ (# realizations). 4.4 Order-of-magnitude estimates How well does the localized engine perform? We estimate its power and power density, then compare the values with three competitors’ performances. Localized engine: Localization has been achieved in solid-state systems.8 Consider silicon doped with phosphorus [44]. A distance of ∼ 10 nm may separate phosphorus impurities. Let our engine cycle’s shallowly localized regime have a localization length of ξ > ∼ 10 sites, or 100 nm. The work-outputting degrees of freedom will be electronic. The localized states will correspond to energies E ∼ 1 eV.   2 Each subengine’s half-filling Hilbert space has dimensionality N = 10 5 ∼ 10 . √ Hence each subengine has an effective average gap hδi ∼ ENN ∼ 110eV2 ∼ 10 meV. The cold-bath bandwidth must satisfy hδi  Wb . We set Wb to be an order of magnitude down from hδi: Wb ∼ 1 meV ∼ 10 K. The cold-bath bandwidth approximates the work outputted by one subengine per cycle:9 hWtot i ∼ Wb ∼ 1 meV [Eq. (4.12)]. What volume does a localized subengine fill? Suppose that the engine is threedimensional (3D).10 A little room should separate the subengines. Classical-control 8 This localization is single-particle, or Anderson [72], rather than many-body. Section 4.5 extends the MBL Otto engine to an Anderson-localized Otto engine. 9 The use of semiconductors would require corrections to our results. (Dipolar interactions would couple the impurities’ spins. Energy eigenfunctions would decay as power laws with distance.) But we aim for just a rough estimate. 10 Until now, we have supposed that the engine is 1D. Anderson localization, which has been realized in semiconductors, exists in all dimensionalities. Yet whether MBL exists in dimensionalities D > 1 remains an open question. Some evidence suggests that MBL exists in D ≥ 2 [7, 9, 11]. But attributing a 3D volume to the engine facilitates comparisons with competitors. We imagine 10-nm- 142 equipment requires more room. Also, the subengine needs space to connect to the baths. We therefore associate each subengine with a volume of V ≈ (100 nm) 3 . The last element needed is the cycle time, τcycle . We choose for δ − to be a little smaller than Wb —of the same order: δ − ∼ Wb ∼ 1 meV. In the extreme case −15 2 s)(1 eV) 2 ~E 2 allowed by Ineq. (4.34), τcycle ∼ W ~E ∼ (W ∼ (10 (1eV ∼ 1 µs. (δ ) 2 )3 meV) 3 b − b −16 W. b The localized engine therefore operates with a power P ∼ τW ∼ 11meV µs ≈ 10 cycle Interestingly, this P is one order of magnitude greater than a flagellar motor’s [79] power, according to our estimates. We can assess the engine by calculating not only its power, but also its power den10 −16 W 3 sity. The localized engine packs a punch at P V ∼ (10 −7 m) 3 = 100 kW/m . Car engine: The quintessential Otto engine powers cars. A typical car engine 100 kW outputs P ∼ 100 horsepower ∼ 100 kW . A car’s power density is P V ∼ 100 L = 1 MW/ m3 (wherein L represents liters). The car engine’s P V exceeds the MBL engine’s by only an order of magnitude, according to these rough estimates. Array of quantum dots: MBL has been modeled with quasilocal bits [2, 80]. A string of ideally independent bits or qubits, such as quantum dots, forms a natural competitor. A qubit Otto engine’s gap is shrunk, widened, and shrunk [81–85]. A realization could consist of double quantum dots [86, 87]. The scales in [86, 87] suggest that a quantum-dot engine could output an amount Wtot ∼ 10 meV of work per cycle. We approximate the cycle time τcycle with the spin relaxation time: τcycle ∼ 1 µs. (The energy eigenbasis need not rotate, unlike for the MBL engine. Hence diabatic hops do not lower-bound the ideal-quantum-dot τcycle .) The −15 W. The quantum-dot engine’s power tot ∼ 101meV power would be P ∼ τWcycle µs ∼ 10 exceeds the MBL engine’s by an order of magnitude. However, the quantum dots must be separated widely. Otherwise, they will interact, as an ETH system. (See [61] for disadvantages of interactions in another quantum thermal machine. Spin-spin couplings cause “quantum friction,” limiting the temperatures to which a refrigerator can cool.) We compensate by attributing a volume 3 V ∼ (1 µm) 3 to each dot. The power density becomes P V ∼ 1 kW/m , two orders of magnitude less than the localized engine’s. Localization naturally implies near independence of the subengines. long 1D strings of sites. Strings are arrayed in a plane, separated by 10 nm. Planes are stacked atop each other, separated by another 10 nm. 143 4.5 Formal comparisons with competitor engines The Otto cycle can be implemented with many media. Why use MBL? How does the “athermality” of MBL level correlations advantage our engine? We compare our engine with five competitors: an ideal thermodynamic gas, a set of ideally noninteracting qubits (e.g., quantum dots), a many-body system whose bandwidth is compressed and expanded, an MBL engine tuned between equal-disorder-strength disorder realizations, and an Anderson-localized Otto engine. An MBL Otto engine whose cold bath has an ordinary bandwidth Wb > hδi is discussed in Suppl. Mat. C.6. Ideal-gas Otto engine The conventional thermodynamic Otto engine consists of an ideal gas. Its efficiency, η Otto , approximately equals the efficiency η MBL of an ideal mesoscopic MBL enb gine: η Otto ≈ η MBL . More precisely, for every MBL parameter ratio W hδi , and for every ideal-gas heat-capacity ratio γ = CCPv , there exists a compression ratio r := VV21 Wb 1 = 1 − 2hδi ≈ η MBL . such that η Otto = 1 − r γ−1 However, scaling up the mesoscopic MBL engine to the thermodynamic limit requires a lower bound on the tuning speed v (Sec. 4.2). The lower bound induces b diabatic jumps that cost work hWdiab i, detracting from η MBL by an amount ∼ W hδi (Suppl. Mat. C.1). (For simplicity, we have assumed that TC = 0 and TH = ∞ and have kept only the greatest terms.) The ideal-gas engine suffers no such diabatic jumps. However, the MBL engine’s hWdiab i is suppressed in small parameters δ− b √v (W hδi , hδi , hδi  1). Hence the thermodynamically large MBL engine’s efficiency true ≈ η lies close to the ideal-gas engine’s efficiency: η MBL Otto . Moreover, the thermodynamically large MBL engine may be tuned more quickly than the ideal-gas engine. The MBL engine is tuned nearly quantum-adiabatically. The ideal-gas engine is tuned quasistatically. The physics behind the quantum adiabatic theorem differs from the physics behind the quasistatic condition. Hence the engines’ speeds v are bounded with different functions of the total system size Nmacro . The lower bound on the MBL engine’s v remains constant as Nmacro grows: v  E 2 e−3ξ > /ξ (t) 2−2.5ξ > [Ineq. (4.27)]. Rather than Nmacro , the fixed localization length ξ > governs the bound on v.11 In contrast, we expect an ideal-gas engine’s 1 . The quasistatic condition requires that the engine respeed to shrink: v ∼ Nmacro main in equilibrium. The agent changes the tuning parameter α by a tiny amount 11 Cold thermalization of the MBL engine lasts longer than one tuning stroke: τ  E (Sec. 4.2). th v But even τth does not depend on Nmacro . 144 ∆α, waits until the gas calms, then changes α by ∆α. The changes are expected to propagate as waves with some speed c. The wave reaches the engine’s far edge in a c time ∼ Nmacro c . Hence v < E Nmacro . However, the ideal-gas engine is expected to output more work per unit volume than the MBL engine. According to our order-of-magnitude estimates (Sec. 4.4), the 3 ideal-gas engine operates at a power density of P V ∼ 1 MW/m ; and the localized 3 engine, at P V ∼ 100 kW/m . An order of magnitude separates the estimates. Quantum-dot engine Section 4.4 introduced the quantum-dot engine, an array of ideally independent bits or qubits. We add to the order-of-magnitude analysis two points about implementations’ practicality. The MBL potential’s generic nature offers an advantage. MBL requires a random disorder potential {h(αt )h j }, e.g., a “dirty sample,” a defect-riddled crystal. This “generic” potential contrasts with the pristine background required by quantum dots. Imposing random MBL disorder is expected to be simpler. On the other hand, a quantum-dot engine does not necessarily need a small-bandwidth cold bath, Wb  hδi. Bandwidth engine Imagine eliminating the scaling factor Q(h(αt )) from the Hamiltonian (4.35). The energy band is compressed and expanded as the disorder strength h(αt ) is ramped down and up. The whole band, rather than a gap, contracts and widens as in Fig. 4.2, between a size ∼ E Nmacro h(α0 ) and a size ∼ E Nmacro h(α1 )  E Nmacro h(α0 ). The engine can remain in one phase throughout the cycle. The cycle does not benefit from the “athermality” of local level correlations. Furthermore, this accordion-like motion requires no change of the energy eigenbasis’s form. Tuning may proceed quantum-adiabatically: v ≈ 0. The ideal engine suffers no diabatic jumps, losing hWdiab imacro = 0. But this engine is impractical: Consider any perturbation V that fails to commute with the ideal Hamiltonian H (t): [V, H (t)] , 0. Stray fields, for example, can taint an environment. As another example, consider cold atoms in an optical lattice. The disorder strength is ideally Eh(αt ). One can strengthen the disorder by strengthening the lattice potential Ulattice . Similarly, one can raise the hopping frequency (ideally E) by raising the pressure p. Strengthening Ulattice and p while achieving t) = h(αt ) requires fine control. If the ratio the ideal disorder-to-hopping ratio Eh(α E 145 changes from h(αt ), the Hamiltonian H (t) acquires a perturbation V that fails to commute with other terms. This V can cause diabatic jumps that cost work hWdiab imacro . Jumps suppress the √ scaling of the average work outputted per cycle by a factor of Nmacro (Suppl. Mat. C.6). The MBL Otto engine may scale more robustly: The net work extracted scales as Nmacro [Eq. (4.24)]. Furthermore, diabatic jumps cost work hWdiab imacro √ v suppressed small parameters such as hδi . Engine tuned between equal-disorder-strength disorder realizations The disorder strength h(αt ) in Eq. (4.35) would remain  1 and constant in t, while the random variables h j would change. Let S̃ denote this constant-h(αt ) engine, and let S denote the MBL engine. S̃ takes less advantage of MBL’s “athermality,” as S̃ is not tuned between level-repelling and level-repulsion-free regimes. Yet S̃ outputs the amount hWtot i of work outputted by S per cycle, on average. Because Wb is small, cold thermalization drops S̃ across only small gaps δ0  hδi. S̃ traverses a trapezoid, as in Fig. 4.2, in each trial. However, the MBL engine has two advantages: greater reliability and fewer worst-case (negative-work-outputted) trials. Both the left-hand gap δ and the right-hand gap δ0 traversed by S̃ are Poissondistributed. Poisson-distributed gaps more likely assume extreme values than GOE(E) (E) distributed gaps: PMBL (δ) > PGOE (δ) if δ ∼ 0 or δ  hδi [46]. The left-hand gap δ traversed by S is GOE-distributed. Hence the Wtot outputted by S̃ more likely assumes extreme values than the Wtot outputted by S. The greater reliability of S may suit S better to “one-shot statistical mechanics” [17, 18, 20, 21, 23, 24, 88– 93]. In one-shot theory, predictability of the work Wtot extractable in any given trial serves as a resource. S suffers fewer worst-case trials than S̃. We define as worst-case a trial in which the engine outputs net negative work, Wtot < 0. Consider again Fig. 4.2. Consider a similar figure that depicts the trapezoid traversed by S̃ in some trial. The left-hand (E) gap, δ, is distributed as the right-hand gap, δ0, is, according to PMBL (δ). Hence δ 0 0 has a decent chance of being smaller than δ : δ < δ . S̃ would output Wtot < 0 in such a trial. We estimate worst-case trials’ probabilities in Suppl. Mat. C.6. Each trial un 2 b of yielding dergone by one constant-h(αt ) subengine has a probability ∼ W hδi Wtot < 0 . An MBL subengine has a worst-case probability one order of magnitude 146  W 3 lower: ∼ hδib . Hence the constant-h(αt ) engine illustrates that local MBL level correlations’ athermality suppresses worst-case trials and enhances reliability. Anderson-localized engine Anderson localization follows from removing the interactions from MBL (Suppl. Mat. C.2). One could implement our Otto cycle with an Anderson insulator because Anderson Hamiltonians exhibit Poissonian level statistics (4.1). But strokes 1 and 3 would require the switching off and on of interactions. Tuning the interaction, as well as the disorder-to-interaction ratio, requires more effort than tuning just the latter. Also, particles typically interact in many-body systems. MBL particles interact; Anderson-localized particles do not. Hence one might eventually expect less difficulty in engineering MBL engines than in engineering Anderson-localized engines. 4.6 Outlook The realization of thermodynamic cycles with quantum many-body systems was proposed very recently [36, 38, 39, 94–98]. MBL offers a natural platform, due to its “athermality” and to athermality’s resourcefulness in thermodynamics. We designed an Otto engine that benefits from the discrepancy between many-bodylocalized and “thermal” level statistics. The engine illustrates how MBL can be used for thermodynamic advantage. Realizing the engine may provide a near-term challenge for existing experimental set-ups. Possible platforms include cold atoms [4, 5, 7, 8, 11]; nitrogen-vacancy centers [9]; ion traps [10]; and doped semiconductors [44], for which we provided order-of-magnitude estimates. Realizations will require platform-dependent corrections due to, e.g., variable-range hopping induced by particle-phonon interactions. As another example, semiconductors’ impurities suffer from dipolar interactions. The interactions extend particles’ wave functions from decaying exponentially across space to decaying as power laws. Reversing the engine may pump heat from the cold bath to the hot, lowering the cold bath’s temperature. Low temperatures facilitate quantum computation and lowtemperature experiments. An MBL engine cycle might facilitate state preparation and coherence preservation in quantum many-body experiments. Experiments motivate explicit modeling of the battery. We have defined as work the energy outputted during Hamiltonian tunings. A work-storage device, or battery, 147 must store this energy. We have refrained from specifying the battery’s physical form, using an implicit battery model. An equivalent explicit battery model could depend on the experimental platform. Quantum-thermodynamics batteries have been modeled abstractly with ladder-like Hamiltonians [99]. An oscillator battery for our engine could manifest as a cavity mode. MBL is expected to have thermodynamic applications beyond this Otto engine. A localized ratchet, which leverages information to transform heat into work, is under investigation. The paucity of transport in MBL may have technological applications beyond thermodynamics. Dielectrics, for example, prevent particles from flowing in certain directions. Dielectrics break down in strong fields. To survive, a dielectric must insulate well—as does MBL. In addition to suggesting applications of MBL, this work identifies an opportunity within quantum thermodynamics. Athermal quantum states (e.g., ρ , e−H/T /Z) are usually regarded as resources in quantum thermodynamics [16, 17, 19, 20, 22–27, 100–103]. Not only athermal states, we have argued, but also athermal energy-level statistics, offer thermodynamic advantages. Generalizing the quantumthermodynamics definition of “resource” may expand the set of goals that thermodynamic agents can achieve. Optimization offers another theoretical opportunity. We have shown that the engine works, but better protocols could be designed. For example, we prescribe nearly quantum-adiabatic tunings. Shortcuts to adiabaticity (STA) avoid both diabatic transitions and exponentially slow tunings [29, 52, 61, 73–75]. STA have been used to reduce other quantum engines’ cycle times [29, 52, 75]. STA might be applied to the many-body Otto cycle, after being incorporated in to MBL generally. References [1] N. Yunger Halpern, C. D. White, S. Gopalakrishnan, and G. Refael, ArXiv e-prints (2017), 1707.07008v1. [2] D. A. Huse, R. Nandkishore, and V. Oganesyan, Phys. Rev. B 90, 174202 (2014). [3] J. A. Kjäll, J. H. Bardarson, and F. Pollmann, Phys. Rev. Lett. 113, 107204 (2014). [4] M. Schreiber et al., Science 349, 842 (2015). 148 [5] S. S. Kondov, W. R. McGehee, W. Xu, and B. DeMarco, Phys. Rev. Lett. 114, 083002 (2015). [6] M. Ovadia et al., Scientific Reports 5, 13503 EP (2015), Article. [7] J.-y. Choi et al., Science 352, 1547 (2016). [8] H. P. Lüschen et al., Phys. Rev. X 7, 011034 (2017). [9] G. Kucsko et al., ArXiv e-prints (2016), 1609.08216. [10] J. Smith et al., Nat Phys 12, 907 (2016), Letter. [11] P. Bordia et al., ArXiv e-prints (2017), 1704.03063. [12] J. M. Deutsch, Phys. Rev. A 43, 2046 (1991). [13] M. Srednicki, Phys. Rev. E 50, 888 (1994). [14] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007). [15] R. Nandkishore and D. A. Huse, Annual Review of Condensed Matter Physics 6, 15 (2015), 1404.0686. [16] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and T. Beth, Int. J. Theor. Phys. 39, 2717 (2000). [17] O. C. O. Dahlsten, R. Renner, E. Rieper, and V. Vedral, New J. Phys. 13, 053015 (2011). [18] J. Åberg, Nat. Commun. 4, 1925 (2013). [19] F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Physical Review Letters 111, 250404 (2013). [20] M. Horodecki and J. Oppenheim, Nat. Commun. 4, 1 (2013). [21] D. Egloff, O. C. O. Dahlsten, R. Renner, and V. Vedral, New Journal of Physics 17, 073001 (2015). [22] J. Goold, M. Huber, A. Riera, L. del Río, and P. Skrzypczyk, Journal of Physics A: Mathematical and Theoretical 49, 143001 (2016). [23] G. Gour, M. P. Müller, V. Narasimhachar, R. W. Spekkens, and N. Yunger Halpern, Physics Reports 583, 1 (2015), The resource theory of informational nonequilibrium in thermodynamics. [24] N. Yunger Halpern and J. M. Renes, Phys. Rev. E 93, 022126 (2016). [25] N. Yunger Halpern, Journal of Physics A: Mathematical and Theoretical 51, 094001 (2018), 10.1088/1751-8121/aaa62f. 149 [26] S. Deffner, J. P. Paz, and W. H. Zurek, Phys. Rev. E 94, 010103 (2016). [27] H. Wilming and R. Gallego, ArXiv e-prints (2017), 1701.07478. [28] J. E. Geusic, E. O. Schulz-DuBios, and H. E. D. Scovil, Phys. Rev. 156, 343 (1967). [29] A. del Campo, J. Goold, and M. Paternostro, Scientific Reports 4 (2014). [30] N. Brunner et al., Phys. Rev. E 89, 032115 (2014). [31] F. C. Binder, S. Vinjanampathy, K. Modi, and J. Goold, New Journal of Physics 17, 075015 (2015). [32] M. P. Woods, N. Ng, and S. Wehner, ArXiv e-prints (2015), 1506.02322. [33] D. Gelbwaser-Klimovsky and A. Aspuru-Guzik, Journal of Physical Chemistry Letters 6, 3477 http://dx.doi.org/10.1021/acs.jpclett.5b01404, PMID: 26291720. The (2015), [34] Q. Song, S. Singh, K. Zhang, W. Zhang, and P. Meystre, ArXiv e-prints (2016), 1607.00119. [35] H. Terças, S. Ribeiro, M. Pezzutto, and Y. Omar, ArXiv e-prints (2016), 1604.08732. [36] M. Perarnau-Llobet, A. Riera, R. Gallego, H. Wilming, and J. Eisert, New Journal of Physics 18, 123035 (2016). [37] R. Kosloff and Y. Rezek, Entropy 19, 136 (2017). [38] J. Lekscha, H. Wilming, J. Eisert, and R. Gallego, ArXiv e-prints (2016), 1612.00029. [39] J. Jaramillo, M. Beau, and A. del Campo, New Journal of Physics 18, 075019 (2016). [40] D. Basko, I. Aleiner, and B. Altshuler, Annals of Physics 321, 1126 (2006). [41] V. Oganesyan and D. A. Huse, Phys. Rev. B 75, 155111 (2007). [42] A. Pal and D. A. Huse, Phys. Rev. B 82, 174411 (2010). [43] M. Serbyn and J. E. Moore, Phys. Rev. B 93, 041424 (2016). [44] B. Kramer and A. MacKinnon, Reports on Progress in Physics 56, 1469 (1993). [45] U. Sivan and Y. Imry, Phys. Rev. B 35, 6074 (1987). [46] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, Advances in Physics 65, 239 (2016), http://dx.doi.org/10.1080/00018732.2016.1198134. 150 [47] Y. Imry and S.-k. Ma, Phys. Rev. Lett. 35, 1399 (1975). [48] S. V. Syzranov, A. V. Gorshkov, and V. Galitski, ArXiv e-prints (2017), 1704.08442. [49] D. Quattrochi, The internal combustion engine (otto cycle), 2006. [50] M. O. Scully, Phys. Rev. Lett. 88, 050602 (2002). [51] O. Abah et al., Phys. Rev. Lett. 109, 203006 (2012). [52] J. Deng, Q.-h. Wang, Z. Liu, P. Hänggi, and J. Gong, Phys. Rev. E 88, 062122 (2013). [53] Y. Zheng and D. Poletti, Phys. Rev. E 90, 012145 (2014). [54] B. Karimi and J. P. Pekola, Phys. Rev. B 94, 184503 (2016), 1610.02776. [55] S. Vinjanampathy and J. Anders, Contemporary Physics 0, 1 (0), http://dx.doi.org/10.1080/00107514.2016.1201896. [56] S.-Z. Lin and S. Hayami, Phys. Rev. B 93, 064430 (2016). [57] P. Corboz, Phys. Rev. B 94, 035133 (2016). [58] S. Gopalakrishnan, M. Knap, and E. Demler, Phys. Rev. B 94, 094201 (2016). [59] R. Kosloff and T. Feldmann, Phys. Rev. E 65, 055102 (2002). [60] T. D. Kieu, Phys. Rev. Lett. 93, 140403 (2004). [61] R. Kosloff and T. Feldmann, Phys. Rev. E 82, 011134 (2010). [62] S. Çakmak, F. Altintas, A. Gençten, and Ö. E. Müstecaplıoğlu, The European Physical Journal D 71, 75 (2017). [63] C. De Grandi and A. Polkovnikov, Adiabatic Perturbation Theory: From Landau-Zener Problem to Quenching Through a Quantum Critical Point, in Lecture Notes in Physics, Berlin Springer Verlag, edited by A. K. K. Chandra, A. Das, and B. K. K. Chakrabarti, , Lecture Notes in Physics, Berlin Springer Verlag Vol. 802, p. 75, 2010, 0910.2236. [64] D. A. Huse, R. Nandkishore, F. Pietracaprina, V. Ros, and A. Scardicchio, Phys. Rev. B 92, 014203 (2015). [65] A. De Luca and A. Rosso, Phys. Rev. Lett. 115, 080401 (2015). [66] E. Levi, M. Heyl, I. Lesanovsky, and J. P. Garrahan, Phys. Rev. Lett. 116, 237203 (2016). 151 [67] M. H. Fischer, M. Maksymenko, and E. Altman, Phys. Rev. Lett. 116, 160401 (2016). [68] A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. Lett. 88, 186802 (2002). [69] S. Gopalakrishnan and R. Nandkishore, Phys. Rev. B 90, 224203 (2014). [70] S. A. Parameswaran and S. Gopalakrishnan, Phys. Rev. B 95, 024201 (2017). [71] S. Shevchenko, S. Ashhab, and F. Nori, Physics Reports 492, 1 (2010). [72] P. W. Anderson, Phys. Rev. 109, 1492 (1958). [73] X. Chen et al., Phys. Rev. Lett. 104, 063002 (2010). [74] E. Torrontegui et al., Advances in Atomic Molecular and Optical Physics 62, 117 (2013), 1212.6343. [75] O. Abah and E. Lutz, ArXiv e-prints (2016), 1611.09045. [76] V. Khemani, R. Nandkishore, and S. L. Sondhi, Nature Physics 11, 560 (2015), 1411.2616. [77] H. Kim and D. A. Huse, Phys. Rev. Lett. 111, 127205 (2013). [78] E. Lieb and D. Robinson, Commun. Math. Phys. 28, 251 (1972). [79] M. T. Brown, Bacterial flagellar motor: Biophysical studies, in Encyclopedia of Biophysics, edited by G. C. K. Roberts, pp. 155–155, Springer Berlin Heidelberg, Berlin, Heidelberg, 2013. [80] A. Chandran, I. H. Kim, G. Vidal, and D. A. Abanin, Phys. Rev. B 91, 085425 (2015). [81] E. Geva and R. Kosloff, The Journal of Chemical Physics 96, 3054 (1992), http://dx.doi.org/10.1063/1.461951. [82] E. Geva and R. Kosloff, The Journal of Chemical Physics 97, 4398 (1992), http://dx.doi.org/10.1063/1.463909. [83] T. Feldmann, E. Geva, R. Kosloff, and P. Salamon, American Journal of Physics 64, 485 (1996), http://dx.doi.org/10.1119/1.18197. [84] J. He, J. Chen, and B. Hua, Phys. Rev. E 65, 036145 (2002). [85] G. Alvarado Barrios, F. Albarrán-Arriagada, F. A. Cárdenas-López, G. Romero, and J. C. Retamal, ArXiv e-prints (2017), 1707.05827. [86] J. R. Petta et al., Science 309, 2180 http://science.sciencemag.org/content/309/5744/2180.full.pdf. (2005), 152 [87] J. Petta et al., Physica E: Low-dimensional Systems and Nanostructures 34, 42 (2006), Proceedings of the 16th International Conference on Electronic Properties of Two-Dimensional Systems (EP2DS-16). [88] L. del Río, J. Aberg, R. Renner, O. Dahlsten, and V. Vedral, Nature 474, 61 (2011). [89] O. C. O. Dahlsten, Entropy 15, 5346 (2013). [90] F. Brandão, M. Horodecki, Woods, N. Ng, J. Oppenheim, and S. Wehner, 112, 3275 (2015). [91] G. Gour, Phys. Rev. A 95, 062314 (2017). [92] K. Ito and M. Hayashi, ArXiv e-prints (2016), 1612.04047. [93] R. van der Meer, N. H. Y. Ng, and S. Wehner, ArXiv e-prints (2017), 1706.03193. [94] M. Campisi and R. Fazio, Nature Communications 7, 11895 EP (2016), Article. [95] R. Modak and M. Rigol, ArXiv e-prints (2017), 1704.05474. [96] W. Verstraelen, D. Sels, and M. Wouters, Phys. Rev. A 96, 023605 (2017). [97] D. Ferraro, M. Campisi, V. Pellegrini, and M. Polini, ArXiv e-prints (2017), 1707.04930. [98] Y.-H. Ma, S.-H. Su, and C.-P. Sun, 1705.08625. Phys. Rev. E 96, 022143 (2017), [99] P. Skrzypczyk, A. J. Short, and S. Popescu, 1302.2811. ArXiv e-prints (2013), [100] M. Lostaglio, D. Jennings, and T. Rudolph, Nature Communications 6, 6383 (2015), 1405.2188. [101] M. Lostaglio, D. Jennings, and T. Rudolph, New Journal of Physics 19, 043008 (2017). [102] N. Yunger Halpern, P. Faist, J. Oppenheim, and A. Winter, Nature Communications 7, 12051 (2016), 10.1038/ncomms12051. [103] Y. Guryanova, S. Popescu, A. J. Short, R. Silva, and P. Skrzypczyk, Nature Communications 7, 12049 (2016), 1512.01190. 153 Chapter 5 NON-ABELIAN THERMAL STATE: THE THERMAL STATE OF A QUANTUM SYSTEM WITH NONCOMMUTING CHARGES This chapter was published as [1]. Recently reignited interest in quantum thermodynamics has prompted informationtheoretic approaches to fundamental questions. have enjoyed particular interest. [2– 5]. The role of entanglement, for example, has been clarified with canonical typicality [6–9]. Equilibrium-like behaviors have been predicted [10–13] and experimentally observed in integrable quantum gases [14, 15]. Thermodynamic resource theories offer a powerful tool for analyzing fundamental properties of the thermodynamics of quantum systems. Heat exchanges with a bath are modeled with “free states” and “free operations” [16–19]. These resource theories have been extended to model exchanges of additional physical quantities, such as particles and angular momentum [19–23]. A central concept in thermodynamics and statistical mechanics is the thermal state. The thermal state has several important properties. First, typical dynamics evolve the system toward the thermal state. The thermal state is the equilibrium state. Second, consider casting statistical mechanics as an inference problem. The thermal state is the state which maximizes the entropy under constraints on physical quantities [24, 25]. Third, consider the system as interacting with a large bath. The system-and-bath composite occupies a microcanonical state. Physical observables of the composite, such as the total energy and total particle number, have sharply defined values. The system’s reduced state is the thermal state. Finally, in a resource theory, the thermal state is the only completely passive state. No work can be extracted from any number of copies of the thermal state [26, 27]. If a small system exchanges heat and particles with a large environment, the system’s thermal state is a grand canonical ensemble: e− β(H−µN ) /Z. The system’s Hamiltonian and particle number are represented by H and N. β and µ denote the environment’s inverse temperature and chemical potential. The partition function Z normalizes the state. The system-and-bath dynamics conserves the total energy and total particle number. More generally, subsystems exchange conserved quantities, 154 or “charges,” Q j , j = 1, 2, . . . c. To these charges correspond generalized chemical potentials µ j . The µ j ’s characterize the bath. We address the following question. Suppose that the charges fail to commute with each other: [Q j ,Q k ] , 0. What form does the thermal state have? We call this state “the Non-Abelian Thermal State” (NATS). Jaynes applied the Principle of Maximum Entropy to this question [25]. He associated fixed values v j with the charges’ expectation values. He calculated the state that, upon satisfying these constraints, maximizes an entropy. This thermal state has a generalized Gibbs form: γv := 1 − Pcj=0 µ j Q j e , Z (5.1) wherein the the v j ’s determine the µ j ’s. Our contribution is a mathematical, physically justified derivation of the thermal state’s form for systems whose dynamics conserve noncommuting observables. We recover the state (5.1) via several approaches, demonstrating its physical importance. We address puzzles raised in [22, 28] about how to formulate a resource theory in which thermodynamic charges fail to commute. Closely related, independent work was performed by Guryanova et al. [29]. We focus primarily on the nature of passive states. Guryanova et al., meanwhile, focus more on the resource theory for multiple charges and on tradeoffs amongst types of charge extractions. In this paper, we derive the NATS’s form from a microcanonical argument. A simultaneous eigenspace of all the noncommuting physical charges might not exist. Hence we introduce the notion of an approximate microcanonical subspace. This subspace consists of the states in which the charges have sharply defined values. We derive conditions under which this subspace exists. We show that a small subsystem’s reduced state lies, on average, close to γv . Second, we invoke canonical typicality [8, 9]. If the system-and-bath composite occupies a random state in the approximate microcanonical subspace, we argue, a small subsystem’s state likely lies close to the NATS. Typical dynamics are therefore expected to evolve a wellbehaved system’s state towards the NATS. Third, we define a resource theory for thermodynamic exchanges of noncommuting conserved charges. We extend existing resource theories to model the exchange of noncommuting quantities. We show that the NATS is the only possible free state that renders the theory nontrivial: Work cannot be extracted from any number of copies of γv . We show also that the NATS is the only state preserved by free operations. From this preservation, we derive “second laws” that govern state transformations. This work provides a 155 well-rounded, and novelly physical, perspective on equilibrium in the presence of quantum noncommutation. This perspective opens truly quantum avenues in thermodynamics. 5.1 Results Overview We derive the Non-Abelian Thermal State’s form via three routes: from a microcanonical argument, from a dynamical argument built on canonical typicality, and from complete passivity in a resource theory. Details appear in Appendices D.1– D.3. Microcanonical derivation In statistical mechanics, the form e− β(H−µN ) /Z of the grand canonical ensemble is well-known to be derivable as follows. The system of interest is assumed to be part of a larger system. Observables of the composite have fixed values v j . For example, the energy equals E0 , and the particle number equals N0 . The microcanonical ensemble is the whole-system state spread uniformly across these observables’ simultaneous eigenspace. Tracing out the environmental degrees of freedom yields the state e− β(H−µN ) /Z. We derive the NATS’s form similarly. Crucially, however, we adapt the above strategy to allow for noncommuting observables. Observables might not have welldefined values v j simultaneously. Hence a microcanonical ensemble as discussed above, suitable for commuting observables, may not exist. We overcome this obstacle by introducing an approximate microcanonical ensemble Ω. We show that, for every state satisfying the conditions of an approximate microcanonical ensemble, tracing out most of the larger system yields, on average, a state close to the NATS. We exhibit conditions under which an approximate microcanonical ensemble exists. The conditions can be satisfied when the larger system consists of many noninteracting replicas of the system. An important step in the proof consists of reducing the noncommuting case to the commuting one. This reduction relies on a result by Ogata [30, Theorem 1.1]. A summary appears in Fig. 5.1. Set-up: Let S denote a system associated with a Hilbert space H ; with a Hamiltonian H ≡ Q0 ; and with observables (which we call “charges”) Q1 ,Q2 , . . . ,Q c . The charges do not necessarily commute with each other: [Q j ,Q k ] , 0. Consider N replicas of S, associated with the composite system Hilbert space H ⊗N . 156 Non-Abelian Thermal State total spin approximately fixed to values . Figure 5.1: Non-Abelian Thermal State: We derive the form of the thermal state of a system that has charges that might not commute with each other. Example charges include the components Ji of the spin J. We derive the thermal state’s form by introducing an approximate microcanonical state. An ordinary microcanonical ensemble could lead to the thermal state’s form if the charges commuted: Suppose, for example, that the charges were a Hamiltonian H and a particle number N that satisfied [H, N] = 0. Consider many copies of the system. The composite system could have a well-defined energy Etot and particle number Ntot simultaneously. Etot and Ntot would correspond to some eigensubspace HEtot, Ntot shared by the total Hamiltonian and the total-particle-number operator. The (normalized) projector onto HEtot, Ntot would represent the composite system’s microcanonical state. Tracing out the bath would yield the system’s thermal state. But the charges Ji under consideration might not commute. The charges might share no eigensubspace. Quantum noncommutation demands a modification of the ordinary microcanonical argument. We define an approximate microcanonical subspace M. Each state in M simultaneously has almost-well-defined values of noncommuting whole-system charges: Measuring any such whole-system charge has a high probability of outputting a value close to an “expected value” analogous to Etot and Ntot . We derive conditions under which the approximate microcanonical subspace M exists. The (normalized) projector onto M represents the whole system’s state. Tracing out most of the composite system yields the reduced state of the system of interest. We show that the reduced state is, on average, close to the Non-Abelian Thermal State (NATS). This microcanonical derivation of the NATS’s form links Jaynes’s information-theoretic derivation to physics. 157 m blocks Figure 5.2: Noncommuting charges: We consider a thermodynamic system S that has conserved charges Q j . These Q j ’s might not commute with each other. The system occupies a thermal state whose form we derive. The derivation involves an approximate microcanonical state of a large system that contains the system of interest. Consider a block of n copies of S. Most copies act, jointly, similarly to a bath for the copy of interest. We define Q̃ j as the average of the Q j ’s of the copies in the block. Applying results from Ogata [30], we find operators Ỹj that are close to the Q̃ j ’s and that commute with each other. Next, we consider m such blocks. This set of m blocks contains N = mn copies of S. Averaging the Q̃ j ’s over the blocks, for a fixed j-value, yields a global observable Q̄ j . The Q̄ j ’s are approximated by Ȳj ’s. The Ȳj ’s are the corresponding averages of the Ỹj ’s. The approximate global charges Ȳj commute with each other. The commuting Ȳj ’s enable us to extend the concept of a microcanonical ensemble from the well-known contexts in which all charges commute to truly quantum systems whose charges do not necessarily commute. We average each charge Q j over the N copies: N −1 Q̄ j := 1 X ⊗` I ⊗ Q j ⊗ I⊗(N −1−`) . N `=0 (5.2) The basic idea is that, as N grows, the averaged operators Q̄ j come increasingly to commute. Indeed, there exist operators operators Ȳj that commute with each other and that approximate the averages [30, Theorem 1.1]. An illustration appears in Fig. 5.2. Derivation: Since the Ȳj ’s commute mutually, they can be measured simultaneously. More importantly, the joint Hilbert space H ⊗n contains a subspace on which each Q̄ j has prescribed values close to v j . Let M denote the subspace. Perhaps unsurprisingly, because the Ȳj ’s approximate the Q̄ j ’s, each state in M has a nearly welldefined value of Q̄ j near v j . If Q̄ j is measured, the distribution is sharply peaked around v j . We can also show the opposite: every state with nearly well-defined values v j of all Q̄ j ’s has most of its probability weight in M. These two properties show that M is an approximate microcanonical subspace for the Q̄ j ’s with values v j . The notion of the approximate microcanonical subspace is the first major contribution of our work. It captures the idea that, for large N, we can approximately fix the values of the noncommuting charges Q j . An approximate 158 microcanonical subspace M is any subspace consisting of the whole-system states whose average observables Q̄ j have nearly well-defined values v j . More precisely, a measurement of any Q̄ j has a high probability of yielding a value near v j if and only if most of the state’s probability weight lies in M. Normalizing the projector onto M yields an approximate microcanonical ensemble, Ω. Tracing out every copy of S but the ` th yields the reduced state Ω` . The distance between Ω` and the NATS γv can be quantified by the relative entropy D(Ω` kγv ) := −S(Ω` ) − Tr(Ω` log γv ). (5.3) Here, S(Ω` ) := −Tr(Ω` log Ω` ) is the von Neumann entropy. The relative entropy D is bounded by the trace norm k.k1 , which quantifies the distinguishability of Ω` and γv [31]: D(Ω` kγv ) ≥ 1 kΩ` − γv k 21 . 2 (5.4) Our second main result is that, if Ω is an approximate microcanonical ensemble, then the average, over systems `, of the relative entropy D between Ω` and γv is small: N −1 1 X D(Ω` kγv ) ≤ θ + θ 0 . N `=0 (5.5) √ The parameter θ = (const.)/ N vanishes in the many-copy limit. θ 0 depends on the number c of charges, on the approximate expectation values v j , on the eigenvalues of the charges Q j , and on the (small) parameters in terms of which M approximates a microcanonical subspace. Inequality (5.5) capstones the derivation. The inequality follows from bounding each term in Eq. (5.3), the definition of the relative entropy D. The entropy S(Ω` ) is bounded with θ. This bound relies on Schumacher’s Theorem, which quantifies the size of a high-probability subspace like M with an entropy S(γv ) [32]. We bound the second term in the D definition with θ 0. This bound relies on the definition of M: Outcomes of measurements of the Q̄ j ’s are predictable up to parameters on which θ 0 depends. Finally, we present conditions under which the approximate microcanonical subspace M exists. Several parameters quantify the approximation. The parameters are shown to be interrelated and to approach zero simultaneously as N grows. In particular, the approximate microcanonical subspace M exists if N is great enough. 159 This microcanonical derivation offers a physical counterpoint to Jaynes’s maximumentropy derivation of the NATS’s form. We relate the NATS to the physical picture of a small subsystem in a vast universe that occupies an approximate microcanonical state. This vast universe allows the Correspondence Principle to underpin our argument. In the many-copy limit as N → ∞, the principle implies that quantum behaviors should vanish, as the averages of the noncommuting charges Q j come to be approximated by commuting Ȳj ’s. Drawing on Ogata’s [30, Theorem 1.1], we link thermality in the presence of noncommutation to the physical Correspondence Principle. Dynamical considerations The microcanonical and maximum-entropy arguments rely on kinematics and information theory. But we wish to associate the NATS with the fixed point of dynamics. The microcanonical argument, combined with canonical typicality, suggests that the NATS is the equilibrium state of typical dynamics. Canonical typicality enables us to model the universe’s state with a pure state in the approximate microcanonical subspace M. If a large system occupies a randomly chosen pure state, the reduced state of a small subsystem is close to thermal [6–9]. Consider, as in the previous section, N copies of the system S. By Ω, we denoted the composite system’s approximately microcanonical state. We denoted by Ω` the reduced state of the ` th copy, formed by tracing out most copies from Ω. Imagine that the whole system occupies a pure state |ψi ∈ M. Denote by ρ` the reduced state of the ` th copy. ρ` is close to Ω` , on average, by canonical typicality [8]: hk ρ` − Ω` k1 i ≤ √ d . DM (5.6) The average h.i is over pure states |ψi ∈ M. The trace norm is denoted by k.k1 ; d := dim(H ) denotes the dimensionality of the Hilbert space H of one copy of S; and D M := dim(M) denotes the dimensionality of the approximate microcanonical subspace M. We have bounded, using canonical typicality, the average trace norm between ρ` and Ω` . We can bound the average trace norm between Ω` and the NATS γv , using our microcanonical argument. [Equation (??) bounds the average relative entropy D between Ω` and γv . Pinsker’s Inequality, Ineq. (5.4), lower bounds D in terms of the trace norm.] Combining these two trace-norm bounds via the Triangle Inequality, 160 we bound the average distance between ρ` and γv : −1  1 NX p d k ρ` − γv k1 ≤ √ + 2(θ + θ 0 ). N `=0 D  (5.7) If the whole system occupies a random pure state |ψi in M, the reduced state ρ` of a subsystem is, on average, close to the NATS γv . Sufficiently ergodic dynamics is expected to evolve the whole-system state to a |ψi that satisfies Ineq. (5.7): Suppose that the whole system begins in a pure state |ψ(t=0)i ∈ M. Suppose that the system’s Hamiltonian commutes with the charges: [H,Q j ] = 0 for all j = 1, . . . , c. The dynamics conserves the charges. Hence most of the amplitude of |ψ(t)i remains in M for appreciable times. Over sufficient times, ergodic dynamics yields a state |ψ(t)i that can be regarded as random. Hence the reduced state is expected be close to Ω` ≈ γv for most long-enough times t. Exploring how the dynamics depends on the number of copies of the system offers promise for interesting future research. Resource theory A thermodynamic resource theory is an explicit characterization of a thermodynamic system’s resources, free states, and free operations with a rigorous mathematical model. The resource theory specifies what an experimenter considers valuable (e.g., work) and what is considered plentiful, or free (e.g., thermal states). To define a resource theory, we specify allowed operations and which states can be accessed for free. We use this framework to quantify the resources needed to transform one state into another. The first resource theory was entanglement theory [33]. The theory’s free operations are local operations and classical communication (LOCC). The free states are the states which can be easily prepared with LOCC, the separable states. Entangled states constitute valuable resources. One can quantify entanglement using this resource theory. We present a resource theory for thermodynamic systems that have noncommuting conserved charges Q j . The theory is defined by its set of free operations, which we call “Non-Abelian Thermal Operations” (NATO). NATO generalize thermal operations [16, 19]. How to extend thermodynamic resource theories to conserved quantities other than energy was noted in [19, 21, 22]. The NATO theory is related to the resource theory in [28]. 161 We supplement these earlier approaches with two additions. First, a battery has a work payoff function dependent on chemical potentials. We use this payoff function to define chemical work. Second, we consider a reference system for a non-Abelian group. The reference system is needed to resolve the difficulty encountered in [22, 28]: There might be no nontrivial operations which respect all the conservation laws. The laws of physics require that any operation performed by an experimenter commutes with all the charges. If the charges fail to commute with each other, there might be no nontrivial unitaries which commute with all of them. In practice, one is not limited by such a stringent constraint. The reason is that an experimenter has access to a reference frame [34–36]. A reference frame is a system W prepared in a state such that, for any unitary on a system S which does not commute with the charges of S, some global unitary on W S conserves the total charges and approximates the unitary on S to arbitrary precision. The reference frame relaxes the strong constraint on the unitaries. The reference frame can be merged with the battery, in which the agent stores the ability to perform work. We refer to the composite as “the battery.” We denote its state by ρW . The battery has a Hamiltonian HW and charges Q jW , described below. Within this resource theory, the Non-Abelian Thermal State emerges in two ways: 1. The NATS is the unique state from which work cannot be extracted, even if arbitrarily many copies are available. That is, the NATS is completely passive. 2. The NATS is the only state of S that remains invariant under the free operations during which no work is performed on S. Upon proving the latter condition, we prove second laws for thermodynamics with noncommuting charges. These laws provide necessary conditions for a transition to be possible. In some cases, we show, the laws are sufficient. These second laws govern state transitions of a system ρS , governed by a Hamiltonian HS , whose charges Q jS can be exchanged with the surroundings. We allow the experimenter to couple ρS to free states ρR . The form of ρR is determined by the Hamiltonian HR and the charges Q jR attributable to the free system. We will show that these free states have the form of the NATS. As noted above, no other state could be free. If other states were free, an arbitrarily large amount of work could be extracted from them. 162 Before presenting the second laws, we must define “work.” In textbook examples about gases, one defines work as δW = p dV , because a change in volume at a fixed pressure can be translated into the ordinary notion of mechanical work. If a polymer is stretched, then δW = F dx, wherein x denotes the polymer’s linear displacement and F denotes the restoring force. If B denotes a magnetic field and M denotes a medium’s magnetization, δW = B dM. The definition of “work” can depend on one’s ability to transform changes in thermodynamic variables into a standard notion of “work,” such as mechanical or electrical work. Our approach is to define a notion of chemical work. We could do so by modelling explicitly how the change in some quantity Q j can be used to extract µ j δQ j work. Explicit modelling would involve adding a term to the battery Hamiltonian HW . Rather than considering a specific work Hamiltonian or model of chemical work, however, we consider a work payoff function, W= c X µ j Q jW . (5.8) j=0 The physical situation could determine the form of this W. For example, the µ j ’s could denote the battery’s chemical potentials. In such a case, W would denote the battery’s total Hamiltonian, which would depend on those potentials. We choose a route conceptually simpler than considering an explicit Hamiltonian and battery system, however. We consider Eq. (5.8) as a payoff function that defines the linear combination of charges that interests us. We define the (chemical) work expended or distilled during a transformation as the change in the quantum expectation value hWi. The form of W is implicitly determined by the battery’s structure and by how charges can be converted into work. For our purposes, however, the origin of the form of W need not be known. W will uniquely determine the µ j ’s in the NATS. Alternatively, we could first imagine that the agent could access, for free, a particular NATS. This NATS’s form would determine the work function’s form. If the charges commute, the corresponding Gibbs state is known to be the unique state that is completely passive with respect to the observable (5.8). In App. D.3, we specify the resource theory for noncommuting charges in more detail. We show how to construct allowable operations, using the reference frame and battery. From the allowable operations, we derive a zeroth law of thermodynamics. 163 Complete passivity and zeroth law: This zeroth law relates to the principle of complete passivity, discussed in [26, 27]. A state is complete passive if, an agent cannot extract work from arbitrarily many copies of the state. In the resource theory for heat exchanges, completely passive states can be free. They do not render the theory trivial because no work can be drawn from them [18]. In the NATO resource theory, we show, the only reasonable free states have the NATS’s form. The free states’ chemical potentials equal the µ j ’s in the payoff function W, at some common fixed temperature. Any other state would render the resource theory trivial: From copies of any other state, arbitrary much work could be extracted for free. Then, we show that the NATS is preserved by NATO, the operations that perform no work on the system. The free states form an equivalence class. They lead to notions of temperature and chemical potentials µ j . This derivation of the free state’s form extends complete passivity and the zeroth law from [18] to noncommuting conserved charges. The derivation further solidifies the role of the Non-Abelian Thermal State in thermodynamics. Second laws: The free operations preserve the NATS. We therefore focus on contractive measures of states’ distances from the NATS. Contractive functions decrease monotonically under the free operations. Monotones feature in “second laws” that signal whether NATO can implement a state transformation. For example, the α-Rényi relative entropies between a state and the NATS cannot increase. Monotonicity allows us to define generalized free energies as Fα ( ρS , γS ) := k BT Dα ( ρS kγS ) − k BT log Z , (5.9) wherein β ≡ 1/(k BT ) and k B denotes Boltzmann’s constant. γS denotes the NATS with respect to the Hamiltonian HS and the charges Q jS of the system S. The partition function is denoted by Z. Various classical and quantum definitions of the Rényi relative entropies Dα are known to be contractive [18, 37–40]. The free energies Fα decrease monotonically if no work is performed on the system. Hence the Fα ’s characterize natural second laws that govern achievable transitions. For example, the classical Rényi divergences Dα ( ρS kγS ) are defined as   X sgn(α) α 1−α log  pk qk  , Dα ( ρS kγS ) := α−1 k (5.10) 164 wherein pk and qk denote the probabilities of ρS and of γS in the W basis. The Dα ’s lead to second laws that hold even in the absence of a reference frame and even outside the context of the average work. The Fα ’s reduce to the standard free energy when averages are taken over large numbers. Consider the asymptotic (“thermodynamic”) limit in which many copies ( ρS ) ⊗n of ρS are transformed. Suppose that the agent has some arbitrarily small probability ε of failing to implement the desired transition. ε can be incorporated into the free energies via a technique called “smoothing” [18]. The average, over copies of the state, of every smoothed Fαε approaches F1 [18]:  1 ε Fα ( ρS ) ⊗n , (γS ) ⊗n = F1 n→∞ n = k BT D( ρS kγS ) − k BT log(Z ) c X = hHS i ρS − k BT S( ρS ) + µ j hQ jS i. lim (5.11) (5.12) (5.13) j=1 We have invoked the relative entropy’s definition,     D( ρS kγS ) := Tr ρS log( ρS ) − Tr ρS log(γS ) . (5.14) Note the similarity between the many-copy average F1 in Eq. (5.13) and the orP dinary free energy, F = E − T dS + j µ j dN j . The monotonic decrease of F1 constitutes a necessary and sufficient condition for a state transition to be possible in the presence of a reference system in the asymptotic limit. In terms of the generalized free energies, we formulate second laws: Proposition 1. In the presence of a heat bath of inverse temperature β and chemical potentials µ j , the free energies Fα ( ρS , γS ) decrease monotonically: Fα ( ρS , γS ) ≥ Fα ( ρ0S , γS0 ) ∀α ≥ 0, (5.15) wherein ρS and ρ0S denote the system’s initial and final states. The system’s Hamiltonian and charges may transform from HS and Q jS to HS0 and Q0jS . The NATSs associated with the same Hamiltonians and charges are denoted by γS and γS0 . If [W, ρ0S ] = 0 and Fα ( ρS , γS ) ≥ Fα ( ρ0S , γS0 ) ∀α ≥ 0, some NATO maps ρS to ρ0S . (5.16) 165 As in [18], additional laws can be defined in terms of quantum Rényi divergences [37– 40]. This amounts to choosing, in Proposition 1, a definition of the Rényi divergence which accounts for the possibility that ρS and ρ0S have coherences relative to the WS eigenbasis. Several measures are known to be contractive [37–40]. They, too, provide a new set of second laws. Extractable work: In terms of the free energies Fα , we can bound the work extractable from a resource state via NATO. We consider the battery W separately from the system S of interest. We assume that W and S occupy a product state. (This assumption is unnecessary if we focus on average work.) Let ρW and ρ0W denote the battery’s initial and final states. For all α, Fα ( ρS ⊗ ρW , γSW ) ≥ Fα ( ρ0S ⊗ ρ0W , γSW ). (5.17) Since Fα ( ρS ⊗ ρW , γSW ) = Fα ( ρS , γS ) + Fα ( ρW , γW ),   Fα ρ0W , γW − Fα ( ρW , γW ) ≤ Fα ( ρS , γS ) − Fα ( ρ0S , γS ). (5.18) The left-hand side of Ineq. (5.18) represents the work extractable during one implementation of ρS → ρ0S . Hence the right-hand side bounds the work extractable during the transition. Consider extracting work from many copies of ρS (i.e., extracting work from ρS⊗n ) in each of many trials. Consider the average-over-trials extracted work, defined as Tr(W[ρ0W − ρW ]). The average-over-trials work extracted per copy of ρS is 1 0 n Tr(W[ρW − ρW ]). This average work per copy has a high probability of lying close to the change in the expectation value of the system’s work function, 1 0 0 n Tr(W[ρW − ρW ]) ≈ Tr(W[ρS − ρS ]), if n is large. Averaging over the left-hand side of Ineq. (5.18) yields the average work δhW i extracted per instance of the transformation. The average over the right-hand side approaches the change in F1 [Eq. (5.13)]: δhW i ≤ δhHS i ρS − T δS( ρS ) + c X µ j δhQ jS i. (5.19) j=1 This bound is achievable with a reference system, as shown in [41, 42]. We have focused on the extraction of work defined by W. One can extract, instead, an individual charge Q j . The second laws do not restrict single-charge extraction. 166 But extracting much of one charge Q j precludes the extraction of much of another charge, Q k . In App. D.3, we discuss the tradeoffs amongst the extraction of different charges Q j . 5.2 Discussion We have derived, via multiple routes, the form of the thermal state of a system that has noncommuting conserved charges. First, we regarded the system as part of a vast composite that occupied an approximate microcanonical state. Tracing out the environment yields a reduced state that lies, on average, close to a thermal state of the expected form. This microcanonical argument, with canonical typicality, suggests that the NATS is the fixed point of typical dynamics. Defining a resource theory, we showed that the NATS is the only completely passive state and is the only state preserved by free operations. These physical derivations buttress Jaynes’s information-theoretic derivation from the Principle of Maximum Entropy. Our derivations also establish tools applicable to quantum noncommutation in thermodynamics. In the microcanonical argument, we introduced an approximate microcanonical state Ω. This Ω resembles the microcanonical ensemble associated with a fixed energy, a fixed particle number, etc. but accommodates noncommuting charges. Our complete-passivity argument relies on a little-explored resource theory for thermodynamics, in which free unitaries conserve noncommuting charges. We expect that the equilibrium behaviors predicted here may be observed in experiments. Quantum gases have recently demonstrated equilibrium-like predictions about integrable quantum systems [12, 14]. From a conceptual perspective, our work shows that notions previously considered relevant only to commuting charges—for example, microcanonicals subspace— extend to noncommuting charges. This work opens fully quantum thermodynamics to analysis with familiar, but suitably adapted, technical tools. References [1] N. Yunger Halpern, P. Faist, J. Oppenheim, and A. Winter, Nature Communications 7, 12051 (2016). [2] J. Gemmer, M. Michel, M. Michel, and G. Mahler, Quantum thermodynamics: Emergence of thermodynamic behavior within composite quantum systems (Springer Verlag, 2009). [3] C. Gogolin and J. Eisert, Reports on Progress in Physics 79, 056001 (2016). 167 [4] J. Goold, M. Huber, A. Riera, L. del Rio, and P. Skrzypczyk, Journal of Physics A: Mathematical and Theoretical 49, 143001 (2016). [5] S. Vinjanampathy and J. Anders, Quantum Thermodynamics. Preprint at http://arxiv.org/abs/1508.06099 (2015). [6] S. Goldstein, J. L. Lebowitz, R. Tumulka, and N. Zanghí, Physical review letters 96, 050403 (2006). [7] J. Gemmer, M. Michel, and G. Mahler, 18 Equilibrium Properties of Model Systems (Springer, 2004). [8] S. Popescu, A. J. Short, and A. Winter, Nature Physics 2, 754 (2006). [9] N. Linden, S. Popescu, A. J. Short, and A. Winter, Phys. Rev. E 79, 061103 (2009). [10] E. Fermi, J. Pasta, and S. Ulam, Los Alamos Report LA-1940 (1955). [11] T. Kinoshita, T. Wenger, and D. S. Weiss, Nature 440, 900 (2006). [12] M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007). [13] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Rev. Mod. Phys. 83, 863 (2011). [14] T. Langen et al., Science 348, 207 (2015). [15] T. Langen, R. Geiger, and J. Schmiedmayer, Annual Review of Condensed Matter Physics 6, 201 (2015). [16] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and T. Beth, Int. J. Theor. Phys. 39, 2717 (2000). [17] F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Physical Review Letters 111, 250404 (2013). [18] F. G. S. L. Brandao, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner, Proc. Natl. Acad. Sci. 112, 3275 (2015). [19] M. Horodecki and J. Oppenheim, Nature Communications 4, 2059 (2013). [20] J. A. Vaccaro and S. M. Barnett, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences 467, 1770 (2011). [21] N. Yunger Halpern and J. M. Renes, Phys. Rev. E 93, 022126 (2016). [22] N. Y. Halpern, 094001 (2018). Journal of Physics A: Mathematical and Theoretical 51, 168 [23] M. Weilenmann, L. Krämer, P. Faist, and R. Renner, Axiomatic relation between thermodynamic and information-theoretic entropies. Preprint at http: //arxiv.org/abs/1501.06920 (2015). [24] E. T. Jaynes, Phys. Rev. 106, 620 (1957). [25] E. T. Jaynes, Phys. Rev. 108, 171 (1957). [26] W. Pusz and S. L. Woronowicz, Comm. Math. Phys. 58, 273 (1978). [27] A. Lenard, J. Stat. Phys. 19, 575 (1978). [28] M. Lostaglio, D. Jennings, and T. Rudolph, 1511.04420. ArXiv e-prints (2015), [29] Y. Guryanova, S. Popescu, A. J. Short, R. Silva, and P. Skrzypczyk, Nature Communications 7, 12049 (2016), 1512.01190. [30] Y. Ogata, Journal of Functional Analysis 264, 2005 (2013). [31] F. Hiai, M. Ohya, and M. Tsukada, Pacific J. Math. 96, 99 (1981). [32] B. Schumacher, Phys. Rev. A 51, 2738 (1995). [33] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009). [34] Y. Aharonov and L. Susskind, Phys. Rev. 155, 1428 (1967). [35] A. Kitaev, D. Mayers, and J. Preskill, Phys. Rev. A 69, 052326 (2004). [36] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Reviews of Modern Physics 79, 555 (2007). [37] F. Hiai, M. Mosonyi, D. Petz, and C. Bény, Rev. Math. Phys. 23, 691 (2011). [38] M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, Journal of Mathematical Physics 54 (2013). [39] M. M. Wilde, A. Winter, and D. Yang, Communications in Mathematical Physics 331, 593 (2014). [40] V. Jaksic, Y. Ogata, Y. Pautrat, and C.-A. Pillet, Entropic fluctuations in quantum statistical mechanics. an introduction, in Quantum Theory from Small to Large Scales: Lecture Notes of the Les Houches Summer School, edited by J. Fröhlich, S. Manfred, M. Vieri, W. De Roeck, and L. F. Cugliandolo, , Lecture Notes of the Les Houches Summer School Vol. 95, Oxford University Press, 2012. [41] J. Åberg, Phys. Rev. Lett. 113, 150402 (2014). [42] K. Korzekwa, M. Lostaglio, J. Oppenheim, and D. Jennings, New Journal of Physics 18 (2016). 169 Appendix A APPENDICES FOR “JARZYNSKI-LIKE EQUALITY FOR THE OUT-OF-TIME-ORDERED CORRELATOR” A.1 Weak measurement of the combined quantum amplitude à ρ à ρ [Eq. (2.4)] resembles the Kirkwood-Dirac quasiprobability for a quantum state [1– 3]. This quasiprobability has been inferred from weak-measurement experiments [4– 8]. Weak measurements have been performed on cold atoms [9], which have been proposed as platforms for realizing scrambling and quantum chaos [10–12]. à ρ can be inferred from many instances of a protocol Pweak . Pweak consists of a state preparation, three evolutions interleaved with three weak measurements, and a strong measurement. The steps appear in Sec. 2.2. I here flesh out the protocol, assuming that the system, S, begins in the infinitetemperature Gibbs state: ρ = 1/d. à ρ simplifies as in Eq. (2.5). The final factor becomes pw3 ,α w3 = 1/d. The number of weak measurements in Pweak reduces to two. Generalizing to arbitrary ρ’s is straightforward but requires lengthier calculations and more “background” terms. Each trial in the simplified Pweak consists of a state preparation, three evolutions interleaved with two weak measurements, and a strong measurement. Loosely, one performs the following protocol: Prepare |w3 , α w3 i. Evolve S backward under U † . Measure |v1 , λ v1 ihv1 , λ v1 | weakly. Evolve S forward under U. Measure |w2 , α w2 ihw2 , α w2 | weakly. Evolve S backward under U † . Measure |v2 , λ v2 ihv2 , λ v2 | strongly. Let us analyze the protocol in greater detail. The |w3 , α w3 i preparation and backward evolution yield |ψi = U † |w3 , α w3 i. The weak measurement of |v1 , λ v1 ihv1 , λ v1 | is implemented as follows: S is coupled weakly to an ancilla A a . The observable Ṽ of S comes to be correlated with an observable of A a . Example A a observables include a pointer’s position on a dial and a component σ` of a qubit’s spin (wherein ` = x, y, z). The A a observable is measured projectively. Let x denote the measurement’s outcome. x encodes partial information about the system’s state. We label by (v1 , λ v1 ) the Ṽ eigenvalue most reasonably attributable to S if the A a measurement yields x. 170 The coupling and the A a measurement evolve |ψi under the Kraus operator [13] Mx = p pa (x) 1 + ga (x) |v1 , λ v1 ihv1 , λ v1 | . (A.11) Equation (A.11) can be derived, e.g., from the Gaussian-meter model [3, 14] or the qubit-meter model [15]. The projector can be generalized to a projector Πv1 onto a degenerate eigensubspace. The generalization may decrease exponentially the number of trials required [16]. By the probabilistic interpretation of quantum channels, the baseline probability pa (x) denotes the likelihood that, in any given trial, S fails to couple to A a but the A a measurement yields x nonetheless. The detector is assumed, for convenience, to be calibrated such that Z dx · x pa (x) = 0 . (A.12) The small tunable parameter ga (x) quantifies the coupling strength. The system’s state becomes |ψ 0i = Mx U † |w3 , α w3 i, to within a normalization factor. S evolves under U as |ψ 0i 7→ |ψ 00i = U Mx U † |w3 , α w3 i , (A.13) to within normalization. |w2 , α w2 ihw2 , α w2 | is measured weakly: S is coupled weakly to an ancilla A b . W̃ comes to be correlated with a pointer-like variable of A b . The pointer-like variable is measured projectively. Let y denote the outcome. The coupling and measurement evolve |ψ 00i under the Kraus operator My = p pb (y) 1 + gb (y) |w2 , α w2 ihw2 , α w2 | . (A.14) The system’s state becomes |ψ 000i = My U Mx U † |w3 , α w3 i, to within normalization. The state evolves backward under U † . Finally, Ṽ is measured projectively. Each trial involves two weak measurements and one strong measurement. The probability that the measurements yield the outcomes x, y, and (v2 , λ v2 ) is Pweak (x, y, (v2 , λ v2 )) = |hv2 , λ v2 |U † My U Mx U † |w3 , α w3 i| 2 . Integrating over x and y yields Z I := dx dy · x y Pweak (x, y, (v2 , λ v2 )) . (A.15) (A.16) 171 We substitute in for Mx and My from Eqs. (A.11) and (A.14), then multiply out. We approximate to second order in the weak-coupling parameters. The calibration condition (A.12) causes terms to vanish: Z h p I= dx dy · x y pa (x) pb (y) ga (x) gb (y) · d i Z p × Ã1/d (w, v, α w , λ v ) + c.c. + dx dy · x y pa (x) pb (y) h × ga∗ (x) gb (y) hv2 , λ v2 |U † |w2 , α w2 ihw2 , α w2 |w3 , α w3 i i × (hv2 , λ v2 |v1 , λ v1 ihv1 , λ v1 |U † |w3 , α w3 i) ∗ + c.c. + O(ga (x) 2 gb (y)) + O(ga (x) gb (y) 2 ) . (A.17) The baseline probabilities pa (x) and pb (x) are measured during calibration. Let us focus on the second integral. By orthonormality, hw2 , α w2 |w3 , α w3 i = δw2 w3 δ α w2 α w3 , and hv2 , λ v2 |v1 , λ v1 i = δv2 v1 δ λ v2 λ v1 . The integral vanishes if (w3 , α w3 ) , (w2 , α w2 ) or if (v2 , λ v2 ) , (v1 , λ v1 ). Suppose that (w3 , α w3 ) = (w2 , α w2 ) and (v2 , λ v2 ) = (v1 , λ v1 ). The second integral becomes Z h p dx dy · x y pa (x) pb (y) ga∗ (x) gb (y) i × |hv2 , λ v2 |U † |w3 , α w3 i| 2 + c.c. . (A.18) The square modulus, a probability, can be measured via Born’s rule. The experimenter controls ga (x) and gb (y). The second integral in Eq. (A.17) is therefore known. From the first integral, we infer about Ã1/d . Consider trials in which the couplings are chosen such that Z p (A.19) α := dx dy · x y pa (x) pb (y) ga (x) gb (y) ∈ R . The first integral becomes 2α d <( Ã1/d (w, v, α w , λ v )). From these trials, one infers the real part of Ã1/d . Now, consider trials in which i α ∈ R. The first bracketed term becomes 2|α| d =( Ã1/d (w, v, α w , λ v )) . From these trials, one infers the imaginary part of Ã1/d . α can be tuned between real and imaginary in practice [4]. Consider a weak measurement in which the ancillas are qubits. An ancilla’s σ y can be coupled to a system observable. Whether the ancilla’s σ x or σ y is measured dictates whether α is real or imaginary. 172 The combined quantum amplitude à ρ can therefore be inferred from weak measurements. à ρ can be measured alternatively via interference. A.2 Interference-based measurement of the combined quantum amplitude à ρ I detail an interference-based scheme for measuring à ρ (w, v, α w , λ v ) [Eq. (2.4)]. The scheme requires no reversal of the time evolution in any trial. As implementing time reversal can be difficult, the absence of time reversal can benefit OTOCmeasurement schemes [11, 17]. I specify how to measure an inner product z := ha|U |bi, wherein a, b ∈ {(w` , α w` ), (vm , λ vm )} and U ∈ {U,U † }. Then, I discuss measurements of the state-dependent factor in Eq. (2.4). The inner product z is measured as follows. The system S is initialized to some fiducial state | f i. An ancilla qubit A is prepared in the state √1 (|0i + |1i). The 2 +1 and −1 eigenstates of σ z are denoted by |0i and |1i. The composite system AS begins in the state |ψi = √1 (|0i| f i + |1i| f i). 2 A unitary is performed on S, conditioned on A: If A is in state |0i, then S is brought to state |bi, and U is applied to S. If A is in state |1i, S is brought to state |ai. The global state becomes |ψ 0i = √1 [|0i(U |bi) + |1i|ai)] . A unitary e−iθσ x rotates the 2 ancilla’s state through an angle θ about the x-axis. The global state becomes ! " 1 θ θ 00 |ψ i = √ cos |0i − i sin |1i (U |bi) 2 2 2 ! # θ θ + −i sin |0i + cos |1i |ai . (A.21) 2 2 The ancilla’s σ z is measured, and the system’s {|ai} is measured. The probability that the measurements yield +1 and a is ! 1 2 θ 2 2 θ P (+1, a) = (1 − sin θ) cos |z| − sin θ =(z) + sin . (A.22) 4 2 2 The imaginary part of z is denoted by =(z). P (+1, a) can be inferred from the outcomes of multiple trials. The |z| 2 , representing a probability, can be measured independently. From the |z| 2 and P (+1, a) measurements, =(z) can be inferred. <(z) can be inferred from another set of interference experiments. The rotation about x̂ is replaced with a rotation about ŷ. The unitary e−iφσy implements this 173 rotation, through an angle φ. Equation (A.21) becomes 1 h φ  φ | ψ̃ 00i = √ cos |0i + sin |1i (U |bi) 2 2 2  φ  i φ + − sin |0i + cos |1i |ai . 2 2 (A.23) The ancilla’s σ z and the system’s {|ai} are measured. The probability that the measurements yield +1 and a is P̃ (+1, a) = 1 φ (1 − sin φ) cos2 |z| 2 4 2 ! 2 φ − sin φ <(z) + sin . 2 (A.24) One measures P̃ (+1, a) and |z| 2 , then infers <(z). The real and imaginary parts of z are thereby gleaned from interferometry. Equation (2.4) contains the state-dependent factor M := hv1 , λ v1 | ρU † |w3 , α w3 i. This factor is measured easily if ρ shares its eigenbasis with W̃ (t) or with Ṽ . In these cases, M assumes the form ha|U † |bi p. The inner product is measured as above. The probability p is measured via Born’s rule. In an important subcase, ρ is the infinite-temperature Gibbs state 1/d. The system’s size sets p = 1/d. Outside of these cases, M can be inferred from quantum tomography [18]. Tomography requires many trials but is possible in principle and can be realized with small systems. 174 BIBLIOGRAPHY [1] J. G. Kirkwood, Physical Review 44, 31 (1933). [2] P. A. M. Dirac, Reviews of Modern Physics 17, 195 (1945). [3] J. Dressel, Phys. Rev. A 91, 032116 (2015). [4] J. S. Lundeen, B. Sutherland, A. Patel, C. Stewart, and C. Bamber, Nature 474, 188 (2011). [5] J. S. Lundeen and C. Bamber, Phys. Rev. Lett. 108, 070402 (2012). [6] C. Bamber and J. S. Lundeen, Phys. Rev. Lett. 112, 070405 (2014). [7] M. Mirhosseini, O. S. Magaña Loaiza, S. M. Hashemi Rafsanjani, and R. W. Boyd, Phys. Rev. Lett. 113, 090402 (2014). [8] J. Dressel, M. Malik, F. M. Miatto, A. N. Jordan, and R. W. Boyd, Rev. Mod. Phys. 86, 307 (2014). [9] G. A. Smith, S. Chaudhury, A. Silberfarb, I. H. Deutsch, and P. S. Jessen, Phys. Rev. Lett. 93, 163602 (2004). [10] B. Swingle, G. Bentsen, M. Schleier-Smith, and P. Hayden, ArXiv e-prints (2016), 1602.06271. [11] N. Y. Yao et al., ArXiv e-prints (2016), 1607.01801. [12] I. Danshita, M. Hanada, and M. Tezuka, ArXiv e-prints (2016), 1606.02454. [13] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2010). [14] Y. Aharonov, D. Z. Albert, and L. Vaidman, Phys. Rev. Lett. 60, 1351 (1988). [15] T. C. White et al., npj Quantum Information 2 (2016). [16] J. Dressel, B. Swingle, and N. Yunger Halpern, in prep. [17] G. Zhu, M. Hafezi, and T. Grover, ArXiv e-prints (2016), 1607.00079. [18] M. Paris and J. Rehacek, editors, Quantum State Estimation, Lecture Notes in Physics Vol. 649 (Springer, Berlin, Heidelberg, 2004). 175 Appendix B APPENDICES FOR “THE QUASIPROBABILITY BEHIND THE OUT-OF-TIME-ORDERED CORRELATOR” B.1 Mathematical properties of P(W,W 0 ) Summing à ρ , with constraints, yields P(W,W 0 ) [Eq. (3.37)]. Hence properties of à ρ (Sec. 3.5) imply properties of P(W,W 0 ). Property 8. P(W,W 0 ) is a map from a composition of two sets of complex numbers to the complex numbers: P : {W } × {W 0 } → C. The range is not necessarily real: C ⊃ R. Summing quasiprobability values can eliminate nonclassical behavior: Interference can reduce quasiprobabilities’ nonreality and negativity. Property 6 consists of an example. One might expect P(W,W 0 ), a sum of à ρ (.) values, to be real. Yet P(W,W 0 ) is nonreal in many numerical simulations (Sec. 3.3). Property 9. Marginalizing P(W,W 0 ) over one argument yields a probability if ρ shares the Ṽ eigenbasis or the W̃ (t) eigenbasis. Consider marginalizing Eq. (3.37) over W 0. The (w2 , α w2 ) and (v1 , λ v1 ) sums can be performed explicitly: X P(W ) := P(W,W 0 ) (B.11) W0 = X hw3 , α w3 |U |v2 , λ v2 ihv2 , λ v2 | ρU † |w3 , α w3 i (v2 ,λ v2 ), (w3 ,α w3 ) × δW (w3∗ v2∗ ) . (B.12) The final expression is not obviously a probability. But suppose that ρ shares its eigenbasis with Ṽ or with W̃ (t). Suppose, for example, that ρ has the form in Eq. (3.35). Equation (B.12) simplifies: X P(W ) = p(v2 , λ v2 ; w3 , α w3 ) δW (w3∗ v2∗ ) . (B.13) (v2 ,λ v2 ), (w3 ,α w3 ) 176 The p(v2 , λ v2 ; w3 , α w3 ) := |hw3 , α w3 |U |v2 , λ v2 i| 2 pv2 ,λ v2 denotes the joint probability that a Ṽ measurement of ρ yields (v2 , λ v2 ) and, after a subsequent evolution under U, a W̃ measurement yields (w3 , α w3 ). Every factor in Eq. (B.13) is nonnegative. Summing over W yields a sum over the P arguments of à ρ (.). The latter sum equals one, by Property 6: W P(W ) = 1. Hence P(W ) ∈ [0, 1]. Hence P(W ) behaves as a probability. We can generalize Property 9 to arbitrary Gibbs states ρ = e−H/T /Z, using the regulated quasiprobability (3.110). The regulated OTOC (3.108) equals a moment of the complex distribution X Preg (W,W 0 ) := (B.14) (v1 ,λ v1 ),(w2 ,α w2 ),(v2 ,λ v2 )(w3 ,α w3 ) reg à ρ (v1 , λ v1 ; w2 , α w2 ; v2 , λ v2 ; w3 , α w3 ) δW (w3∗ v2∗ ) δW 0 (w2 v1 ) . The proof is analogous to the proof of Theorem 1 in [1]. P Summing over W 0 yields Preg (W ) := W 0 Preg (W,W 0 ). We substitute in from reg Eq. (B.14), then for à ρ from Eq. (3.110). We perform the sum over W 0 explicitly, then the sums over (w2 , α w2 ) and (v1 , λ v1 ): X |hw3 , α w3 |Ũ |v2 , λ v2 i| 2 δW (w3∗ v2∗ ) . (B.15) Preg (W ) = (v2 ,λ v2 ) (w3 ,α w3 ) This expression is real and nonnegative. Preg (W ) sums to one, as P(W ) does. Hence Preg (W ) ∈ [0, 1] acts as a probability. Property 10 (Degeneracy of every P(W,W 0 ) associated with ρ = 1/d and with eigenvalue-(±1) operators W and V ). Let the eigenvalues of W and V be ±1. For example, let W and V be Pauli operators. Let ρ = 1/d be the infinite-temperature Gibbs state. The complex distribution has the degeneracy P(1, −1) = P(−1, 1). Property 10 follows from (1) Eq. (3.52) and (2) Property 7 of Ã(1/d) . Item (2) can be replaced with the trace’s cyclicality. We reason as follows: P(W,W 0 ) is defined in Eq. (3.37). Performing the sums over the degeneracies yields A˜(1/d) . Substituting in from Eq. (3.52) yields   1 X P(W,W 0 ) = Tr ΠwW3 (t) ΠvV2 ΠwW2 (t) ΠvV1 d v ,w ,v ,w 1 2 2 3 × δW (w3∗ v2∗ ) δW 0 (w2 v1 ) . (B.16) 177 (W,W 0 ) (1, 1) (1, −1) (−1, 1) (−1, −1) (v1 , w2 , v2 , w3 ) (1, 1, 1, 1), (1, 1, −1, −1), (−1, −1, 1, 1), (−1, −1, −1, −1) (−1, 1, 1, 1), (−1, 1, −1, −1), (1, −1, 1, 1), (1, −1, −1, −1) (1, 1, −1, 1), (1, 1, 1, −1), (−1, −1, −1, 1), (−1, −1, 1, −1) (−1, 1, −1, 1), (−1, 1, 1, −1), (1, −1, −1, 1), (1, −1, 1, −1) Table B.1: Correspondence between tuples of composite variables and quadruples of “base” variables: From each weak-measurement trial, one learns about a quadruple (v1 , w2 , v2 , w3 ). Suppose that the out-of-time-ordered-correlator operators W and V have the eigenvalues w` , vm = ±1. For example, suppose that W and V are Pauli operators. The quadruple’s elements are combined into W := w3∗ v2∗ and W 0 := w2 v1 . Each (W,W 0 ) tuple can be formed from each of four quadruples. Consider inferring Ã(1/d) or A˜(1/d) from weak measurements. From one trial, we infer about four random variables: v1 , w2 , v2 and w3 . Each variable equals ±1. The quadruple (v1 , w2 , v2 , w3 ) therefore assumes one of sixteen possible values. These four “base” variables are multiplied to form the composite variables W and W 0. The tuple (W,W 0 ) assumes one of four possible values. Every (W,W 0 ) value can be formed from each of four values of (v1 , w2 , v2 , w3 ). Table B.1 lists the tuplequadruple correspondences. Consider any quadruple associated with (W,W 0 ) = (1, −1), e.g., (−1, 1, 1, 1). Consider swapping w2 with w3 and swapping v1 with v2 . The result, e.g., (1, 1, −1, 1), leads to (W,W 0 ) = (−1, 1). This double swap amounts to a cyclic permutation of the quadruple’s elements. This permutation is equivalent to a cyclic permutation of the argument of the (B.16) trace. This permutation preserves the trace’s value while transforming the trace into P(−1, 1). The trace originally equaled P(1, −1). Hence P(1, −1) = P(−1, 1). B.2 Retrodiction about the symmetrized composite observable Γ̃ := i(K . . . A− A...K) Section 3.5 concerns retrodiction about the symmetrized observable Γ := K . . . A+ A . . . K . The product K . . . A is symmetrized also in Γ̃ := i(K . . . A − A . . . K ). One can retrodict about Γ̃, using K -extended KD quasiprobabilities Ã(ρK ) , similarly to in Theorem 2. The value most reasonably attributable retrodictively to the time-t 0 value of Γ̃ is given by Eqs. (3.111), (3.112), and (3.114). The conditional quasiprobabilities on 178 the right-hand sides of Eqs. (3.113) and (3.115) become p̃→ (a, . . . , k, f | ρ) = −=(h f 0 |kihk | . . . |aiha| ρ0 | f 0i) h f 0 | ρ0 | f 0i (B.21) p̃← (k, . . . , a, f | ρ) = =(h f 0 |aiha| . . . |kihk | ρ0 | f 0i) . h f 0 | ρ0 | f 0i (B.22) and The extended KD distributions become K) Ã(ρ,→ ( ρ, a, . . . , k, f ) = ih f 0 |kihk | . . . |aiha| ρ0 | f 0i (B.23) K) Ã(ρ,← ( ρ, k, . . . , a, f ) = −ih f 0 |aiha| . . . |kihk | ρ| f 0i . (B.24) and To prove this claim, we repeat the proof of Theorem 2 until reaching Eq. (3.123). The definition of Γ̃ requires that an i enter the argument of the first < and that a −i enter the argument of the second <. The identity <(iz) = −=(z), for z ∈ C, implies Eqs. (B.21)–(B.24). 179 BIBLIOGRAPHY [1] N. Yunger Halpern, Phys. Rev. A 95, 012120 (2017). 180 Appendix C APPENDICES FOR “MBL-MOBILE: MANY-BODY-LOCALIZED ENGINE” C.1 Quantitative assessment of the mesoscopic MBL Otto engine We asses the mesoscopic engine introduced in Sec. 4.2. Section C.1 reviews and introduces notation. Section C.1 introduces small expansion parameters. Section C.1 reviews the partial swap [1, 2], used to model cold thermalization (stroke 2). The average heat hQ2 i absorbed during stroke 2 is calculated in Sec. C.1; the average heat hQ4 i absorbed during stroke 4, in Sec. C.1; the average per-trial power hWtot i, in Sec. C.1; and the efficiency η MBL , in Sec. C.1. The foregoing calculations rely on adiabatic tuning of the Hamiltonian. Six diabatic corrections are estimated in Sec. C.1. Notation We focus on one mesoscopic engine S of N sites. The engine corresponds to a Hilbert space of dimensionality N ∼ 2 N . We drop the subscript from the Hamiltonian Hmeso (t). H (t) is tuned between HGOE , which obeys the ETH, and HMBL , which governs an MBL system. Unprimed quantities often denote properties of HGOE ; and primed quantities, properties of HMBL : E j denotes the j th -greatest energy of HGOE ; and E 0j , the j th -greatest energy of HMBL . δ j denotes the gap just below E j ; and δ0j , the gap just below E 0j . When approximating the spectra as continuous, we replace E j with E and E 0j with E 0. Though the energies form a discrete set, they can approximated as continuous. ETH and MBL Hamiltonians have Gaussian DOSs: µ(E) = √ N 2πN E e−E /(2N E ) , 2 2 (C.11) R∞ normalized to −∞ dE µ(E) = N . The unit of energy, or energy density per site, is E. We often extend energy integrals’ limits to ±∞, as the Gaussian peaks sharply 1 about E = 0. The local average gap hδiE = µ(E) and the average gap hδi := √ 2 πN E N = (footnote 2). N dE µ2 (E) −∞ R∞ The average HGOE gap, hδi, equals the average HMBL gap, by construction. hδi sets 181 the scale for work and heat quantities. Hence we cast Q’s and W ’s as (number)(function of small parameters) hδi . (C.12) The system begins the cycle in the state ρ(0) = e− βH HGOE /Z. The partition function   Z := Tr e− βH HGOE normalizes the state. Wb denotes the cold bath’s bandwidth. We set ~ = k B = 1 . t H (t) is tuned at a speed v := E dα dt , wherein αt denotes the dimensionless tuning parameter. v has dimensions of energy2 , as in [3]. Though our v is not defined identically to the v in [3], ours is expected to behave similarly. Small parameters We estimate low-order contributions to hWtot i and to η MBL in terms of small parameters: b 1. The cold bath has a small bandwidth: W hδi  1. 2. The cold bath is cold: βCWb > 0. 3. Also because the cold bath is cold, 1  e− βCWb ≈ 0, and βC1,hδi  1. √ 4. The hot bath is hot: N βH E  1. This inequality prevents βH from contaminating leading-order contributions to heat and work quantities. ( βH depen2 dence manifests in factors of e−N ( βH E) /4 .) Since βH E  √1 and hδi E  1, N  hδi  βH hδi = ( βH E) E  √1 . N We focus on the parameter regime in which √ TC  Wb  hδi and N βH E  1 . (C.13) The numerical simulations (Sec. 4.3) took place in this regime. We approximate to 2 b second order in βC1hδi , W hδi , and N ( βH E) . We approximate to zeroth order in the much smaller e− βCWb . The diabatic corrections to hWtot i involve three more small parameters. H (t) is √ v tuned slowly: hδi  1. The MBL level-repulsion scale δ − (Appendix C.2) is very δ− E small: hδi  1 . The third parameter, hδi E  1, follows from hδi ∼ N . 182 Partial-swap model of thermalization Classical thermalization can be modeled with a probabilistic swap, or partial swap, or p-SWAP [1, 2]. Let a column vector ~v represent the state. The thermalization is broken into time steps. At each step, a doubly stochastic matrix Mp operates on ~v . The matrix’s fixed point is a Gibbs state ~g . Mp models a probabilistic swapping out of ~v for ~g : At each time step, the system’s state has a probability 1 − p of being preserved and a probability p ∈ [0, 1] of being replaced by ~g . This algorithm gives Mp the form Mp = (1 − p) 1 + pG. Every column in the matrix G equals the Gibbs state ~g . We illustrate with thermalization across two levels. Let 0 and ∆ label the levels,  −β∆  e 1 such that ~g = 1+e , − β∆ 1+e −β∆ : e −β∆  1 − p 1− β∆ p 1+e − β∆ 1+e Mp =   . −β∆ e 1 1 − p p −β∆ −β∆ 1+e 1+e   (C.14) The off-diagonal elements, or transition probabilities, obey detailed balance [4, 5]: P(0→∆) − β∆ . P(∆→0) = e  n Repeated application of Mp maps every state to ~g [4]: limn→∞ Mp ~v = ~g . The parameter p reflects the system-bath-coupling strength. We choose p = 1: The system thermalizes completely at each time step. (If p , 1, a more sophisticated model may be needed for thermalization across > 2 levels.) Average heat hQ2 i absorbed during stroke 2 We calculate hQ2 i in four steps, using the density operator’s statistical interpretation (see the caption of Fig. 4.2). Section C.1 focuses on one trial. We average over two distributions in Sec. C.1: (1) the probabilities that cold thermalization changes or (E) preserves the engine’s energy and (2) the Poisson gap distribution, PMBL (δ). We − β H H GOE average with respect to the initial density operator, ρ(0) = e /Z, in Sec. C.1. Heat Q2 absorbed during one trial Let j denote the HGOE level on which the engine begins. Stroke 1 (adiabatic tuning) ( j) preserves the occupied level’s index. Let Q2 denote the heat absorbed during cold thermalization. Suppose that the gap just above level j is smaller than the cold bath’s bandwidth: δ0j+1 < Wb . The engine might jump upward, absorbing heat ( j) Q2 = δ0j+1 . Suppose that the gap just below level j is small enough: δ0j < Wb . The 183 ( j) engine might drop downward, absorbing Q2 = −δ0j . The engine absorbs no heat if it fails to hop:    δ0j+1 , engine jumps     ( j) Q2 =  −δ0j , engine drops . (C.15)      0 ,  cold thermalization preserves engine’s energy Averages with respect to cold-thermalization probabilities and gap distributions The discrete E j becomes a continuous E:   hQ2 (E)i cold therm. gaps = Z Wb 0 + dδ0j+1 δ0j+1 P (S jumps | δ0j+1 < Wb ) P (δ0j+1 < Wb ; S does not drop) Z Wb 0 dδ0j (−δ0j ) P (S drops | δ0j < Wb ) P (δ0j < Wb ; S does not jump) . (C.16) Each P (a) denotes the probability that event a occurs. P (a|b) denotes the conditional probability that, if an event b has occurred, a will occur. P (a; b) denotes the joint probability that a and b occur. The p-SWAP model (Suppl. Mat. C.1) provides the conditional probabilities. The Poisson distribution provides the probability that a gap is small enough. Each joint probability factorizes, e.g., P (δ0j+1 < Wb ; S does not drop) = P (δ0j+1 < Wb ) P (S does not drop). The engine refrains from dropping if (1) the gap below level j is too large or if (2) the gap below j is small but cold thermalization fails to drop the engine’s state:   P (S does not drop) = P (δ0j > Wb ) + P S does not drop | δ0j+1 < Wb P (δ0j+1 < Wb ) . (C.17)   b The gap has a probability P (δ0j > Wb ) = 1 + O W hδi of being too large and a probW  ability P (δ0j+1 < Wb ) = O hδib of being small enough.1 The detailed-balance 1 Any given gap’s probability of being small enough to thermalize equals P (δ ≤ Wb ) = 1 N Z Emax E min dE µ(E) Z Wb 0 (E) dδ PMBL (δ) ≈ 1 N Z ∞ h i dE µ(E) 1 − e − µ(E)Wb . −∞ (C.18) The first term evaluates to one. We Taylor-expand the exponential to first order, then integrate term 184   W  probability P S does not drop | δ0j+1 < Wb is too small to offset the O hδib scal  b ing of P (δ0j+1 < Wb ). Hence the O W hδi terms are negligible here: Each multiplies, in Eq. (C.16), a δ0j+1 that will average to ∼ Wb and a P (S jumps | δ0j+1 < Wb ) that  W 2  W 3 b b b . Each such compound term ∼ W = will average to ∼ W hδi b hδi hδi hδi . We evalb uate quantities only to second order in W hδi  1 . Hence Eq. (C.17) approximates to one. A similar argument concerns the final factor in Eq. (C.16). Equation (C.16) becomes 0 Z Wb   e− βC δ j+1 (E) 0 0 = dδ j+1 δ j+1 PMBL (δ0j+1 ) hQ2 (E)i cold 0 − β δ C j+1 therm. gaps 0 1+e  " # 3 Z Wb Wb 1 (E) 0 0 0  . PMBL (δ j ) + µ(E)O  − dδ j δ j − βC δ 0j hδi  0 1+e (C.111) Computing the integrals is tedious but is achievable by techniques akin to the Sommerfeld expansion [6]. The calculation appears in [7, App. G 4] and yields (      3 π 2 µ(E) 1 2 + µ(E) O µ(E) W = − µ(E) (Wb ) + hQ2 (E)i cold b 2 6 ( βC ) 2 therm. gaps " ! # 3 ) µ(E) µ(E) − βCWb  . (C.112) + O [µ(E) Wb ] e + O  βC βC  We have assumed that the engine cannot cold-thermalize down two adjacent small gaps (from level j+1 to level j −1, wherein δ0j , δ0j−1 < Wb ). Such a gap configuration appears with probability ∝ µ(E)[µ(E)Wb ]2 . Each gap contributes energy ∼ Wb to the heat. Hence double drops contribute to (C.112) at third order in µ(E)Wb . by term: " P (δ ≤ Wb ) ≈ 1 − 1 N Z ∞ dE µ(E) − −∞ " # 2 Wb Wb  . = + O  hδi hδi  Wb N Z ∞ −∞ dE µ2 (E) + O (Wb ) 2 N Z ∞ !# dE µ3 (E) −∞ (C.19) (C.110) 185 Thermal average with respect to ρ(0) We integrate Eq. (C.112) over energies E, weighted by the initial-state Gibbs distribution: *  + (C.113) hQ2 i := hQ2 (E)i cold therm. gaps ρ(0) (Wb ) 2 π 2 1 = − + 2 6 ( βC ) 2 " # 3 ) µ(E)  . + O  βC  ! (  " # 3 − βH E e Wb − βCWb Wb 2  +O e dE µ (E) + hδi O  Z hδi  hδi −∞ !Z ∞ (C.114) The DOS’s sharp peaking about E = 0 justifies our approximation of the energy integral as extending between ±∞. We substitute in for the DOS from Eq. (C.11): !Z ∞ 2 2 (Wb ) 2 π 2 1 N2 1 dE e−E /N E e− βH E + O(.) . − + hQ2 i = 2 2 2 6 ( βC ) 2πN E Z −∞ (C.115) √ 2 We have abbreviated the correction terms. The integral evaluates to πN E e N ( βH E) /4 . The partition function is Z ∞ 2 Z= dE µ(E)e− βH E = N e N ( βH E) /2 . (C.116) −∞ Substituting into Eq. (C.115) yields ! 2 (Wb ) 2 π 2 1 + − e−N ( βH E) /4 + O(.) (C.117) hQ2 i = √ 2 2 6 ( βC ) 2 πN E ! ! (  " # 3 1 µ(E) − βCWb Wb (Wb ) 2 π 2 −N ( βH E) 2 /4  + O [µ(E) Wb ] = − + e e + hδi O  2 hδi 6 ( βC ) 2 hδi βC hδi  " # 3  h√ i 4 ) µ(E)  +O + O  N βH E . (C.118) βC  N 1 via Eq. (4.10). The prefactor was replaced with hδi Equation (C.117) is compared with numerical simulations in Fig. C.1. In the appropriate regime (wherein Wb  hδi and TC  Wb ), the analytics agree well with the numerics, to within finite-size effects. In terms of small dimensionless parameters, #  1 Wb ! 2 π 2  " 1 N 2 + hQ2 i = hδi −  1 − ( βH E) + O(.) . 2 hδi 6 ( βC hδi) 2  4   (C.119) 186 1.4 0.0025 1.2 0.00002 0.0020 √ hQ2i/E N hQ2i/hδi hQ2i/hδi 1.0 0.0015 0.8 0.6 0.0010 0.4 0.0005 0.0000 0.2 0.4 0.6 0.8 1.0 1.2 Wb/hδi (a) hQ2 i vs. Wb at TC = 0 and TH = ∞ 0.00000 −0.00001 0.2 0.0 0.0 0.00001 −0.00002 0.00 0.02 0.04 0.06 TC/hδi 0.0 0.5 1.0 √ βH E N 1.5 2.0 (b) hQ2 i vs. TC at TH = ∞ (c) hQ2 i vs. βH at TC = 0 and Wb = 2 −4 hδi and Wb = 2 −4 hδi Figure C.1: Magnitude | hQ2 i | of the average heat absorbed during cold thermalization (stroke 2) as a function of the cold-bath bandwidth Wb (C.1a), the cold-bath temperature TC (C.1b), and the hot-bath temperature TH = 1/ βH (C.1c): The blue lines represent the magnitude of the analytical prediction (C.117). See Sec. 4.3 for other parameters and definitions. The analytics match the numerics’ shapes, and the agreement is fairly close, in Wb  1 and TC / hδi  1, in the gray shaded regions). The the appropriate limits (where hδi analytics systematically underestimate hQ2 i at fixed Wb , due to the small level repulsion at finite N. The analytical prediction (C.117) substantially underestimates hQ2 i when the cold-bath bandwidth is large, Wb & hδi. Such disagreement is expected: The analytics rely Wb on hδi  1, neglecting chains of small gaps δ 0j , δ 0j+1 · · · < Wb . Such chains proliferate as Wb grows. A similar reason accounts for the curve’s crossing the origin in Fig. C.1b: We analytically compute hQ2 i only to second order in TC / hδi. The leading-order term is second-order. So is the βC correction; but ( β 1hδi)2  C  W 2 b hδi , by assumption [Eq. (C.13)]. The βH correction is fourth-order—too small to include. To lowest order, (Wb ) 2 . hQ2 i ≈ − 2 hδi (C.120) Average heat hQ4 i absorbed during stroke 4 The hQ4 i calculation proceeds similarly to the hQ2 i calculation. When calculating hQ2 i, however, we neglected contributions from the engine’s cold-thermalizing  2 b down two small gaps. Two successive gaps have a probability ∼ W of being hδi < Wb each. Thermalizing across each gap produces  h iheat  ≤ Wb . Each such pair Wb 3 therefore contributes negligibly to hQ2 i, as hδi O hδi . We cannot neglect these pairs when calculating hQ4 i. Each typical small gap widens, during stroke 3, to size ∼ hδi . These larger gaps are thermalized across  h during i  Wb 2 stroke 4, contributing at the nonnegligible second order, as ∼ hδi O hδi to hQ4 i . Chains of ≥ 3 small MBL gaps contribute negligibly. 187 The calculation is tedious, appears in [7, App. G 5], and yields hQ4 i ≈ D E D E Wb 2 ln 2 (Wb ) 2 Q4n=1 + Q4n=2 ≈ Wb − + + 4 ln 2 . βC 2 hδi βC hδi (C.121) The leading-order term, Wb , is explained heuristically below Eq. (4.12). The leading-order βC correction, − 2 βlnC2 , shows that a warm cold bath lowers the heat required to reset the engine. Suppose that the cold bath is maximally cold: TC = 0. Consider any trial that S begins just above a working gap (an ETH gap δ > Wb that narrows to an MBL gap δ0 < Wb ). Cold thermalization drops S deterministically to the lower level. During stroke 4, S must absorb Q4 > 0 to return to its start-of-trial state. Now, suppose that the cold bath is only cool: TC & 0. Cold thermalization might leave S in the upper level. S needs less heat, on average, to b offsets the reset than if TC = 0. A finite TC detracts from hQ4 i. The +4 ln 2 βW C hδi detracting. However, the positive correction is smaller than the negative correction, b as W hδi  1 . A similar argument concerns TH < ∞. But the βH correction is too small to include 2 −( βH E) 2 /4 . b) in Eq. (C.121): hQ4 i ≈ Wb − 2 βlnC2 + (W 2hδi e Figure C.2 shows Eq. (C.121), to lowest order in TC , as well as the βH dependence of hQ4 i. The analytical prediction is compared with numerical simulations. The agreement is close, up to finite-size effects, in the appropriate regime (TC  Wb  hδi). Per-cycle power hWtot i By the first law of thermodynamics, the net work outputted by the engine equals the net heat absorbed. Summing Eqs. (C.121) and (C.120) yields the per-trial power, or average work outputted per engine cycle: hWtot i = hQ2 i + hQ4 i ≈ Wb − 2 ln 2 Wb + 4 ln 2 . βC βC hδi (C.122) The leading-order βH correction is negative and too small to include—of order  W 2 b N ( βH E) 2 . Equation (C.122) agrees well with the numerics in the approhδi priate limits (TC  Wb  hδi) and beyond, as shown in Fig. C.3. The main text contains the primary analysis of Eq. (C.122). Here, we discuss the hQ2 i correction, limiting behaviors, and scaling. 2 The negative hQ2 i = − (Whδib ) detracts little from the leading term Wb of hQ4 i: (Wb ) 2 Wb hδi  Wb , since hδi  1. The hQ 2 i cuts down on the per-trial power little. 188 2.5 0.06 2.0 0.00026 0.05 √ hQ4i/E N 0.00025 hQ4i/hδi hQ4i/hδi 0.04 1.5 0.00024 0.03 1.0 0.00023 0.02 0.5 0.01 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.00 0.02 0.04 0.06 TC/hδi Wb/hδi (a) hQ4 i vs. Wb at TC = 0 and TH = ∞ 0.00022 0.00 0.0 0.5 1.0 √ βH E N 1.5 2.0 (b) hQ4 i vs. TC at TH = ∞ (c) hQ4 i vs. βH at TC = 0 and Wb = 2 −4 hδi and Wb = 2 −4 hδi Figure C.2: Average heat hQ4 i absorbed during hot thermalization (stroke 4) as a function of the cold-bath bandwidth Wb , the cold-bath temperature TC , and the hot-bath temperature TH = 1/ βH : The blue lines represent the analytical prediction (C.121), to lowest order in TC , with the βH dependence of hQ4 i, too small a correction to include in Eq. (C.121): 2 −(β H E ) 2 /4 . See Sec. 4.3 for other parameters and definitions. b) hQ4 i ≈ Wb − 2βlnC2 + (W 2hδi e The analytics’ shapes agree with the numerics’, and the fit is fairly close, in the Wb appropriate limits (where e −βC Wb  1, βC1hδi  1, and hδi  1, in the gray shaded regions). The predictions underestimate hQ4 i; see the Fig. C.1 caption. Figure C.2c suggests that the numerics deviate significantly from the analytics: The numerics appear to depend on βH via a linear term absent from the hQ4 i prediction. This seeming mismatch appears symptomatic of finite sample and system sizes. The limiting behavior of Eq. (C.122) makes sense: Consider the limit as Wb → 0. The cold bath has too small a bandwidth to thermalize the engine. The engine should output no work. Indeed, the first and third terms in Eq. (C.122) vanish, being proportional to Wb . The second term vanishes because βC → ∞ more quickly than Wb → 0 , by Eq. (C.13): The cold bath is very cold. √ Equation (C.122) scales with the system size N no more quickly than N/2 N , by √ the assumption Wb  hδi ∼ N/2 N . This scaling makes sense: The engine outputs work because the energy eigenvalues meander upward and downward in Fig. 4.2 as H (t) is tuned. In the thermodynamic limit, levels squeeze together. Energy eigenvalues have little room in which to wander, and S outputs little work. Hence our parallelization of fixed-length mesoscopic subengines in the thermodynamic limit (Sec. 4.2). Efficiency η MBL in the adiabatic approximation The efficiency is defined as η MBL := hWtot i . hQin i (C.123) The numerator is averaged separately from the denominator because averaging Wtot over runs of one mesoscopic engine is roughly equivalent to averaging over simul- 189 1.4 0.00025 hWtoti/hδi hWtoti/hδi √ hWtoti/E N 0.05 1.0 0.04 0.8 0.00024 0.03 0.6 0.00023 0.02 0.4 0.00022 0.01 0.2 0.0 0.0 0.00026 0.06 1.2 0.2 0.4 0.6 0.8 1.0 0.00 0.00 1.2 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.00021 0.0 0.5 TC/hδi Wb/hδi 1.0 √ βH E N 1.5 2.0 (a) hWtot i vs. Wb at TC = 0 (b) hWtot i vs. TC at TH = ∞ (c) hWtot i vs. βH at TC = 0 and Wb = 2 −4 hδi and Wb = 2 −4 hδi and TH = ∞ Figure C.3: Per-cycle power hWtot i as a function of the cold-bath bandwidth Wb , the cold-bath temperature TC , and the hot-bath temperature TH = 1/ βH : The blue lines Wb represent the analytical prediction hWtot i ≈ Wb − 2βlnC2 : Eq. (C.122), to first order in hδi and 1 in βC hδi . See Sec. 4.3 for other parameters and definitions. The analytics largely agree Wb TC with the numerics in the appropriate regime: hδi  1, hδi  1 (in the gray shaded region). Outside that regime, the analytics underestimate hWtot i; see Fig. C.1 for analysis. Figure C.3c suggests that the numerics depend on βH via a linear term absent from the analytical prediction; see the caption of Fig. C.2c. tot i taneous runs of parallel subengines in one macroscopic engine. hW hQin i may therefore tot be regarded as the W Qin of one macroscopic-engine trial. Having calculated hWtot i, we must identify hQin i . In most trials, the engine expels heat −Q2 > 0 during cold thermalization and absorbs Q4 > 0 during hot thermalization. The positive-heat-absorbing-stroke is stroke 4, in the average trial: ! hQ2 i = hWtot i (1 + φ) , (C.124) hQin i = hQ4 i = hWtot i − hQ2 i = hWtot i 1 − hWtot i wherein (Wb ) 2 1 Wb hQ2 i φ := − ≈ ≈ . 2 hδi Wb 2 hδi hWtot i (C.125) Substituting from Eq. (C.124) into Eq. (C.123) yields η MBL ≈ Wb hWtot i ≈1−φ= 1− . 2 hδi hWtot i (1 + φ) (C.126) Using suboptimal baths diminishes the efficiency. Addding βC -dependent terms from Eq. (C.122) to hWtot i yields φ0 = Wb ln 2 Wb 1 + − 2 ln 2 . 2 hδi βC hδi hδi βC hδi (C.127) Wb The βH correction, 1 − 2hδi e−N ( βH E) /4 , is too small to include. The correction shares the sign of βH : A lukewarm hot bath lowers the efficiency. 2 190 0.97 0.990 0.9 0.96 0.8 0.95 ηMBL 0.985 ηMBL ηMBL 1.0 0.7 0.94 0.980 0.975 0.6 0.93 0.5 0.4 0.0 0.970 0.00 0.2 0.4 0.6 0.8 1.0 Wb/hδi 1.2 0.02 0.04 TC/hδi 0.06 0.965 0.0 0.5 1.0 √ βH E N 1.5 2.0 (b) η MBL vs. TC√at TH = ∞ (c) η MBL vs. βH at TC = 0 (a) η MBL vs. Wb at TC = 0 and Wb ≈ 10 −4 N E ≈ and Wb = 2 −4 hδi and TH = ∞ 0.04 hδi Figure C.4: Efficiency η MBL as a function of the cold-bath bandwidth Wb , the cold-bath temperature TC , and the hot-bath temperature TH = 1/ βH : The blue lines represent the analytical predictions (C.126) and (C.127). Figure (C.4c) shows the leading-order βH dependence of η MBL , a correction too small to include in Eq. (C.127): 2 Wb 1 − 2hδi e − N (βH E ) /4 . See Sec. 4.3 for other parameters and definitions. The analytics Wb TC agree with the numerics fairly well in the appropriate regime ( hδi  1, hδi  1, and √ N TH E  1). The analytics underestimate η MBL ; see the Fig. C.1 caption. Expressions (C.126) and (C.127) are compared with results from numerical simulations in Fig. C.4. The analytics agree with the numerics in the appropriate regime (TC  Wb  hδi). Diabatic corrections We have approximated strokes 1 and 3 as quantum-adiabatic. But the strokes prot ceed at a finite speed v := E dα dt . The engine may “hop” diabatically between energy eigenstates. We estimate the work costs from three types of diabatic transitions, introduced in Sec. 4.2. APT transitions are analyzed in Sec. C.1; LandauZener (LZ) transitions, in Sec. C.1; and fractional-LZ transitions, in Sec. C.1. The efficiency η MBL is diabatically corrected in Sec. C.1. We neglect the variation of the local average gap hδiE with energy. The approximations facilitate this appendix’s calculations, which can require heavier machinery than the adiabatic approximation. We aim to estimate just diabatic corrections’ sizes and scalings. Average work costs of APT transitions in the ETH phase: WAPT,1 and WAPT,3 Consider tuning H (t) near the start of stroke 1, withinin the ETH phase. (An analogous argument concerns the end of stroke 3.) Let |Em (t)i denote the instantaneous mth eigenstate of H (t). The perturbation couples together eigenstates of the initial 191 Hamiltonian. S can hop from its initial state, |Em (t i )i, to some other energy eigenstate n , m. The transition probability is denoted by PAPT (n|m). These transitions cost, on average, work WAPT,1 during stroke 1 and work WAPT,3 during stroke 3. We estimate PAPT (n|m) from an APT calculation in [8]. We estimate WAPT,1 , then argue that WAPT,3 ≈ WAPT,1 . Diabatic-hopping probability PAPT (Ef − Ei ) from APT: In this section, we generalize from the engine S to a closed quantum system S̃. Let H (t) denote a timedependent Hamiltonian. The mth instantaneous energy eigenstate is denoted by |Em (t)i. Let S̃ begin in the state |Em (t i )i. Let V denote the term “turned on” in H (t). V couples H (t i ) eigenstates together. The coupling transfers S̃ to some |En (t f )i with probability PAPT (n|m). De Grandi and Polkovnikov calculate [8, Eq. (20), p. 4] PAPT (n|m) ≈  v  2 " hEn (t)|∂α t |Em (t)i|α t E [En (t i ) − Em (t i )]2 −2 2 i + hEn (t)|∂α t |Em (t)i|α t f [En (t f ) − Em (t f )]2 hEn (t)|∂α t |Em (t)i|α i hEn (t)|∂α t |Em (t)i|α f En (t i ) − Em (t i ) En (t f ) − Em (t f ) 2 # cos(∆Θnm ) . (C.128) De Grandi and Polkovnikov’s λ is our Hamiltonian-tuning parameter αt . Their speed δ, which has dimensions of energy, equals our Ev . The ∆Θnm denotes a difference between two phase angles. The final term in Eq. (C.128) results from interference. This term often oscillates quickly and can be neglected [8]. Furthermore, we will integrate PAPT (n|m) over energies. The integration is expected to magnify cancellations. The second term in Eq. (C.128) shares the first term’s form. The first term is evaluated at t = t i ; the second term, at t = t f . The quantities evaluated at t i are close their t f counterparts, as H (t) obeys the ETH at all t ∈ [t i ,t f ]. Equation (C.128) approximates to2 PAPT (n|m) ∼ 2  v  2 hEn (t)|∂α t |Em (t)i|α t E [En (t i ) − Em (t i )]2 2 i . (C.129) 2 Equation (C.130) accounts for the greater frequency with which APT transitions occur in the ETH phase than in the MBL phase. In the ETH phase, |hEn |V |Em i| has a considerable size, ∼ √1 , for most (n, m) pairs [9]. In the MBL phase, few pairs correspond to a large numerator: N |hEn |V |Em i| ∼ N1 [10]. The corresponding energies tend to lie far apart: |En − Em |  |hEn |V |Em i|. Most APT transition probabilities are therefore suppressed [11]. 192 The perturbation-matrix element comes from the Chain Rule and from [8, Eq. (10)]: + * ! E ∂t ∂ E v hEn (t)|V |Em (t)i hEn (t)|∂α t |Em (t)i = En (t) Em (t) = hEn (t)|∂t |Em (t)i = − . ∂αt ∂t v v E En (t) − Em (t) (C.130) √ The modulus |hEn (t)|V |Em (t)i| scales as 1/ N for ETH Hamiltonians [9].3 We E . Substiintroduce an E for dimensionality: |hEn (t)|∂α t |Em (t)i| ∼ √ N |En (t)−Em (t)| tuting into Eq. (C.129) yields PAPT (n|m) ∼ v2 . N [En (t i ) − Em (t i )]4 (C.131) We have dropped a two, due to our focus on scaling. We will drop the time arguments. This probability is an even function of the signed gap En − Em : Only the gap’s size, not its direction, affects the hopping probability. PAPT (n|m) is normalized to one, so the right-hand side of Eq. (C.131) makes sense only when < 1. The right-hand side diverges if En lies close to Em . But energies rarely lie close together in the ETH phase, due to level repulsion. Furthermore, slow tuning of H (t) impedes diabatic transitions. Hence we introduce a regularization factor R: PAPT (n|m) ∼ v2  2 . N (En − Em ) 2 + R2 (C.132) In the worst case—when the right-hand side of Eq. (C.132) is largest—|En − Em | 2 is small. The right-hand side then approximates to NvR4 , which must < 1. The regularization must obey √ v R > 1/4 . (C.133) N How to choose a form for R is unclear. We therefore leave R unspecified temporarily. We will compute hWAPT i in terms of R, then survey the possible forms of R. We will choose the worst-case form for R—the form that maximizes the average work cost hWAPT i—consistent with Ineq. (C.133) and with the smallness of v. 3 One might worry that, when this mesoscale engine functions as a component of a macroscopic engine, the Hamiltonian will not obey the ETH. Rather, H (t) will be MBL at all times t. However, for the purposes of level-spacing statistics and operator expectation values on length scales of the order of the localization length, L ∼ ξ > , H (t) can be regarded as roughly ETH. The shallowly localized Hamiltonian’s key feature is some nontrivial amount of level repulsion. The ETH gap distribution, encoding level repulsion, suffices as an approximation. However, |hEn (t)|V |Em (t)i| ∼ 1 N for a mesoscale subengine in the macroscopic engine [10]. 193 Average work cost WAPT,1 of stroke-1 APT transitions: S begins stroke 1 in a temperature-TH Gibbs state. We focus on TH < ∞. Most of the state’s weight  − βH HGOE  lies below the energy band’s center: hEm i ≡ Tr e Z HGOE < 0. More levels lie above hEm i than below. Hence S more likely hops upward than drops. APT transitions draw the state toward maximal mixedness. Let S begin on the energy-Em level. A conditional density of states contributes to the probability that S hops to the energy-En level. Em has a negligible chance of lying within < hδi of En , due to level repulsion: µ(n|m) ∼ µ(En ) p |En − Em | (En − Em ) 2 + hδi2 . (C.134) We approximate sums with integrals, replacing Em with E and En with E 0: Z ∞ Z ∞

e − βH E µ (E) dE 0 µ(E 0 |E) PAPT E 0 |E · E 0 − E . WAPT,1 ∼ dE Z −∞ −∞ (C.135) The partition function appears in Eq. (C.116); the DOS, in Eq. (C.11); and the APT hopping probability, in Eq. (C.132): ! Z ∞ 2 N e − βH E − (E ) 2 e 2N E WAPT,1 ∼ (C.136) dE √ 2 Z −∞ 2πN E   ! Z ∞ 0 − E| (E 0 ) 2 |E v2 N − 0 2  e 2N E p (E 0 − E) . × dE  √ 0 − E) 2 + R2 ]2 2 2 N [(E 0 2 −∞ 2πN E (E − E) + hδi We change variables from E and E 0 to x := E − E 0 and y := E + E 0. As E = 21 (x + y) and E 0 = 12 (y − x), Z ∞ Z e− βH y/2 −y2 /4N E 2 ∞ |x|x 1 v2 N − βH x/2 −x 2 /4N E 2 dy e dx e e . WAPT,1 ∼ − p 2 8π N E 2 −∞ Z 2 2 2 2 −∞ x + hδi (x + R ) (C.137) We focus first on the x integral, I. The regularization factor, R, is small. (Later, √ we will see that all reasonable options for R ≤ v, which  hδi by assumption.) Therefore, the integral peaks sharply around x = 0. We Taylor-approximate the 2 2 slowly varying numerator exponentials to first order in x: e− βH x/2 e−x /4N E ∼     2 1 − β2H x 1 − 4Nx E 2 . The zeroth-order term vanishes by parity: βH I∼− hδi 2 βH dx p =− 2 hδi −∞ x 2 + hδi (x 2 + R2 ) 2 Z ∞ x 2 |x| Z ∞ 0 dx p x3 x 2 + hδi2 (x 2 + R2 ) 2 (C.138) . 194 The final equality follows from the integrand’s evenness. The square-root’s behavior varies between two regimes:   h i 2 x 1   + O , x  hδi  1 hδi hδi   h i 2 . = p hδi  , x  hδi x 2 + hδi2  1 + O  x (C.139) x We therefore split the integral: 2 βH 1 I∼− hδi hδi Z hδi 0 x3 dx 2 + (x + R2 ) 2 Z ∞ ! 1 dx 2 . x hδi (C.140) We have dropped the +R2 from the second integral’s denominator: Throughout the integration range, x  hδi, which  R. Integrating yields ! βH hδi2 log . (C.141) I≈− hδi R2 √ We have evaluated the x integral in Eq. (C.137). The y integral evaluates to 2 NπN Ee−N ( βH E) /4 . Substituting into Eq. (C.137) yields  !" !#  √ 2 2 βH 2 πN 1 v2 N hδi2  − log Ee−N ( βH E) /4  (C.142) WAPT,1 ∼ − 8π N E 2 N hδi R2 ! 1 1 v 2 βH hδi2 −N ( βH E)2 /4 e . (C.143) = √ √ log R2 2 π N E hδi The regularization R appears only in the logarithm. Hence the form of R barely impacts WAPT,1 . Which forms can R assume? R should be small in v and should have dimensions of energy. The only other relevant energy scales are hδi and E.4 We choose the “worst-case” R, which leads to the greatest hWAPT , 1i consistent with Ineq. (C.133) and with the smallness of v. WAPT,1 is large when R is small. √ v The possible regularizations small in v are v, hδi , and Ev . Consider substituting each value into Ineq. (C.133). If R ∝ v, Ineq. (C.133) lower-bounds v. Diabatic transitions should upper-bound, not lower-bound, the speed. We therefore disregard √ v v 1 hδi and E . Substituting R = v into Ineq. (C.133) yields 1 > N 1/4 , which is true. We therefore choose √ R= v. 4 δ (C.144) − is irrelevant, being a property of MBL systems. This calculation concerns the ETH phase. 2 195 Consequently, WAPT,1 ! 1 v 2 βH hδi2 −N ( βH E)2 /4 ∼√ log e . v N E hδi (C.145) Average work cost WAPT,3 of stroke-3 APT transitions: In the lowest-order approximation, (i) the stroke-1 tuning is adiabatic, and (ii) cold thermalization transfers S across just one gap. Let P↓ (P↑) denote the engine’s probability of dropping (rising) during cold thermalization. The average work cost is X e − βH E m X ( Z 0 dδ0 PMBL (|δ0 |) P↓ (|δ0 |) PAPT (n|m − 1)(En − Em−1 ) WAPT,3 ≈ Z −Wb n m Z Wb + dδ0 PMBL (δ0 ) P↑ (δ0 ) PAPT (n|m + 1)(En − Em+1 ) 0 " # Z 0 Z Wb 0 0 0 0 0 0 + 1− dδ PMBL (|δ |) P↓ (|δ |) − dδ PMBL (δ ) P↑ (δ ) −Wb 0 ) × PAPT (n|m)(En − Em ) . (C.146) We have artificially extended the gap variable δ0 to negative values: δ0 < 0 denotes a size-|δ0 | gap just below level m. The bracketed factor [1 − . . .] represents the probability that cold thermalization preserves the engine’s energy. Let us analyze WAPT,3 physically. Consider the TC = 0 limit, for simplicity. On average over trials, the engine’s state barely changes between strokes 1 and 3. Tiny globules of weight drop across single gaps. Hence most stroke-3 APT transitions look identical, on average over trials, to the stroke-1 APT transitions: WAPT,3 ≈ WAPT,1 + (correction). The correction comes from the probability-weight globules. APT transitions hop some globules off the bottoms of “working gaps” (Fig. 4.2), derailing trials that would have outputted Wtot ∼ hδi. But other globules, which began stroke 3 elsewhere in the spectrum, hop onto the bottoms of working gaps. The globules hopping off roughly cancel with the globules hopping on: WAPT,3 ≈ WAPT,1 , and hWAPT i ≈ WAPT,1 . Average work costs of Landau-Zener diabatic jumps: WLZ,1 and WLZ,3 Consider H (t) within the MBL phase (near, but not quite at, the end of stroke 1 or the start of stroke 3). Two energy levels can wiggle toward each other and apart. 196 Eigenenergies (E) Zoom ETH MBL Hamiltonian [H(t)] Figure C.5: Fractional-Landau-Zener transition: The straight solid green lines represent two eigenenergies. The engine ideally occupies the upper level throughout stroke 1. At the end of stroke 1, the energies approach each other. Zooming in on the approach shows that the lines are not straight, but wiggle slightly. A full Landau-Zener transition could occur if the approaching lines came very close together and then separated. The green dotted lines illustrate the hypothetical separation. Since the approaching energies do not separate, the engine may undergo an approximate fractional-Landau-Zener transition. The wiggling has a probability PLZ (∆) ≈ e−2π(δ − ) /v 2 (C.147) of inducing a Landau-Zener transition [3]. δ − roughly equals the size of the Hamiltonianperturbation matrix element that couples the wiggling-together states. The average work cost vanishes by parity. The engine’s probability of hopping upward equals its probability of dropping, by Eq. (C.147). Only hops to nearest neighbors have significant probabilities. Hence the existence of more levels above hH (t)i than below has no impact on WLZ,1 .5 The upward hops’ work cost cancels, on average, with the drops’ work cost: WLZ,3 = WLZ,1 = 0 . Average work costs of fractional-Landau-Zener diabatic jumps: Wfrac-LZ,1 and Wfrac-LZ,3 A Landau-Zener transition can occur when two energies begin far apart, come together, suffer a mixing of eigenstates, and separate. Eliminating the first or last step can induce a fractional-Landau-Zener transition. Such transitions can occur at the hWAPT i’s in Sec. C.1. There, we Taylor-approximated e −βH x to 0 first order in x := E − E , because S could hop across several levels. The zeroth-order term vanished 5 The imbalance impacted the by parity. The LZ calculation may be thought of as a truncation of the APT calculation at zeroth order, because S can hop only one gap. Put another way, in the APT calculation, the E integral affected the ∆ integral, preventing parity from sending the ∆ integral to zero. Here, the integrals decouple. 197 end of stroke 1 (Fig. C.5) or the start of stroke 3. We apply to these strokes the model in [8]. Modeling fractional-Landau-Zener transitions: De Grandi and Polkovnikov model an arbitrary portion of the LZ process using APT [8, Sec. II A]. We conjugate their Hamiltonian [their Eq. (21)] by the Hadamard √1 (σ x + σ z ): 2 Hfrac-LZ = δ − σ z + vt σ x . (C.148) This Hamiltonian captures the basic physics of growing energies and rotating eigenstates. De Grandi and Polkovnikov’s speed δ translates into our v.6 De Grandi and Polkovnikov’s time parameter t ∈ [t i ,t f ]. In the ordinary LandauZener problem, t ∈ (−∞, ∞). We approximate t ∈ (−∞, 0] at the end of stroke 1 and t ∈ [0, ∞) at the start of stroke 3. The qubit’s probability of hopping between eigenstates is [8, Eq. (29)]    1   2  3   − i (δ − ) 2 + vt f   v 2 (δ − ) 2 1 1  =
+
 . 16 Initial gap 6 Final gap 6 Pfrac-LZ ≈ 1 v 2 (δ − ) 2  + h  16  (δ ) 2 + (vt ) 2 i 3  (C.149) (C.150) We focus on stroke 3, which dominates hWfrac-LZ i. The second fraction vanishes, since t f = ∞. Let ∆0 denote the gap with which stroke 3 starts. We can no longer approximate the MBL “working gaps” as ∆0 ∈ [0,Wb ]: To avoid the ∆0 = 0 divergence, we refine our model. In which trials do fractional-LZ transitions cost Wfrac-LZ > 0? The trials that otherwise—in the absence of the transitions—would output Wtot > 0.7 Most otherwise-successful trials involve gaps ∆0 ∼ Wb . Hence δ  − we integrate ∆0 from Wb to Wb , wherein  ∈ W ,1 . b Simple approximation of exp Wfrac-LZ and associated v bound: The engine has 0 b a probability ∼ W hδi of neighboring an MBL gap ∆ . Wb . In the worst case, 6 The significance of δ changes between the general APT discussion and the fractional-LZ discussion in [8]. In the latter discussion, δ has dimensions of time2 . 7 A fractional-LZ transition costs work of two types. To describe them concretely, we suppose that the transition boosts the engine’s energy at the start of stroke 3: (1) S absorbs energy from the battery while hopping. (2) After hopping, typically, S slides up an energy level, like the top green line in Fig. 4.2. The sliding “undoes” the stroke-1 work extraction. The average type-(1) work cost ≈ Wb . The average type-(2) work cost ≈ hδi  Wb . Hence hWfrac-LZ i ≈ the type-(2) work. 198 whenever the engine neighbors such a gap, the engine suffers a stroke-3 fractionalLZ transition. Suppose, for simplicity, that TC = 0. Each such transition costs work ∼ hδi (the work that the trial would have outputted in the transition’s absence). Hence gaps ∆0 . Wb cost, at most, work Wb · hδi = Wb , hδi (C.151) on average. This bound shows that hWfrac-LZ i is small. Approximating dominant initial gaps with ∼ Wb implies a condition on v under which Eq. (C.150) is justified. The probability Pfrac-LZ must be normalized, so v(δ − ) 2 Pfrac-LZ ∼ 16(W ≤ 1. Solving for the speed yields )6 b v≤ 4(Wb ) 3 . δ− (C.152) We can bound v, instead, by (1) estimating hWfrac-LZ i and (2) demanding that fractional-LZ transitions cost less work than the engine outputs per ideal average cycle: hWfrac-LZ i  hWtot i. This inequality implies Ineq. (C.152), up to prefactors, we will find. Hence (C.150) leads to a self-consistent argument. Average work cost hWfrac-LZ,1 i of fractional-Landau-Zener diabatic transitions at the end of stroke 1: These transitions cost zero work, on average, by symmetry: Wfrac-LZ,1 = 0 . Suppose that S starts stroke 1 on the j th energy level. At the end of stroke 1, level j as likely approaches level j − 1 as it approaches level j + 1. A fractional-LZ transition as likely costs W > 0 as it costs W < 0. Hence hW i = 0. This symmetry is absent from Wfrac-LZ,3 , due to cold thermalization. Average work cost hWfrac-LZ,3 i of fractional-Landau-Zener diabatic transitions at the start of stroke 3: S starts the trial of interest with the ETH eigenenergy E, which tuning maps to the MBL E 0. No diabatic transitions occur during stroke 1, in the lowest-order approximation. E 0 neighbors at most one small gap, to lowest order. Cold thermalization hops S upward/downward with probability 1+e±β1 C |∆0 | . As stroke 3 begins, S reverses across the gap with probability Pfrac-LZ (∆0 ). Cold R0 thermalization has a probability 1 − P↓ − P↑ ≡ 1 − −W d∆0 PMBL (|∆0 |) 1+e −1βC |∆0 | − b R Wb 1 0 P 0) d∆ (∆ of preserving the engine’s energy. In this case, any 0 MBL 0 1+e β C ∆ 199 stroke-3 fractional-LZ transition costs Wfrac-LZ,1 = 0. Hence " Z −Wb Z ∞ e − βH E 1 0 Wfrac-LZ,3 ≈ dE µ(E) d∆0 PMBL (|∆0 |) 0 | Pfrac-LZ (∆ ) − β |∆ C Z 1+e −∞ −Wb Z ∞ × d∆ · ∆ PGOE (∆) 0 # Z Wb Z 0 0 e − βC ∆ 0 0 0 + d∆ · ∆ PGOE (|∆|) d∆ PMBL (∆ ) 0 Pfrac-LZ (∆ ) 1 + e − βC ∆ −∞ Wb + (1 − P↓ − P↑ ) Wfrac-LZ,1 + Wb . (C.153) The final term is consistent with (C.151). Computing the integral [7, App. G 8 iii] yields Wfrac-LZ,3 ≈ 1 v 2 (δ − ) 2 + Wb . 80 5 (Wb ) 5 (C.154) By assumption,  < 1. We will often assume that  ≈ 13 . Hence the final term in Eq. (C.154) is smaller than hWtot i ∼ Wb . Equation (C.154) implies an upper bound on v of the form in Ineq. (C.152). The Hamiltonian must be tuned slowly enough that fractional-LZ transitions cost less work than an ideal cycle outputs, on average: hWfrac-LZ i  hWtot i. The right-hand side roughly equals Wb , by Eq. (4.12). We substitute in for the left-hand side from √ 3 Eq. (C.154). Solving for the speed yields v  80 5 (Wδ −b ) . For every tolerance  ∈ (0, 1), there exist speeds v such that the inequality is satisfied. For simplicity, we suppose that  ≈ 31 , such that the overall constant ≈ 1. The bound reduces to v (Wb ) 3 . δ− (C.155) This bound has the form of Ineq. (C.152). Our approximation (C.150) leads to a self-consistent argument. Diabatic correction to the efficiency η MBL The efficiency has the form η MBL := hWtot i hWtot i hWtot i   = = hQin i hWtot i − hQ2 i hWtot i 1 − hQ2 i (C.156) hWtot i Here, hWtot i denotes the net work extracted per trial, on average over trials. [Earlier, hWtot i denoted the average net work extracted per trial in which H (t) is tuned adiD E adiab the adiabatic abatically.] We Taylor-approximate to first order, relabel as Wtot 200 approximation (C.122), and denote by hWdiab i the average total per-cycle diabatic D E adiab  hW 2i work cost: η MBL = 1 + W adiabhQ−hW . Invoking W diab i , we Taylortot h tot i diab i   2i diab i approximate again: η MBL ≈ 1 + WhQadiab 1 + hW . adiab i h tot i hWtot adiab the adiabatic estimate (C.126) of the efficiency: η adiab We relabel as η MBL MBL ≈ η MBL + 2i . Substituting in from Eq. (C.120), and substituting in the leadinghWdiab i hQ adiab 2 hWtot i order term from Eq. (C.122), yields adiab η MBL ≈ η MBL − hWdiab i adiab ≡ η MBL − φdiab . 2 hδi (C.157) For simplicity, we specialize to TH = ∞ and TC = 0 . The correction becomes φdiab = 1 hWdiab i = WAPT,3 2 hδi TC =0,TH =∞ 2 hδi TC =0,TH =∞ ≈ 1 v 2 (δ − ) 2  Wb + , 5 5 160 (Wb ) hδi 2 hδi (C.158) by Eqs. (C.145) and (C.154). As expected, work-costing diabatic jumps detract from the efficiency slightly. The √ v δ− first term is suppressed in in hδi  1 and in hδi  1. The second term is suppressed Wb 1  in hδi  1 and in a constant 2 ≈ 6 . C.2 Phenomenological model for the macroscopic MBL Otto engine The macroscopic MBL Otto engine benefits from properties of MBL (Sec. 4.2): localization and local level repulsion. We understand these properties from (1) Anderson insulators [12] and (2) perturbation theory. Anderson insulators are reviewed in Sec. C.2. Local level repulsion in Anderson insulators [13] in the strong-disorder limit is reviewed in Sec. C.2. Section C.2 extends local level repulsion to MBL. Local level repulsion’s application to the MBL engine is discussed in Sec. C.2. Throughout this section, N denotes the whole system’s length. Anderson localization Consider a 1D spin chain or, equivalently, lattice of spinless fermions. An Andersonlocalized Hamiltonian HAnd has almost the form of Eq. (4.35), but three elements are removed: (1) the t-dependence, Q(h(αt )), and the interaction. [The σ j · σ j+1 is   replaced with σ +j σ −j+1 + σ −j σ +j+1 . The site- j raising and lowering operators are     y y denoted by σ +j := 12 σ xj + iσ j and σ −j := 21 σ xj − iσ j .] Let |0i denote some reference state in which all the spins point downward (all the fermionic orbitals are empty). In this section, we focus, for concreteness, on the 201 properties of single-spin excitations relative to |0i [12, 13]. The ` th excitation is P represented, in fermionic notation, as x ψ` (x) σ +x ` |0i. The single-excitation wave functions ψ` (x) are localized: x ` denotes the point at which the probability density |ψ` (x)| 2 peaks. The wave function decays exponentially with the distance |x − x ` | from the peak: s 2 ψ` (x) ≈ e−|x−x ` |/ξAnd . (C.21) ξAnd The localization length varies with the Hamiltonian parameters as [7, App. H 2] ξAnd ∼ 1 . ln h (C.22) Local level repulsion in Anderson insulators We begin with the infinitely localized limit h → ∞. We take E → 0 to keep the Hamiltonian’s energy scale finite. The hopping terms can be neglected, and particles on different sites do not repel. Single-particle excitations are localized on single sites. The site-i excitation corresponds to an energy 2Ehhi . Since the on-site potentials h · hi are uncorrelated, neighboring-site excitations’ energies are uncorrelated. Let us turn to large but finite h. Recall that h · hi is drawn uniformly at random from [−h, h]. The uniform distribution has a standard deviation of √h  1 . Therefore, 3 h|hi − hi+1 |  1 for most pairs of neighboring sites. The hopping affects these sites’ wave functions and energies weakly. But with a probability ∼ 1h , neighboring sites have local fields h · hi and h · hi+1 such that h|hi − hi+1 |  1. The hopping hybridizes such sites. The hybridization splits the sites’ eigenvalues by an amount p ∼ h2 (hi − hi+1 ) 2 + E 2 ≥ E. Consider, more generally, two sites separated by a distance L . Suppose that the sites’ disorder-field strengths are separated by < 1/h L . (The upper bound approximates the probability amplitude associated with a particle’s hopping the L intervening sites). The sites’ excitation energies and energy eigenfunctions are estimated perturbatively. The expansion parameter is 1/h . To zeroth order, the energies are uncorrelated and (because h|hi − hi+L | < 1/h L ) are split by < E/h L . The eigenfunctions are hybridized at order L . The perturbed energies are split by ≥ E/h L ∼ Ee−L/ξAnd . [Recall that ξAnd ∼ 1/ ln h, by Eq. (C.22).] Hence eigenstates localized on nearby sites have correlated energies: The closer together sites lie in real space, the lower the probability that they correspond to 202 similar energies. This conclusion agrees with global Poisson statistics: Consider a large system of N  1 sites. Two randomly chosen single-particle excitations are typically localized a distance ∼ N apart. The argument above implies only that the energies lie > Ee−N/ξAnd apart. This scale is exponentially smaller (in N) than the average level spacing ∼ Eh N between single-particle excitations.8 We can quantify more formally the influence of hybridization on two energies separated by ω and associated with eigenfunctions localized a distance L apart. The level correlation function is defined as 1 X + |h0|σi+ |ni| 2 |h0|σi+L |n0i| 2 δ(En − En0 − ω) − µ̃(ω) 2 . (C.23) R(L,ω) := 2 N i,n,n0 The spatially averaged density of states at frequency ω is denoted by µ̃(ω) := 1 P + 2 0 n |h0|σi |ni| δ(En −ω) . |ni and |n i denote eigenstates, corresponding to singleN particle excitations relative to |0i, associated with energies En and En0 . In the Anderson insulator, R(L,ω) ≈ 0 when ω  Ee−L/ξAnd : Levels are uncorrelated when far apart in space and/or energy. When energies are close (ω  Ee−L/ξAnd ), R(L,ω) is negative. These levels repel (in energy space). Generalization to many-body localization The estimates above can be extended from single-particle Anderson-localized systems to MBL systems initialized in arbitrary energy eigenstates (or in position-basis product states). R(L,ω) is formulated in terms of matrix elements h0|σi+ |ni of local operators σi+ . The local operators relevant to Anderson insulators have the forms of the local operators relevant to MBL systems. Hence R(L,ω) is defined for MBL as for Anderson insulators. However, |0i now denotes a generic many-body state. Let us estimate the scale JL of the level repulsion between MBL energies, focusing on exponential behaviors. The MBL energy eigenstates result from perturbative expansions about Anderson energy eigenstates. Consider representing the Hamiltonian as a matrix M with respect to the true MBL energy eigenbasis. Off-diagonal matrix elements couple together unperturbed states. These couplings hybridize the unperturbed states, forming corrections. The couplings may be envisioned as rearranging particles throughout a distance L. 8 The average level spacing between single-particle excitations scales as ∼ 1/N for the following reason. The reference state |0i consists of N downward-pointing spins. Flipping one spin upward yields a single-particle excitation. N single-particle-excitation states exist, as the chain contains N sites. Each site has an energy ∼ ±Eh, to zeroth order, as explained three paragraphs ago. The excitation energies therefore fill a band of width ∼ Eh . An interval ∼ ENh therefore separates singleparticle-excitation energies, on average. 203 MBL dynamics is unlikely to rearrange particles across considerable distances, due to localization. Such a rearrangement is encoded in an off-diagonal element Mi j of M. This Mi j must be small—suppressed exponentially in L. Mi j also forces the eigenstates’ energies apart, contributing to level repulsion [7, App. F]. Hence the level-repulsion scale is suppressed exponentially in L: JL ∼ Ee−L/ζ , (C.24) for some ζ . At infinite temperature, ζ must < ln12 for the MBL phase to remain stable [14]. Substituting into Eq. (C.24) yields JL < 2EL . The level-repulsion scale is smaller than the average gap. The size and significance of JL depend on the size of L. At the crossover distance ξ, the repulsion JL (between energy eigenfunctions localized a distance ξ apart) becomes comparable to the average gap ∼ 2Eξ between the eigenfunctions in the same length-ξ interval: Ee−ξ/ζ ∼ 1e 2Eξ . Solving for the crossover distance yields ξ∼ 1 1 ζ − ln 2 . (C.25) Relation (C.25) provides a definition of the MBL localization length ξ . [This ξ differs from the Anderson localization length ξAnd , Eq. (C.22).] Solving for ζ yields ζ∼ 1 1 ξ + ln 2 . (C.26) The MBL Otto cycle involves two localization lengths in the thermodynamic limit. In the shallowly localized regime, ξ = ξ > . Each eigenfunction has significant weight on ξ > ∼ 12 sites, in an illustrative example. In the highly localized regime, ξ = ξ < . Eigenfunctions peak tightly, ξ < ∼ 1 . Suppose that the particles are rearranged across a large distance L  ξ. The levelrepulsion scale JLξ ∼ Ee−L/ξ 2−L . (C.27) In the MBL engine’s very localized regime, wherein ξ = ξ < , if L = ξ > equals one subengine’s length, JLξ = δ − . Now, suppose that particles are rearranged across a short distance L . ξ. Randommatrix theory approximates this scenario reasonably (while slightly overestimating 204 the level repulsion). We can approximate the repulsion between nearby-eigenfunction energies with the average gap hδi(L) in the energy spectrum of a length-L system: JL≤ξ ∼ hδi(L) ∼ E . 2L (C.28) Application of local level repulsion to the MBL Otto engine in the thermodynamic limit Consider perturbing an MBL system locally. In the Heisenberg picture, the perturbing operator spreads across a distance L(t) ∼ ζ ln(Et) [15]. (See also [16].) The longer the time t for which the perturbation lasts the farther the influence spreads. Consider tuning the Hamiltonian infinitely slowly, to preclude diabatic transitions: t → ∞ . Even if the Hamiltonian consists of spatially local terms, the perturbation to each term spreads across the lattice. The global system cannot be subdivided into independent subengines.9 The global system’s average gap vanishes in the thermodynamic limit: hδi → 0 . The average gap sets the scale of one engine’s per-cycle power, hWtot i. Hence the per-cycle power seems to vanish in the thermodynamic limit: hWtot i < hδi ∼ 0 . But consider tuning the Hamiltonian at a finite speed v. Dimensional analysis suggests that the relevant time scale is t ∼ Ev . Local perturbations affect a region of length ∼ L(E/v) ∼ ζ ln(E 2 /v). On a length scale L(E/v), global level correlations govern the engine’s performance less than local level correlations do, i.e., less than R(L(E/v),ω) does. This correlator registers level repulsion at a scale independent of N. Finite-speed tuning enables local level repulsion renders finite the average gap accessible to independent subengines, the hδi that would otherwise close in the thermodynamic limit. Each mesoscale subengine therefore outputs hWtot i > 0 . We can explain the gap’s finiteness differently: Suppose that the engine’s state starts some trial with weight on the j th energy level. The eigenenergies wiggle up and down during stroke 1. The j th energy may approach the ( j − 1) th . Such closetogether energies likely correspond to far-apart subengines. If the levels narrowly avoided crossing, particles would be rearranged across a large distance. Particles 9 Granted, subengines are coupled together even if the Hamiltonian is quenched infinitely quickly: Hsim (t) encodes a nearest-neighbor interaction, for example. That interaction might be regarded as coupling the edge of subengine k with the edge of subengine k + 1 . But subengines’ edges may be regarded as ill-defined. The sites definitively in subengine k, near subengine k’s center, should not couple to the sites near subengine `’s center, for any ` , k , if the subengines are to function mostly independently. Alternatively, one may separate subenegines with “fallow” buffer zones. 205 must not be, as subengines must function independently. Hence the engine must undergo a diabatic transition: The engine’s state must retain its configuration. The engine must behave as though the approaching energy level did not exist. Effectively removing the approaching level from available spectrum creates a gap in the spectrum. One can create such a gap (promote such diabatic transitions) by tuning the Hamiltonian at a finite v (Suppl. Mat. C.4). C.3 Constraint 2 on cold thermalization: Suppression of high-order-in-thecoupling energy exchanges Section 4.2 introduces the dominant mechanism by which the bath changes a subengine’s energy. The subengine energy change by an amount ∼ Wb , at a rate ∼ g. Higherorder processes can change the subengine energy by amounts > Wb and operate at rates O(g` ), wherein ` ≥ 2. The subengine should thermalize across just small gaps. Hence the rate-g` processes must operate much more slowly than the rate-g processes: g must be small. We describe the higher-order processes, upper-bound g, and lower-bound τth . The higher-order processes can be understood as follows. Let Htot = Hmacro (τ) + Hbath + Hint denote the Hamiltonian that governs the engine-and-bath composite. Htot generates the time-evolution operator U (t) := e−iHtot t . Consider Taylorexpanding U (t). The ` th term is suppressed in g` ; contains 2` fermion operators c j and c†j 0 ; and contains ` boson operators bω and bω† 0 . This term encodes the absorption, by the bath, of ` energy quanta of sizes ≤ Wb . The subengine gives the bath a total amount ∼ `Wb of heat. The subengine should not lose so much heat. Hence higher-order processes should occur much more slowly than the rate-g processes: τhigh-ord.  τth . (C.31) Let us construct an expression for the left-hand side. Which processes most urgently require suppressing? Processes that change the subengine’s energy by & hδi. Figure 4.2 illustrates why. If the right-hand leg has length & hδi, the right-hand leg might be longer than the left-hand leg. If the right-hand leg is longer, the trial yields net negative work, Wtot < 0. The bath would absorb energy hδi from a subengine hδi packets of energy ∼ Wb each. Hence the bath would appear to by absorbing ∼ W b hδi need to flip ∼ L = Wb spins to absorb energy ∼ hδi. (We switch from fermion language to spin language for convenience.) However, the length-L spin subchain has a discrete effective energy spectrum. The spectrum might lack a level associated with the amount (initial energy) − hδi of energy. If so, the bath must flip more than 206 hδi suggest that the bath must flip ∼ ξ > spins (Suppl. Wb spins. Local level correlations o n hδi Mat. C.2). Hence L = max Wb , ξ > . Energy is rearranged across the distance L at a rate ∝ g L . Having described the undesirable system-bath interactions, we will bound g via Fermi’s Golden Rule, Eq. (4.32). Let Γf i ∼ 1/τhigh-ord. now denote the rate at which order-g L interactions occur. The bath DOS remains µbath (Ei f ) ∼ W1b . Let us estimate the matrix-element size |h f |V |ii|. The bath flips each spin at a rate g (modulo a contribution from the bath’s DOS). Flipping one spin costs an amount ∼ E of energy, on average. [E denotes the per-site energy density, as illustrated in  L Eq. (4.35).] Hence L spins are flipped at a rate ∼ E Eg . The initial E is included for dimensionality. We substitute into Fermi’s Golden Rule [Eq. (4.32)], then solve for the time: ) ( Wb E 2(L−1) hδi τhigh-ord. ∼ , ξ> . (C.32) wherein L = max Wb g 2L We substitute from Eqs. (C.32) and (4.33) into Ineq. (C.31). Solving for the coupling yields ( ) hδi (L−2)/(L−1) 1/(L−1) gE δ− , wherein L = max , ξ> . (C.33) Wb Substituting back into Eq. (4.33) yields a second bound on τth : τth  Wb E (δ − ) L ! 2/(L−1) ( , wherein hδi L = max , ξ> Wb ) . (C.34) Let us express the bound in terms of localization lengths. We set Wb ∼ hδi 10 , as usual. We approximate L ± 1 ∼ L ∼ ξ > . We substitute in for hδi from Eq. (4.23) and for δ − from Eq. (C.417): τth  1 2ξ > /ξ < 2ξ > 2 e . 10E (C.35) This inequality is looser than Ineq. (4.30): The no-higher-order-processes condition is less demanding than Markovianity. C.4 Optimization of the MBL Otto engine Section 4.2 introduced the macroscopic MBL engine. This section provides background about identifies the engine’s optimal parameter regime. The Hamiltoniantuning speed v is bounded in Sec. C.4; the cycle time τcycle , in Sec. C.4; and the cold-bath bandwidth Wb , in Sec. C.4. 207 We focus on order-of-magnitude estimates and on exponential scaling behaviors. Wb and βH necessitate exceptions. These quantities do not inherently scale in any 1 particular ways, unlike hδi and δ − . We choose Wb ∼ 10 hδi, in the spirit of Sec. 4.4, and βH  √1 , in accordance with Suppl. Mat. C.1. E N The calculations in Suppl. Mat. C.1 concern one length-N mesoscale engine. We translate the calculations into the thermodynamic limit approximately: N is replaced with the subengine length ξ > . Energies such as hWtot i are multiplied by the number of subengines, ∝ Nmacro . Granted, shallowly-localized-MBL energy spectra (E) (δ). This distribution can be replaced with, e.g., the Rosenzweigdo not obey PGOE (E) (δ) captures the crucial physics, some level rePorter distribution [17]. But PGOE pulsion. Bounds on the Hamiltonian-tuning speed v As Hmacro (t) is tuned, the time-t energy eigenstates become linear combinations of the old eigenstates. Levels narrowly avoid crossing. The engine must have high probabilities of (i) transitioning diabatically between energy eigenstates |ψ1 i and |ψ2 i coupled strongly by nonlocal operators, so that subengines barely interact, and (ii) transitionining adiabatically between |ψ1 i and |ψ2 i coupled strongly by local operators, to approximate adiabatic ideal. Requirement (i) lower-bounds v (Sec. C.4), and (ii) upper-bounds v (Sec. C.4). Lower bound on v from subengine independence Figure C.6 illustrates three energy eigenstates. Let L denote the scale of the distance over which energy is rearranged during a transition between |ψ2 i and |ψ3 i. If L ≥ 1.5ξ > , energy is transferred between subengines.10 Subengines should evolve independently. Hence the engine must have a low probability of transitioning from |ψ2 i to |ψ3 i. The crossing must have a high probability of being diabatic. This demand can be rephrased in terms of work. hWtot i denotes the average work D E cost denote the work cost of outputted by one ideal subengine per cycle. Let Wadiab undesirable adiabatic transitions incurred, on average, per subengine per cycle. The cost must be much less than the extracted ideal: D E cost Wadiab  hWtot i . (C.41) 10 One may separate neighboring subengines with “fallow” buffer zones. Buffers would loosen the inequality to L  1.5ξ > + (buffer length). 208 Eigenenergies (Et) | 3i | 2i | 2i | 3i | 1i Shallowly localized Deeply localized Hamiltonian [Hmacro(t)] Figure C.6: Desirable diabatic transition between energy eigenfunctions localized in different subengines: The green, sloping solid lines represent elements |ψ1 i and |ψ2 i of the diabatic basis. (The functional forms of the |ψ` i’s remain constant: Suppose that, at some instant t, |ψ1 i equals some linear combination c1 |↑ . . . ↑i + . . . c2 N |↓ . . . ↓i of tensor products of σ zj eigenstates. |ψ1 i equals that combination at all times.) The dashed, red line represents an energy eigenstate |ψ3 i that turns into |ψ2 i via long-range rearrangements of much energy. The eigenstates’ energies change as the Hamiltonian is tuned. The blue, dotted line represents a state desirable for the engine to occupy. The right-hand side ∼ Wb , to lowest order, by Eq. (4.12). Let us estimate the left-hand side. We label as a “close encounter” an approach, of two levels, that might result in an undesirable adiabatic transition. The left-hand side of Ineq. (C.41) has the form ! ! D E Work cost Prob. of undesirable adiab. transition cost Wadiab ≈ 1 undesirable adiab. transition 1 close encounter (C.42) ! ! # close encounters Avg. # strokes during which can lose work to adiab. transitions × . 1 tuning stroke 1 cycle We estimate the factors individually. We begin with the first factor, assisted by Fig. C.7. Suppose that the engine starts a tuning stroke just above or below a working gap (on a green, solid line). The engine might undesirably transition adiabatically to a red, dashed line. hδi denotes the average gap in the part of the spectrum accessible to an ideal mesoscale subengine [Eq. (4.23)]. The red line likely originated, in the shallowly-MBL regime, a distance ∼ (const.) hδi away. Hence one undesirable adiabatic transition costs ∼ hδi. The Landau-Zener formula gives the second factor in Eq. (C.42) [3]: Padiab = 1 − Pdiab = 1 − e−2πJ /v ≈ 2π 2 J2 . v (C.43) 209 Eigenenergy (Et) ⇠h i ⇠ Wb Shallowly localized Deeply localized Hamiltonian [Hmacro(t)] Figure C.7: “Close encounters” that might result in undesirable adiabatic transitions: The sloping, green solid lines represent the top and bottom of a “working gap.” The red, dashed lines represent other energy levels. Some cross (or anticross with) the working levels. Each such “close encounter” should proceed diabatically. Subengine Subengine Subengine x ⇠loc 1.5⇠loc Figure C.8: Condition forbidding subengines from interacting: The long black line represents the composite engine. Each subengine has size ξ > , the Hamiltonian’s localization length in the shallow-localization regime. Subengines must not interact: Consider particles on one subengine’s left-hand side. Those particles must not shift to the middle of any neighboring subengine, across a distance 1.5ξ > . J denotes the magnitude of the transition-matrix element between the states. J roughly equals the least size JL∼1.5ξ > reasonably attributable to any gap accessible to a subsystem of length L ∼ 1.5ξ > (Suppl. Mat. 4.2). The condition L ∼ 1.5ξ > ensures that the lefthand end of subengine ` fails to interact with the middle of subengine ` ± 1 (Fig. C.8).11 According to Eq. (C.27), J1.5ξ > ∼ Ee−1.5ξ > /ξ (t) 2−1.5ξ > . (C.44) ξ (t) denotes the time-t localization length. Substituting into Eq. (C.43) yields 2 (J1.5ξ > ) 2 −3ξ > /ξ (t) −3ξ > E ∼e 2 . Padiab ∼ v v (C.45) To estimate the third factor in Eq. (C.42), we return to Fig. C.7. How many dashed, 11 If buffers separate the subengines, the condition becomes L > 1.5ξ weakens. > . The lower bound on v 210 red lines cross the bottom green line? Roughly 1 [(# red lines inside the working gap in the shallow-localization regime) 2 − (# red lines inside the working gap in the deep-localization regime)] . (C.46) Let us estimate the first term. When Hmacro (t) is shallowly localized, the working gap is of size ∼ hδi ∼ E2−ξ > [Eq. (4.23)]. The DOS accessible to a size-(1.5ξ > ) 1.5ξ > 1 subsystem is µ (1.5ξ > ) (E) ∼ hδi(1.5ξ ∼ 2 E . Hence roughly hδi × µ (1.5ξ > ) (E) ∼ >) 2ξ > /2 red lines begin inside the working gap. The second term in (C.46) . Wb × µ (1.5ξ > ) (E), as shown in Fig. C.7. By design, Wb  hδi (Suppl. Mat. C.1). Hence the second term in is much less than the first and can be neglected. Hence a subengine suffers about 1 hδi ∼ 2ξ > /2 hδi × µ1.5ξ > (E) ∼ (1.5ξ > ) 2 hδi (C.47) close encounters per stroke. D E cost > Finally, we estimate the last factor in Eq. (C.42). Adiabatic transitions cost Wadiab 0 only during otherwise-successful trials—trials in which the subengine of interest would have outputted Wtot > 0 in the absence of undesirable adiabatic transitions. Why only otherwise-successful trials? Suppose, for simplicity, that TC = 0. First, we argue that inter-subengine adiabatic transitions cost hW i > 0 during otherwise-successful trials. Suppose that the engine starts a trial on the downward-sloping green line in Fig. C.7. During stroke 1, intersubengine adiabatic hops tend to lift the engine to upward-sloping red, dashed lines. Upward hops cost W > 0. During stroke 3, the hops tend to lift the engine to red lines that slope upward from right to left. Such hops cost W > 0. Hence cross-engine adiabatic hops during otherwise-successful trials cost hW i > 0. Now, we argue that intersubengine adiabatic hops incurred during no-ops cost hW i = 0. By “no-op,” we mean a trial during which, in the absence of undesirable hops, the subengine of interest would output Wtot = 0. Suppose that the engine starts some trial on the bottom green line in Fig. C.7. The engine would slide up the bottom green line during stroke 1, then slide downward during stroke 3: Wtot = 0. Interengine adiabatic hops during stroke 1 tend to drop the engine to a red, dashed line, costing W < 0. The hops during stroke 3 tend to raise the engine to a red, 211 dashed line, costing W > 0. The two costs cancel each other, on average, by symmetry. An analogous argument concern no-ops begun on a downward-sloping green line. Hence interengine adiabatic hops during no-ops cost hW i = 0. We can now assemble the final factor in Eq. (C.42): Avg. # strokes during which can lose work to adiab. transitions 1 cycle ! ! 2 strokes Prob. of success ≈ 1 otherwise successful trial 1 hop-free trial Wb Wb ∼ . ≈2 hδi hδi (C.48) (C.49) The final factor was estimated below Eq. (4.12). We have estimated the factors in Eq. (C.42). Substituting in from Eqs. (4.23), (C.45), (C.47), and (C.49) yields D E (J1.5ξ > ) 2 Wb (J1.5ξ > ) 2 Wb hδi hδi cost Wadiab ∼ hδi · · · = . (1.5ξ ) > v v hδi hδi hδi(1.5ξ > ) (C.410) We substitute into Ineq. (C.41) and solve for v: v  (J1.5ξ > ) 2 hδi hδi (1.5ξ > ) ∼ e−3ξ > /ξ (t) 2−2.5ξ > E 2 . (C.411) hδi The bound is twofold small in J1.5ξ >  E and onefold large in hδi(1.5ξ > 1. >) Let us evaluate the bound in the very localized regime, whose ξ (t) ∼ ξ < , and in the shallowly localized regime, whose ξ (t) ∼ ξ > . If ξ > = 12 and ξ < = 1,    10−25 E 2 , very localized v .   10−11 E 2 , shallowly localized  (C.412) Upper bound on the Hamiltonian-tuning speed v D E cost of work, on average Undesirable diabatic transitions cost a total amount Wdiab (Suppl. Mat. C.1).12 One ideal, adiabatic subengine outputs hWtot i ∼ Wb per trial, on average. The requirement D E cost  hWtot i Wdiab 12 The average diabatic work cost was denoted by for emphasis and clarity. (C.413) D E cost earlier. The subscript is added here Wdiab 212 upper-bounds v. APT transitions dominate the left-hand side of Ineq. (C.413) in the shallowly localized regime. Fractional-Landau-Zener transitions dominate in the very localized regime. Upper bound on v in the shallowly localized  2  regime: Substituting from Eq. (C.145) 2 v 2 βH 1 into Ineq. (C.413) yields13 √ Ehδi log hδiv e−N ( βH E) /4  Wb . The √1 and the N N log contribute subdominant (nonexponential) factors. The explicit exponential ≈ 1, √ since N βH E  1 by assumption: s v 2 βH hδi Wb E  Wb ⇒ v . (C.414) E hδi βH √ √ 1 . Since N β E  1 by assumption,  N E. We We approximate Wb ∼ hδi H 10 βH 1 approximate βH ∼ N E. We substitute into Ineq. (C.414) and ignore subdominant factors: v  hδi E . (C.415) This bound is looser than the small-parameter assumption E2 ∼ 2−2ξ > E 2 . (C.416) N2 in Suppl. Mat. C.1. APT transitions do not upper-bound the tuning speed painfully. v  hδi2 ∼ The upper bound (C.416) lies above the lower bound (C.411). The upper bound is suppressed only in 2−2ξ > ; the lower bound, in e−3ξ > /ξ (t) 2−2.5ξ > . The upper bound ∼ 10−7 E 2 , if ξ > = 12. The lower bound ∼ 10−11 E 2 [Ineq. (C.412)]. Therefore, the bounds are consistent with each other. Upper bound on v in the deeply localized regime from fractional-LZ transitions: We have already derived the bound (C.155). We assess the bound’s size by   3 hδi 2 b expressing the right-hand side in terms of small parameters: v  W δ − hδi . hδi hδi hδi b The right-hand side is threefold suppressed in W hδi  1 and is large in δ −  Wb  1. We can express the bound in terms of localization lengths. We substitute in for 1 hδi from Eq. (4.23) and assume that Wb ∼ 10 hδi. A δ − expression follows from substituting ξ = ξ < and L = ξ > into Eq. (C.27): δ − ∼ Ee−ξ > /ξ < 2−ξ > . (C.417) 13 Equation (C.145) follows from the ∼ √1 scaling of a matrix element in the ETH phase. The N ETH phase features in the mesoscale-MBL-engine cycle where shallowly localized MBL features in the thermodynamically-large-MBL-engine cycle. In the MBL phase, the matrix element ∼ N1 √ (footnote 3). Introducing the extra √1 would loosen the bound (C.415) by a factor of N . N 213 Inequality (C.155) becomes v 1 ξ > /ξ < −2ξ > 2 e 2 E . 103 (C.418) Let us check that this upper bound lies above the lower bound, Ineq. (C.411). The lower bound is suppressed in e−3ξ > /ξ < 2−2.5ξ > . The upper bound is suppressed only in 2−2ξ > and is large in e ξ > /ξ < . Hence (lower bound)  (upper bound) by scaling. More concretely, substituting ξ > = 12 and ξ < = 1 into Ineq. (C.155) yields v  10−5 E 2 . (C.419) This upper bound above below the lower bound, v  10−25 E 2 [Ineq. (C.412)]. The bounds are consistent and lie orders of magnitude apart. Time τcycle required to implement a cycle Different cycle segments must satisfy different bounds on v or on implementation time. We (1) compare the bounds and (2) derive bounds on time from bounds on v: 1. To suppress undesirable fractional-Landau-Zener transitions, the tuning speed 3 must satisfy v  (Wδ −b ) ∼ 1013 e ξ > /ξ < 2−2ξ > E 2 in the deeply localized regime [Ineqs. (C.155) and (C.418)]. This bound implies a bound on a time scale. Since dt E t v := E dα dt , dα t = v . Fear of fractional-LZ transitions limits v during some part of stroke 3. Imagine that that part extends throughout stroke 3. αt runs from 1 to 0, so stroke 3 lasts for a time ! Z 0 Z 0 E E dt dαt = − dαt = (C.420) τfrac-LZ = v v 1 1 dαt δ− E  ∼ 103 e−ξ > /ξ < 22ξ > /E . (C.421) (Wb ) 3 If ξ > = 12 and ξ < = 1, τfrac-LZ ∼ 105 /E . 2. Let τAPT denote the time for which fear of APT transitions governs v: v ≤ hδi2 ∼ 2−2ξ > E 2 [Ineq. (C.416)]. τAPT includes stroke 1. Hence τAPT ∼ Ev ∼ 22ξ > E ∼ 107 /E. The final expression follows from ξ > = 12. τAPT  τfrac-LZ , so APT transitions bound the tuning time more stringently than fractional-LZ transitions do, if βH > 0. 2ξ > /ξ < 23ξ > 3. The engine thermalizes with the cold bath for a time τth > W E(δ )2 ∼ 10 Ee b − [Ineqs. (4.30)]. If ξ > = 12 and ξ < = 1, τth > 1022 /E. Cold thermalization lasts much longer than the Hamiltonian tunings, dominating the cycle time: 2 214 τcycle ∼ τth . (Hot thermalization requires less time than cold, involving an ordinary bath bandwidth.) Bounds on the cold-bath bandwidth Wb Wb must be large enough to couple nearby energies, deep in the MBL phase, accessible to a subengine. Hence Wb > δ − , estimated in Eq. (C.417). Wb must be small enough to couple only levels whose energies likely separate during stroke 3, such that subengines output hWtot i > 0. Wb must be less than the average level spacing hδi accessible to a subengine [Eq. (4.23)]. Hence δ − < Wb  hδi , C.5 or Ee−ξ > /ξ < 2−ξ > < Wb  E . 2ξ > (C.422) Numerical simulations of the MBL Otto engine We simulated one 12-site mesoscale engine at half-filling. (We also studied other system sizes, to gauge finite-size effects.) The random-field Heisenberg Hamiltonian (4.35) governed the system. We will drop the subscript from Hsim (t). Call the times at which the strokes end t = τ, τ0, τ00, and τ000. For each of Nreals ∼ 1, 000 disorder realizations, we computed the whole density matrix ρ(t) at t = 0, τ, τ0, τ00, τ000. (See Suppl. Mat. C.5 and C.5 for an explanation of how.) The engine’s time-t internal energy is E(t) = Tr(H (t) ρ(t)) . The quantities of interest are straightforwardly hW1 i = E(0) − E(τ) , hW3 i = E(τ000 ) − E(τ00 ) , hQ2 i = E(τ00 ) − E(τ0 ) , and hQ4 i = E(0) − E(τ000 ) . (C.51) (C.52) We disorder-average these quantities before dividing to compute the efficiency, i+hW3 i η MBL = 1 − hW1hQ . 4i Scaling factor We wish to keep the DOS constant through the cycle. To fix µ(E), we rescale the Hamiltonian by a factor Q(h(αt )). We define Q2 (h(αt )) as the disorder average of the variance of the unrescaled DOS: * 2 Q (h(αt )) := ! ! 2+ * ! 2+ N N 1 X 2 1 X 1 1 2 E − Ej Tr( H̃ (t)) − Tr( H̃ (t)) . = N j=1 j N j=1 N N disorder disorder (C.53) (C.54) 215 The H̃ (t) denotes an unrescaled variation on the random-field Heisenberg Hamiltonian H (t) of Eq. (4.35): −1 N   NX X z   σ j · σ j+1 + h(αt ) hjσj  . H̃ (t) := E    j=1 j=1   (C.55) To compute Q2 (h(αt )), we rewrite the unrescaled Hamiltonian as −1  −1 N   NX X  NX σ +j σ −j+1 + h.c. + σ zj σ zj+1 + h(αt ) h j σ zj  . H̃ (t) = E 2   j=1 j=1 j=1   (C.56) We assume that N is even, and we work at half-filling. The N2 -particle subspace has N  . dimensionality N = N/2 Q Let us calculate some operator traces that we will invoke later. Let X := Nj=1 σ x denote the global spin-flip operator. For any operator A such that X † AX = − A,   Tr( A) = Tr X † AX = −Tr( A) . (C.57) We have used the evenness of N, which implies the invariance of the half-filling subspace under X. Also, Tr( A) = 0. In particular, 0 = Tr(σ zj ) = Tr(σ zj σ zj0 σ zj00 ), if j , j 0 , j 00. Traces of products of even numbers of σ z factors require more thought: Tr(σ zj σ zj+1 ) = (# states j, j + 1 =↑↑) + (# states j, j + 1 =↓↓) − 2(# states j, j + 1 =↑↓) ! ! ! N −2 N −2 N −2 = + −2 N/2 − 2 N/2 N/2 − 1 1 = −N . (C.58) N −1 Similarly,     Tr [σ +j σ −j ][σ −j+1 σ +j+1 ] = Tr [σ −j σ +j ][σ +j+1 σ −j+1 ] N −2 = (# states j, j + 1 =↑↓) = N/2 − 1 N , =N 4(L − 1) ! (C.59) (C.510) 216 and !   4 z z z z 0 0 Tr σ j σ j+1 σ j 0 σ j 0 +1 = (# states j, j + 1, j , j + 1 =↑↑↑↑) + (# states j, j + 1, j 0, j 0 + 1 =↑↑↓↓) 2 + (# states j, j + 1, j 0, j 0 + 1 =↓↓↓↓) ! ! 4 4 0 0 (# states j, j + 1, j 0, j 0 + 1 =↑↓↓↓) (# states j, j + 1, j , j + 1 =↑↑↑↓) − − 1 1 ! ! ! ! ! N −4 N −4 N −4 N −4 N −4 −6 −6 + +6 = N/2 − 1 N/2 − 3 N/2 N/2 − 2 N/2 − 4 3 =N , (C.511) (N − 1)(N − 3) wherein the first equality’s combinatorial factors come from permutations on sites j, j + 1, j 0, and j 0 + 1. P −1  z z  Assembling these pieces, we find Tr( H̃ (t)) = E Nj=1 Tr σ j σ j = −EN . Next, we compute Tr( H̃ 2 (t)): " NX −1 N −1 N −1 X X + − − + − + + − H̃ (t) = E 4 (σ j σ j )(σ j+1 σ j+1 ) + 4 (σ j σ j )(σ j+1 σ j+1 ) + σ zj σ zj+1 σ zj0 σ zj0 +1 2 2 j j,j 0 =1 j + h (αt ) 2 N X # h2j + (traceless terms) (C.512) j=1 " NX −1 N −1 N −1 N −2 X X X =E 4 (σ +j σ −j )(σ −j+1 σ +j+1 ) + 4 (σ −j σ +j )(σ +j+1 σ −j+1 ) + 1+ σ zj σ zj+2 2 j + j N −3 N −1 X X j=1 j 0 = j+2 j=1 σ zj σ zj+1 σ zj0 σ zj0 +1 + h(αt ) 2 (αt ) N X j=1 # h2j + (traceless terms) j=1 (C.513) We take the trace, using Eqs. (C.58), (C.59), and (C.511): # N X N −2 2 2 hj . Tr( H̃ (t)) = N 3N − 1 + +h N −1 j=1 " 2 We disorder-average by taking h2j 7→ D R1 0 dh j h2j = 13 : # N −2 h2 Tr(H (t)) = N 3N − 1 + +N . disorder N −1 3 2 E (C.514) " (C.515) Substituting into Eq. (C.53), we infer the rescaling factor’s square: Q2 (h(αt )) = 3N − 2 + N −2 h2 +N . N −1 3 (C.516) . 217 Our results are insensitive to the details of Q. The width of the DOS in one disorder realization will differ from the disorder average (C.516). Moreover, that difference will vary as we tune h(αt ), because the disorder affects only one term. The agreement between the analytics, in which µ(E) is assumed to remain constant in t, and the numerics is therefore comforting: The engine is robust against small variations in the rescaling. Representing states and Hamiltonians We structured our software to facilitate two possible extensions. First, the Hamiltonian tuning may be generalized to arbitrary speeds. Second, the cold bath might be modeled more realistically, as coupling to the engine only locally. We represent the state of one mesoscopic MBL Otto engine with a density matrix ρ ∈ CN ×N , and the Hamiltonian with a matrix H ∈ CN ×N , relative to the basis {|s1 i, . . . , |sN i} = {|↑ . . . ↑i, . . . , |↓ . . . ↓i} of products of σ z eigenstates. We track the whole density matrix, rather than just the energy-diagonal elements, with an eye toward the coherent superpositionsqthat diabatic corrections create. For an N-site N  2 2N . chain at half-filling, N = N/2 ≃ πN Strokes 1 and 3: Tuning Adiabatic evolution The (l, m) entry of the initial-state density matrix is 1 X − βH E j (0) 1 ρ(0)lm = hsl | e− βH H (0) |s m i = e hsl |E j (0)ihE j (0)|s m i . Z Z j (C.517) The j th eigenstate of H (0), associated with energy E j (0), is denoted by |E j (0)i. We approximate the time evolution from 0 to τ (during stroke 1) as adiabatic. The evolution therefore does not move weight between levels: ρ(τ)lm = 1 X − βH E j (0) e hsl |E j (τ)ihE j (τ)|s m i . Z j (C.518) If we represented our density matrix relative to an instantaneous energy eigenbasis, simulating the time evolution would be trivial: We would reinterpret the diagonal matrix ρ as being diagonal with the same elements in a new basis. However, we wish to represent ρ(t) relative to the σ zj product basis. This representation enhances the code’s flexibility, facilitating future inclusion of diabatic evolutions and a more detailed model of cold thermalization. To represent ρ(t) relative to the σ zj product 218 basis, we note that X ρ(τ)lm = hsl |E j (τ)ihE j (0)| ρ(0)|E j (0)ihE j (τ)|s m i = [U (τ, 0) ρ(0)U (τ, 0) † ]lm . j (C.519) We have defined a time-evolution matrix U (τ, 0) ∈ CN ×N by U (τ, 0)lm = j hsl |E j (τ)ihE j (0)|s m i . This matrix is easily computed via exact diagonalization of H (0) and H (τ). P We can compute the density matrix ρ(τ00 ) at the end of stroke 3 (the tuning from MBL to GOE) from the density matrix ρ(τ0 ) at the end of stroke 2 (the cold-bath thermalization) similarly: ρ(τ00 ) = U (τ00, τ0 ) ρ(τ0 )U (τ00, τ0 ) † . The time-evolution P matrix U (τ00, τ0 ) ∈ CN ×N is given by U (τ00, τ0 )lm = j hsl |E j (0)ihE j (τ)|s m i . [Recall that H (τ00 ) = H (0) and H (τ0 ) = H (τ).] Diabatic (finite-time) evolution We simulate a stepwise tuning—that is, we take α(t) = (δt) bvt/(δt)c . (C.520) To do this, we compute a time-evolution unitary for the whole stroke by chaining together the unitaries for each timestep: so for stroke 1 U (τ, 0; v, δt) = e−iH (τ−δt)δt e−iH (τ−2δt)δt . . . e−iH (0)δt (C.521) with the number of timesteps set by the speed. We use timestep δt = 0.405 hδi, but our results are not sensitive to timestep. In judging the effectiveness of the engine at finite tuning speed, we must estimate the level-repulsion scale δ − . We do this by diagonalizing 106 disorder realizations at the relevant h1 = 20, L = 8, plotting a histogram of the gaps (Fig. C.9, and visually estimating the point at which the distribution turns over. Our results are not sensitive to this value. Stroke 2: Thermalization with the cold bath During stroke 2, the system thermalizes with a bandwidth-Wb cold bath. We make three assumptions. First, the bandwidth cutoff is hard: The bath can transfer only amounts < Wb of energy at a time. Therefore, the cold bath cannot move probability mass between adjacent levels separated by just one gap δ0 > Wb . Second, the bath is Markovian. Third, the system thermalizes for a long time. The bath has time to move weight across sequences of small gaps δ0j , δ0j+1 , . . . < Wb . 219 60000 number of gaps 50000 40000 30000 20000 10000 0 0.00 0.02 0.04 0.06 0.08 δ/hδi Figure C.9: Level-spacing distribution for 106 disorder realizations of the random-field Heisenberg model at field h1 = 20 and system-size L = 8 (blue line), with estimate (vertical black line) for the level-repulsion parameter δ − . Energies E6 Wb E5 E4 E3 E2 E1 Figure C.10: Energies of a cold-thermalized system: We illustrate our implementation of cold thermalization with this example chain of six energies. The cold bath has a bandwidth of size Wb , depicted in green. We can implement thermalization as follows. First, we identify sequences of levels connected by small gaps. Second, we reapportion weight amongst the levels according to a Gibbs distribution. Suppose, for example, that the MBL Hamiltonian H (τ) contains the following chain of six energies, E1 , . . . , E6 , separated from its surrounding levels by large gaps (Fig. C.10): (E2 − E1 ), (E3 − E2 ) < Wb , (E5 − E4 ) < Wb , and (E4 − E3 ), (E6 − E5 ) > Wb . (C.522) We suppress the time arguments to simplify notation. Before thermalization, the P density operator is diagonal with respect to the energy basis: ρ(τ) = j ρ j |E j ihE j | . 220 The weight on level j is denoted by ρ j . Thermalization maps ρ(τ) 7→ ρ(τ0 ) = C.6   ρ1 + ρ2 + ρ3 − βC E1 − βC E2 − βC E3 e |E ihE | + e |E ihE | + e |E ihE | 1 1 2 2 3 3 e− βC E1 + e− βC E2 + e− βC E3   ρ4 + ρ5 − βC E4 − βC E5 + −β E e |E ihE | + e |E ihE | + ρ6 |E6 ihE6 | . 4 4 5 5 e C 4 + e− βC E5 (C.523) Comparison with competitor Otto engines: Details and extensions This section contains elaborates on the bandwidth engine (Sec. C.6) and on an MBL engine tuned between equal-strength disorder realizations (Sec. C.6). Section C.6 compares with an MBL engine thermalized with an ordinary-bandwidth cold bath. Details: Comparison with bandwidth engine Section 4.5 introduced the accordion-like “bandwidth engine.” To work reasonably, we claimed, that engine must not undergo diabatic hops. Can the bandwidth engine not withstand several hops—say, through 0.02Nmacro levels? No, because the ground state pulls away from the rest of the spectrum as Nmacro grows. Suppose, for simplicity, that TC = 0 and TH = ∞. The bandwidth engine starts stroke 1 in ρ(0) = 1/Nmacro . Diabatic hops preserve ρ(t) during stroke 1, on average: The engine as likely hops upward as drops. Cold thermalization drops the engine to the ground state (plus an exponentially small dusting of higher-level states). The ground-state energy is generically extensive. Hence the engine absorbs hQ2 imacro ∼ −Nmacro , on average. Suppose that, during stroke 3, the engine jumps up through 2% of the levels. The engine ends about two standard deviations below √ the spectrum’s center, with average energy ∼ Nmacro . While returning to TH = 0 √ during the average stroke 4, the bandwidth engine absorbs hQ4 imacro ∼ Nmacro . √ The average outputted work hWtot imacro = hQ4 imacro +hQ2 imacro ∼ Nmacro − Nmacro . As Nmacro grows, hWtot imacro shrinks, then goes negative. A few diabatic jumps threaten the bandwidth engine’s ability to output hWtot i > 0. The bandwidth engine’s v must decline as Nmacro grows also because the typical E whole-system gap hδimacro ∼ Nmacro shrinks. The smaller the gaps, the greater the likelihood that a given v induces hops. As hδimacro → 0, v must → 0. The MBL Otto cycle proceeds more quickly, due to subengines’ parallelization (Suppl. Mat. C.4). 221 Details: Comparison with MBL engine tuned between same-strength disorder realizations This engine was introduced in Sec. 4.5. Here, we estimate the probabilities that S and S̃ undergo worst-case trials. Wtot < 0 if an engine traverses, clockwise, a trapezoid whose shorter vertical leg lies leftward of its longer vertical leg (Fig. 4.2). S and S̃ have equal probabilities of traversing trapezoids whose right-hand legs are short. S̃ has a greater probability than S of traversing a trapezoid whose left-hand leg is short: The left-hand S̃ leg represents a gap in a Hamiltonian as localized as the right-hand Hamiltonian. The two S̃ Hamiltonians have the same gap statistics. Hence S̃ has a nontrivial probability of starting any given trial atop a small gap ∆ < Wb that widens to ∆0 ∈ (∆, Wb ). S̃ has a higher probability of traversing a worst-case trapezoid. Suppose, for simplicity, that TH = ∞ and TC = 0. The probability that any given S trial outputs Wtot < 0 is pworst ≈ (Prob. that the left-hand gap < the right-hand gap) (C.61) × (Prob. that the right-hand gap is small enough to be cold-thermalized) Wb ≈ (Prob. that the left-hand gap < Wb ) × . (C.62) hδi (E) The initial factor is modeled by the area of a region under the PGOE (δ) curve. The region stretches from δ = 0 to δ = Wb . We approximate the region as a triangle of Wb − π4 (Wb ) 2 /hδi2 Wb Wb length Wb and height π2 hδi ∼ hδi 2 e 2 , [δ ≈ Wb , Eq. (4.2), and hδi  1].  W 2 Wb b ∼ The triangle has an area of 12 · Wb · π2 hδi 2 hδi . Substituting into Eq. (C.62) yields Wb pworst ∼ hδi !3 . (C.63) Let p̃worst denote the probability that any given S̃ trial outputs Wtot < 0. p̃worst shares the form of Eq. (C.62). The initial factor approximates to the area of a (E) region under the PMBL (δ) curve. The region extends from δ = 0 to δ = Wb . The (E) 1 region resembles a rectangle of height PMBL (0) ≈ hδi . Combining the rectangle’s Wb area, hδi , with Eq. (C.62) yields Wb p̃worst ∼ hδi !2 . (C.64) 222 b Since W hδi  1, pworst  p̃worst .14 Comparison with an MBL Otto engine whose cold bath has an ordinary bandwidth Small-bandwidth baths have appeared elsewhere [18–22]. But suppose that realizing them poses challenges. Let S̃ denote an ordinary-bandwidth MBL subengine; and S, the small-Wb subengine. S̃ has a greater probability of traversing a quadrilateral, as in Fig. 4.2, in any given trial. But during the average quadrilateral traversal, S outputs more work than S̃. An engine can output work only during a quadrilateral traversal, in the adiabatic approximation. An engine fails to traverse a quadrilateral by failing to cold-thermalize. Cold thermalization fails if Wb < δ0, the gap just below the level occupied by the engine at the end of stroke 1. The larger the Wb , the more likely the engine thermalizes. Let us estimate the engines’ quadrilateral-traversal probabilities. Suppose, for simb plicity, that TH = ∞ and TC = 0. S has a probability ≈ W hδi of starting a trial on one side of a gap ∆ that shrinks to a ∆0 < Wb . The TC = 0 bath drops the subengine’s b energy. Hence S’s probability of traversing a quadrilateral ≈ W hδi . S̃ has a probability ≈ 1 of starting a trial just above a gap ∆ that shrinks to a ∆0 < Wb . Hence S̃ much b more likely traverses a quadrilateral than S does: 1  W hδi . But S̃ outputs less work per average quadrilateral traversal. Cold thermalization likely shifts S one level downward. During strokes 1 and 3, S’s energy likely declines more than it rises (Fig. 4.2). S likely outputs Wtot > 0. In contrast, cold thermalization can shift S̃ to any energy level. The energies less likely splay out during stroke 3. If S̃ traverses a quadrilateral, it outputs average work ≈ (average right-hand gap) - (average left-hand gap) ≈ hδi − hδi = 0. References [1] M. Ziman et al., eprint arXiv:quant-ph/0110164 (2001), quant-ph/0110164. [2] V. Scarani, M. Ziman, P. Štelmachovič, N. Gisin, and V. Bužek, Phys. Rev. Lett. 88, 097905 (2002). 14 The discrepancy is exaggerated if the exponent in Eq. (C.63) rises, if the left-hand S Hamil- tonian is modeled with a Gaussian ensemble other than the GOE. The Gaussian unitary ensemble (GUE) contains an exponent of 4; the Gaussian symplectic ensemble (GSE), an exponent of 6. Different ensembles model different symmetries. 223 [3] S. Shevchenko, S. Ashhab, and F. Nori, Physics Reports 492, 1 (2010). [4] N. Yunger Halpern, A. J. P. Garner, O. C. O. Dahlsten, and V. Vedral, New Journal of Physics 17, 095003 (2015). [5] G. E. Crooks, Journal of Statistical Physics 90, 1481 (1998). [6] N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1 ed. (Brooks Cole, 1976). [7] N. Yunger Halpern, C. D. White, S. Gopalakrishnan, and G. Refael, ArXiv e-prints (2017), 1707.07008v1. [8] C. De Grandi and A. Polkovnikov, Adiabatic Perturbation Theory: From Landau-Zener Problem to Quenching Through a Quantum Critical Point, in Lecture Notes in Physics, Berlin Springer Verlag, edited by A. K. K. Chandra, A. Das, and B. K. K. Chakrabarti, , Lecture Notes in Physics, Berlin Springer Verlag Vol. 802, p. 75, 2010, 0910.2236. [9] M. Srednicki, Phys. Rev. E 50, 888 (1994). [10] A. Pal and D. A. Huse, Phys. Rev. B 82, 174411 (2010). [11] M. Serbyn, Z. Papić, and D. A. Abanin, Phys. Rev. X 5, 041047 (2015). [12] P. W. Anderson, Phys. Rev. 109, 1492 (1958). [13] U. Sivan and Y. Imry, Phys. Rev. B 35, 6074 (1987). [14] S. Gopalakrishnan et al., Phys. Rev. B 92, 104202 (2015). [15] R. Nandkishore and D. A. Huse, Annual Review of Condensed Matter Physics 6, 15 (2015), 1404.0686. [16] V. Khemani, R. Nandkishore, and S. L. Sondhi, Nature Physics 11, 560 (2015), 1411.2616. [17] A. Altland, M. Janssen, and B. Shapiro, Phys. Rev. E 56, 1471 (1997). [18] A. V. Khaetskii, D. Loss, and L. Glazman, Phys. Rev. Lett. 88, 186802 (2002). [19] S. Gopalakrishnan and R. Nandkishore, Phys. Rev. B 90, 224203 (2014). [20] S. A. Parameswaran and S. Gopalakrishnan, Phys. Rev. B 95, 024201 (2017). [21] M. P. Woods, N. Ng, and S. Wehner, ArXiv e-prints (2015), 1506.02322. [22] C. Perry, P. Ćwikliński, J. Anders, M. Horodecki, and J. Oppenheim, ArXiv e-prints (2015), 1511.06553. 224 Appendix D APPENDICES FOR “MICROCANONICAL AND RESOURCE-THEORETIC DERIVATIONS OF THE THERMAL STATE OF A QUANTUM SYSTEM WITH NONCOMMUTING CHARGES” The microcanonical, dynamical, and resource-theory arguments are detailed below. D.1 Microcanonical derivation of the NATS’s form Upon describing the set-up, we will define an approximate microcanonical subspace M. Normalizing the projector onto M yields an approximate microcanonical state Ω. Tracing out most of the system from Ω leads, on average, to a state close to the Non-Abelian Thermal State γv . Finally, we derive conditions under which M exists. Set-up: Consider a system S associated with a Hilbert space H of dimension d := dim(H ). Let H ≡ Q0 denote the Hamiltonian. We call observables denoted by Q1 , . . . ,Q c “charges.” Without loss of generality, we assume that the Q j ’s form a linearly independent set. The Q j ’s do not necessarily commute with each other. They commute with the Hamiltonian if they satisfy a conservation law, [H,Q j ] = 0 ∀j = 1, . . . , c. (D.11) This conservation is relevant to dynamical evolution, during which the NATS may arise as the equilibrium state. However, our microcanonical derivation does not rely on conservation. Bath, blocks, and approximations to charges: Consider many copies n of the system S. Following Ogata [1], we consider an average Q̃ j , over the n copies, of each charge Q j (Fig. 5.2 of the main text): n−1 1 X ⊗` I ⊗ Q j ⊗ I⊗(n−1−`) . Q̃ j := n `=0 (D.12) In the large-n limit, the averages Q̃ j are approximated by observables Ỹj that com- 225 mute [1, Theorem 1.1]: k Q̃ j − Ỹj k∞ ≤  O (n) → 0, and (D.13) [Ỹj , Ỹk ] = 0 ∀j, k = 0, . . . , c. (D.14) The Ỹj ’s are defined on H ⊗n , k · k∞ denotes the operator norm, and  O (n) denotes a function that approaches zero as n → ∞. Consider m blocks of n copies of S, i.e., N = nm copies of S. We can view one copy as the system of interest and N − 1 copies as a bath. Consider the average, over N copies, of a charge Q j : N −1 1 X ⊗` I ⊗ Q j ⊗ I⊗(N −1−`) . Q̄ j := N `=0 (D.15) This Q̄ j equals also the average, over m blocks, of the block average Q̃ j : m−1 1 X ⊗λn Q̄ j = I ⊗ Q̃ j ⊗ I⊗[N −n(λ+1)] . m λ=0 (D.16) Let us construct observables Ȳj that approximate the Q̄ j ’s and that commute: [Ȳj , Ȳk ] = 0, and k Q̄ j − Ȳj k∞ ≤  for all m. Since Ỹj approximates the Q̃ j in Eq. (D.16), we may take m−1 Ȳj = 1 X ⊗λn I ⊗ Ỹj ⊗ I⊗[N −n(λ+1)] . m λ=0 (D.17) Approximate microcanonical subspace: Recall the textbook derivation of the form of the thermal state of a system that exchanges commuting charges with a bath. The composite system’s state occupies a microcanonical subspace. In every state in the subspace, every whole-system charge, including the energy, has a welldefined value. Charges that fail to commute might not have well-defined values simultaneously. But, if N is large, the Q̄ j ’s nearly commute; they can nearly have well-defined values simultaneously. This approximation motivates our definition of an approximate microcanonical subspace M. If the composite system occupies any state in M, one has a high probability of being able to predict the outcome of a measurement of any Q̄ j . Definition 2. For η, η 0, , δ, δ0 > 0, an (, η, η 0, δ, δ0 )-approximate microcanonical (a.m.c.) subspace M of H ⊗N associated with observables Q j and with approximate expectation values v j consists of the states ω for which the probability distribution over the possible outcomes of a measurement of any Q̄ j peaks 226 η sharply about v j . More precisely, we denote by Π j the projector onto the direct sum of the eigensubspaces of Q̄ j associated with the eigenvalues in the interval [v j − ηΣ(Q j ), v j + ηΣ(Q j )]. Here, Σ(Q) = λ max (Q) − λ min (Q) is the spectral diameter of an observable Q. M must satisfy the following conditions: 1. Let ω denote any state, defined on H ⊗N , whose support lies in M. A measurement of any Q̄ j is likely to yield a value near v j : supp(ω) ⊂ M ⇒ η Tr(ωΠ j ) ≥ 1 − δ ∀j. (D.18) 2. Conversely, consider any state ω, defined on H ⊗N , whose measurement statistics peak sharply. Most of the state’s probability weight lies in M: η0 Tr(ωΠ j ) ≥ 1 − δ0 ∀j ⇒ Tr(ωP) ≥ 1 − , (D.19) wherein P denotes the projector onto M. This definition merits two comments. First, M is the trivial (zero) subspace if the v j ’s are inconsistent, i.e., if no state ρ satisfies Tr( ρ Q j ) = v j ∀j. Second, specifying (η, η 0, , δ, δ0 ) does not specify a unique subspace. The inequalities enable multiple approximate microcanonical subspaces to satisfy Definition 2. The definition ensures, however, that any two such subspaces overlap substantially. The approximate microcanonical subspace leads to the NATS: Let us show that Definition 2 exhibits the property desired of a microcanonical state: The reduced state of each subsystem is close to the NATS. We denoted by P the projector onto the approximate microcanonical subspace M. 1 Normalizing the projector yields the approximate microcanonical state Ω := Tr(P) P. Tracing out all subsystems but the ` th yields Ω` := Tr0,...,`−1,`+1,...,N −1 (Ω). We quantify the discrepancy between Ω` and the NATS with the relative entropy:   D(Ω` kγv ) := −S(Ω` ) − Tr Ω` log(γv ) . (D.110)   wherein S(Ω` ) := −Tr Ω` log(Ω` ) is the von Neumann entropy. The relative entropy is lower-bounded by the trace norm, which quantifies quantum states’ distinguishability [2]: D(Ω` kγv ) ≥ 1 kΩ` − γv k 21 . 2 (D.111) 227 Theorem 7. Let M denote an (, η, η 0, δ, δ0 )-approximate microcanonical subspace 2 of H ⊗N associated with the Q j ’s and the v j ’s, for N ≥ [2 Q j /(η 2 )] log(2/δ0 ). ∞ The average, over the N subsystems, of the relative entropy between each subsystem’s reduced state Ω` and the NATS is small: N −1 1 X D(Ω` kγv ) ≤ θ + θ 0 . N `=0 (D.112) √ This θ = (const.)/ N is proportional to a constant dependent on , on the v j ’s, and on d. This θ 0 = (c + 1)(const.)(η + 2δ · max j { Q j }) is proportional to a constant ∞ dependent on the v j ’s. Proof. We will bound each term in the definition (D.110) of the relative entropy D. The von Neumann-entropy term S(Ω` ), we bound with Schumacher’s theorem for typical subspaces. The cross term is bounded, by the definition of the approximate microcanonical subspace M, in terms of the small parameters that quantify the approximation. First, we lower-bound the dimensionality of M in terms of , η, η 0, δ, and δ0. Imagine measuring some Q̄ j of the composite-system state γv⊗N . This is equivalent to measuring each subsystem’s Q j , then averaging the outcomes. Each Q j meaj surement would yield a random outcome X` ∈ [λ min (Q j ), λ max (Q j )], for ` = 0, . . . , N − 1. The average of these Q j -measurement outcomes is tightly concentrated around v j , by Hoeffding’s Inequality [3]:  N −1  X   1 j ⊗N η
1 − Tr γv Π j = Pr  (D.113) X` − v j | > ηΣ(Q j )   N    `=0   ≤ 2 exp −2η 2 N (D.114) ≤ δ0 , (D.115) for large enough N. From the second property in Definition 2, it follows that
Tr γv⊗N P ≥ 1 − . Hence M is a high-probability subspace of γv⊗N . By Schumacher’s Theorem, or by the stronger [4, Theorem I.19],   √
S(Ω) = log dim(P) ≥ N S γv − (const.) N (D.116)
= N S γv − N θ, (D.117) √ wherein θ := (const.)/ N. The constant depends on , d, and the charge values P N −1 v j . The entropy’s subadditivity implies that S(Ω) ≤ `=0 S(Ω` ). Combining this 228 inequality with Ineq. (D.117) yields N −1 S γv
1 X −θ ≤ S(Ω` ). N `=0 (D.118) η The support of Ω lies within M: supp(Ω) ⊂ M. Hence Tr(Ω Π j ) = 1 ≥ 1 − δ for P N −1 Ω` . We will bound the many-copy average all j. Let Ω̄ := N1 `=0 N −1 w j := Tr(Q j Ω̄) = 1 X Tr(Ω` Q j ) N `=0 (D.119) = Tr(Ω Q̄ j ). (D.120) Let us bound this trace from both sides. Representing Q̄ j = composition, we upper-bound the following average: X   q Tr(Ω Q̄ j ) = qTr Ω Π j P q q q Π j in its eigende- (D.121) q   η ≤ [v j + ηΣ(Q j )]Tr Ω Π j + Q j ≤ vj + Q j ∞ ∞  h i η Tr Ω I − Π j (η + δ). (D.122) (D.123) We complement this upper bound with a lower bound:   η Tr(Ω Q̄ j ) ≥ [v j − ηΣ(Q j )]Tr Ω Π j − Q j ≥ [v j − ηΣ(Q j )](1 − δ) − Q j ∞ ∞  h i η Tr Ω I − Π j δ. (D.124) (D.125) Inequalities (D.123) and (D.125) show that the whole-system average w j is close to the single-copy average v j : ξ j := w j − v j = Tr(Ω Q̄ j ) − v j ≤ (η + 2δ) Q j ∞ (D.126) . (D.127) Let us bound the average relative entropy. By definition, N −1 N −1  i 1 X 1 Xh S(Ω` ) + Tr Ω` log(γv ) . D (Ω` kγv ) = − N `=0 N `=0 (D.128) Let us focus on the second term. First, we substitute in the form of γv from Eq. (??) of the main text. Next, we substitute in for w j , using Eq. (D.119). Third, we 229 substitute in ξ j , using Eq. (D.126). Fourth, we invoke the definition of S(γv ), which we bound with Ineq. (D.118): − N −1  1 X  Tr Ω` log(γv ) N `=0 (D.129) N −1 c X i 1 Xh log(Z ) + µ j Tr(Ω` Q j ) = N `=0 j=0 = log Z + ≤ log Z + = S(γv ) + c X j=0 c X j=0 c X µjwj µjvj + (D.130) (D.131) c X µj ξj (D.132) j=0 µj ξj (D.133) j=0 N −1 X c X 1 ≤ S(Ω` ) + θ + µj ξj. N `=0 j=0 (D.134) Combining this inequality with Eq. (D.128) yields N −1 c X 1 X D (Ω` kγv ) ≤ θ + µj ξj N `=0 j=0 ! ! ≤ θ + (c + 1) max | µ j | max ξ j j j !" n o# ≤ θ + (c + 1) max | µ j | (η + 2δ) · max kQ j k∞ . j j (D.135) (D.136) (D.137) The final inequality follows from Ineq. (D.127). Since the v j ’s determine the µ j   values, (c + 1) max j | µ j | is a constant determined by the v j ’s. The final term in Ineq. (D.137), therefore, is upper-bounded by θ 0 = (c + 1)(const.)(η + 2δ) · n o max j Q j .  ∞ Existence of an approximate microcanonical subspace: Definition 2 does not reveal under what conditions an approximate microcanonical subspace M exists. We will show that an M exists for , η, η 0, δ, δ0 that can approach zero simultaneously, for sufficiently large N. First, we prove the existence of a microcanonical subspace for commuting observables. Applying this lemma to the Ỹj ’s shows that M exists for noncommuting observables. 230 Lemma 1. Consider a Hilbert space K with commuting observables X j , j =    0, . . . , c. For all , η, δ > 0 and for sufficiently large m, there exists an , η, η 0=η, δ, δ0= c+1 ⊗m approximate microcanonical subspace M of K associated with the observables X j and with the approximate expectation values v j . Proof. Recall that m−1 1 X ⊗λ X̄ j = I ⊗ X j ⊗ I⊗(m−1−λ) m λ=0 (D.138) is the average of X j over the m subsystems. Denote by  η Ξ j := v j − η ≤ X̄ j ≤ v j + η (D.139) the projector onto the direct sum of the X̄ j eigenspaces associated with the eigenη values in [v j − η, v j + η]. Consider the subspace Mcom projected onto by all the η X j ’s. The projector onto Mcom is η η η Pcom := Ξ0 Ξ1 · · · Ξc . (D.140) η Denote by ω any state whose support lies in Mcom . Let us show that ω satisfies η the inequality in (D.18). By the definition of Pcom , supp(ω) ⊂ supp(Ξ j ). Hence  η Tr ωΞ j = 1 ≥ 1 − δ. Let us verify the second condition in Definition 2. Consider any eigenvalue ȳ j of Ȳj , for each j. Consider the joint eigensubspace, shared by the Ȳj ’s, associated with any eigenvalue ȳ1 of Ȳ1 , with any eigenvalue ȳ2 of Ȳ2 , etc. Denote the projector onto this eigensubspace of H ⊗N by Pȳ1 ,··· ,ȳc .   η  Let δ0 = c+1 . Let ω denote any state, defined on H ⊗N , for which Tr ω Ξ j ≥ 1−δ0, for all j = 0, . . . , c. The left-hand side of the second inequality in (D.19) reads, P Tr (ωPcom ). We insert the resolution of identity ȳ0 ,...,ȳc Pȳ0 ... ȳc into the trace. The property P 2 = P of any projector P enables us to square each projector. Because [Pȳ0 ... ȳc , Pcom ] = 0,   X Tr (ωPcom ) = Tr  Pȳ0 ... ȳc ωPȳ0 ... ȳc Pcom  (D.141)   ȳ0 ,...,ȳc
=: Tr ω0 Pcom , (D.142) P wherein ω0 := ȳ0 ,...,ȳc Pȳ0 ... ȳc ωPȳ0 ... ȳc is ω pinched with the complete set {Pȳ0 ȳ1 ... ȳc }     η η of projectors [5]. By this definition of ω0, Tr ω0 Ξ j = Tr ω Ξ j ≥ 1 − δ0, and 231 η [ω0, Ξ j ] = 0. For all j, therefore,   η η ω0 Ξ j = ω0 − ω0 I − Ξ j =: ω0 − ∆ j , (D.143)  h i η Tr(∆ j ) = Tr ω0 I − Ξ j ≤ δ0 . (D.144) wherein Hence  
η η η Tr ω0 Pcom = Tr ω0 Ξ0 Ξ1 · · · Ξc    η η ≥ Tr ω0 − ∆0 Ξ1 · · · Ξc   η η ≥ Tr ω0 Ξ1 · · · Ξc − δ0
≥ Tr ω0 − (c + 1)δ0 = 1 − (c + 1)δ0 = 1 −  . (D.145) (D.146) (D.147) (D.148) (D.149) η  )-approximate microcanoniAs ω satisfies (D.19), Mcom is an (, η, η 0=η, δ, δ0= c+1 cal subspace.  η Lemma 1 proves the existence of an approximate microcanonical subspace Mcom for the Ỹj ’s defined on K = H ⊗n and for sufficiently large n. In the subsequent η discussion, we denote by Υj the projector onto the direct sum of the Ȳj eigenspaces associated with the eigenvalues in [v j − ηΣ(Ỹj ), v j + ηΣ(Ỹj )]. Passing from Ỹj to η Q̃ j to Q j , we now prove that the same Mcom is an approximate microcanonical subspace for the Q j ’s. Theorem 8. Under the above assumptions, for every  > (c + 1)δ0 > 0, η > η 0 > 0, δ > 0, and all sufficiently large N, there exists an (, η, η 0, δ, δ0 )-approximate microcanonical subspace M of H ⊗N associated with the observables Q j and with the approximate expectation values v j . Proof. Let η̂ = (η + η 0 )/2. For a constant CAP > 0 to be determined later, let n be such that  O =  O (n) from Ogata’s result [1, Theorem 1.1] is small enough so that 1/3 1/3 0 η > η̂ + CAP  1/3 O and η < η̂ − CAP  O , as well as such that δ̂ = δ − CAP  O > 0  and such that δ̂0 = δ0 + CAP  1/3 O ≤ c+1 . Choose m in Lemma 1 large enough such that an (, η̂, η̂ 0=η̂, δ̂, δ̂0 )-approximate microcanonical subspace M := Mcom associated with the commuting Ỹj exists, with approximate expectation values v j . 232 Let ω denote a state defined on H ⊗N . We will show that, if measuring the Ȳj ’s of ω yields sharply peaked statistics, measuring the Q̄ j ’s yields sharply peaked statistics. Later, we will prove the reverse (that sharply peaked Q̄ j statistics imply sharply peaked Ȳj statistics). η Recall from Definition 2 that Π j denotes the projector onto the direct sum of the Q̄ j eigenstates associated with the eigenvalues in [v j − ηΣ(Q j ), v j + ηΣ(Q j )]. These eigenprojectors are discontinuous functions of the observables. Hence we look for η better-behaved functions. We will approximate the action of Π j by using    1, x ∈ [−η 0 , η 0 ] f η 0 ,η 1 (x) :=  ,   0, |x| > η 1  (D.150) 1 ∈ R. for η 1 > η 0 > 0. The Lipschitz constant of f is bounded by λ := η 1 −η 0 η The operator f η 0 Σ(Q j ),η 1 Σ(Q j ) (Q̄ j − v j I) approximates the projector Π j 0 . Indeed, as a η matrix, f η 0 Σ(Q j ),η 1 Σ(Q j ) (Q̄ j − v j I) is sandwiched between the projector Π j 0 , associη ated with a width-η 0 interval around v j , and a projector Π j 1 associated with a widthη 1 interval of eigenvalues. f η,η is the indicator function on the interval [−η, η]. η Hence Π j = f ηΣ(Q j ),ηΣ(Q j ) (Q̄ j − v j I). Similarly, we can regard f η 0 Σ(Q j ),η 1 Σ(Q j ) (Ȳj − η η v j I) as sandwiched between Υj 0 and Υj 1 . Because Q̄ j is close to Ȳj , f (Q̄ j ) is close to f (Ȳj ): Let n be large enough so that, by [1, Theorem 1.1], k Q̄ j − Ȳj k∞ ≤  O . By [6, Theorem 4.1], k f η 0 Σ(Q j ),η 1 Σ(Q j ) (Ȳj − v j I) − f η 0 Σ(Q j ),η 1 Σ(Q j ) (Q̄ j − v j I)k∞ ≤ κ λ , (D.151) wherein κ λ = CAP λ 2/3 O and CAP denotes a universal constant. Inequality (D.151) holds because f is λ-Lipschitz and bounded, so the Hölder norm in [6, Theorem 4.1] is proportional to λ. Let us show that, if measuring the Ȳj ’s of ω yields sharply peaked statistics, then measuring the Q̄ j ’s yields sharply peaked statistics, and vice versa. First, we choose −1/3 1/3 η 0 = η, η 1 = η +  1/3 such that κ := κ λ = CAP  O . By the O , and λ =  O “sandwiching,” !  h i η+ 1/3 O Tr ωΠ j ≥ Tr ω f η 0 Σ(Q j ),η 1 Σ(Q j ) Q̄ j − v j I . (D.152) 233 To bound the right-hand side, we invoke Ineq. (D.151): κ ≥ k f η 0 Σ(Q j ),η 1 Σ(Q j ) (Ȳj − v j I) − f η 0 Σ(Q j ),η 1 Σ(Q j ) (Q̄ j − v j I)k∞  ≥ Tr f η 0 Σ(Q j ),η 1 Σ(Q j ) (Ȳj − v j I)  − f η 0 Σ(Q j ),η 1 Σ(Q j ) (Q̄ j − v j I)  h ≥ Tr ω f η 0 Σ(Q j ),η 1 Σ(Q j ) (Ȳj − v j I) i − f η 0 Σ(Q j ),η 1 v (Q̄ j − v j I) . (D.153) (D.154) (D.155) Upon invoking the trace’s linearity, we rearrange terms:   Tr ω f η 0 Σ(Q j ),η 1 Σ(Q j ) (Q̄ j − v j I)   ≥ Tr ω f η 0 Σ(Q j ),η 1 Σ(Q j ) (Ȳj − v j I) − κ  η ≥ Tr ωΥj − κ. (D.156) (D.157) (D.158) The final inequality follows from the “sandwiching” property of f η 0 ,η 1 . Combining Ineqs. (D.152) and (D.158) yields a bound on fluctuations in Q̄ j measurement statistics in terms of fluctuations in Ȳj statistics: ! 1/3  η η+ O Tr ω Π j ≥ Tr ωΥj − κ. (D.159) Now, we bound fluctuations in Ȳj statistics with fluctuations in Q̄ j statistics. If −1/3 1/3 η 0 = η −  1/3 O ; η 1 = η; λ =  O , as before, and κ = κ λ = CAP  O , then !  η η− 1/3 O Tr ωΥj ≥ Tr ωΠ j − κ. (D.160) η̂ Using Ineqs. (D.159) and (D.160), we can now show that M := Mcom is an approximate microcanonical subspace for the observables Q j and the approximate charge values v j . In other words, M is an approximate microcanonical subspace for the observables Q̃ j . η First, we show that M satisfies the first condition in Definition 2. Recall that Mcom   is an , η, η 0=η, δ, δ0= c -approximate microcanonical subspace for the observables Ỹj with the approximate charge values v j , for all , η, δ > 0 and for large enough m (Lemma 1). Recall that N = nm. Choose δ = δ̂ − κ > 0. Let ω denote any η state, defined on H ⊗N , whose support lies in M = Mcom . Let η̂ = η +  1/3 O . By the  η definitions of ω and M, Tr ωΥj = 1 ≥ 1 − δ. By Ineq. (D.159), therefore,    η η̂ (D.161) Tr ωΠ j ≥ Tr ωΥj − κ ≥ 1 − δ − κ = 1 − δ̂. 234 Hence M satisfies Condition 1 in Definition 2. 1/3 , and let δ̂0 = δ0 − κ = c − To show that M satisfies Condition 2, let η̂ 0 = η −  O   0 ⊗N satisfy Tr ωΠ η̂ CAP  1/3 ≥ 1 − δ̂0 for all j. By Ineq. (D.160), j O > 0. Let ω in H  η Tr ωΥj ≥ 1 − δ̂0 − κ = 1 − δ0 . (D.162) η By Condition 2 in the definition of Mcom , therefore, at least fraction 1 −  of the η probability weight of ω lies in Mcom = M: Tr (ωPcom ) ≥ 1 − . As M satisfies Condition 2, M is an (, η̂, η̂ 0, δ̂, δ̂0 )-approximate microcanonical subspace.  This derivation confirms physically the information-theoretic maximum-entropy derivation. By “physically,” we mean, “involving the microcanonical form of a composite system’s state and from the tracing out of an environment.” The noncommutation of the charges Q j required us to define an approximate microcanonical subspace M. The proof of the subspace’s existence, under appropriate conditions, crowns the derivation. The physical principle underlying this derivation is, roughly, the Correspondence Principle. The Q j ’s of one copy of the system S fail to commute with each other. This noncommutation constitutes quantum mechanical behavior. In the many-copy limit, however, averages Q̄ j of the Q j ’s are approximated by commuting Ȳj ’s, whose existence was proved by Ogata [1]. In the many-copy limit, the noncommuting (quantum) problem reduces approximately to the commuting (classical) problem. We stress that the approximate microcanonical subspace M corresponds to a set of observables Q j and a set of values v j . Consider the subspace M 0 associated with a subset of the Q j ’s and their v j ’s. This M 0 differs from M. Indeed, M 0 typically has a greater dimensionality than M, because fewer equations constrain P it. Furthermore, consider a linear combination Q0 = cj=0 µ j Q j . The average Q̄0 of P N copies of Q0 equals cj=0 µ j Q̄ j . The approximate microcanonical subspace M of the whole set of Q j ’s has the property that all states that lie mostly on it have P sharply defined values near v 0 = cj=0 µ j v j . Generally, however, our M is not an approximate microcanonical subspace for Q0, or a selection of Q0, Q00, etc., unless these primed operators span the same set of observables as the Q j ’s. D.2 Dynamical considerations Inequality (??) of the main text is derived as follows: Let us focus on k ρ` − γv k1 . Adding and subtracting Ω` to the argument, then invoking the Triangle Inequality, 235 yields k ρ` − γv k1 ≤ k ρ` − Ω` k1 + kΩ` − γv k1 . (D.21) We average over copies ` and average (via h.i) over pure whole-system states |ψi. The first term on the right-hand side is bounded in Ineq. (??) of the main text: −1  1 NX N `=0 k ρ` − γv k1  * NX + −1 d 1 ≤ √ + kΩ` − γv k1 . N `=0 DM (D.22) To bound the final term, we invoke Pinsker’s Inequality [Ineq. (D.111)], kΩ` − p γv k1 ≤ 2D(Ω` ||γv ). Averaging over ` and over states |ψi yields * + * NX + N −1 −1 p 1 1 X kΩ` − γv k1 ≤ 2D(Ω` ||γv ) N `=0 N `=0 v t N −1 *u + 2 X ≤ D(Ω` ||γv ) , N `=0 (D.23) (D.24) wherein D denotes the relative entropy. The second inequality follows from the square-root’s concavity. Let us double each side of Ineq. (D.112), then take the square-root: v u t N −1 p 2 X (D.25) D(Ω` kγv ) ≤ 2(θ + θ 0 ). N `=0 Combining the foregoing two inequalities, and substituting into Ineq. (D.22), yields Ineq. (??) of the main text. D.3 Derivation from complete passivity and resource theory An alternative derivation of the thermal state’s form relies on complete passivity. One cannot extract work from any number of copies of the thermal state via any energy-preserving unitary [7, 8]. We adapt this argument to noncommuting conserved charges. The Non-Abelian Thermal State is shown to be the completely passive “free” state in a thermodynamic resource theory. Resource theories are models, developed in quantum information theory, for scarcity. Using a resource theory, one can calculate the value attributable to a quantum state by an agent limited to performing only certain operations, called “free operations.” The first resource theory described pure bipartite entanglement [9]. Entanglement theory concerns how one can manipulate entanglement, if able to perform only 236 local operations and classical communications. The entanglement theory’s success led to resource theories for asymmetry [10], for stabilizer codes in quantum computation [11], for coherence [12], for quantum Shannon theory [13], and for thermodynamics, amongst other settings. Resource-theoretic models for heat exchanges were constructed recently [14, 15]. The free operations, called “thermal operations,” conserve energy. How to extend the theory to other conserved quantities was noted in [15]. The commutingobservables version of the theory was defined and analyzed in [16, 17], which posed questions about modeling noncommuting observables. We extend the resource theory to model thermodynamic exchanges of noncommuting observables. The free operations that define this theory, we term “Non-Abelian Thermal Operations” (NATO). This resource theory is related to that in [18]. We supplement earlier approaches with a work payoff function, as well as with a reference frame associated with a non-Abelian group. This section is organized as follows. First, we introduce three subsystems and define work. Next, we define NATO. The NATO resource theory leads to the NATS via two routes: 1. The NATS is completely passive: The agent cannot extract work from any number of copies of γv . 2. The NATS is the state preserved by NATO, the operations that require no work. The latter condition leads to “second laws” for thermodynamics that involves noncommuting conserved charges. The second laws imply the maximum amount of work extractable from a transformation between states. Subsystems: To specify a physical system in this resource theory, one specifies a Hilbert space, a density operator, a Hamiltonian, and operators that represent the system’s charges. To specify the subsystem S of interest, for example, one specifies a Hilbert space H ; a density operator ρS ; a Hamiltonian HS ; and charges Q1S , . . . ,Q cS . Consider the group G formed from elements of the form eiµ·Q . Each Q j can be viewed as a generator. G is non-Abelian if the Q j ’s fail to commute with each other. Following [19], we assume that G is a compact Lie group. The compactness assumption is satisfied if the system’s Hilbert space is finite-dimensional. (We model 237 the reference frame’s Hilbert space as infinite-dimensional for convenience. Finitesize references can implement the desired protocols with arbitrary fidelity [19].) We consider three systems, apart from S: First, R denotes a reservoir of free states. The resource theory is nontrivial, we prove, if and only if the free states have the NATS’s form. Second, a battery W stores work. W doubles as a non-Abelian reference frame. Third, any other ancilla is denoted by A. The Hamiltonian Htot := HS + HR + HW + HA governs the whole system. The j th whole-system charge has the form Q jtot := Q jS + Q jR + Q jW + Q jA . Let us introduce each subsystem individually. Battery: We define work by modeling the system that stores the work. In general, the mathematical expression for thermodynamic work depends on which physical degrees of freedom a system has. A textbook example concerns a gas, subject to a pressure p, whose volume increases by an amount dV . The gas performs an amount dW = p dV of work. If a force F stretches a polymer through a displacement dx, dW = −F dx. If a material’s magnetization decreases by an amount dM in the presence of a strength-B magnetic field, dW = B dM. We model the ability to convert, into a standard form of work, a variation in some physical quantity. The model consists of an observable called a “payoff function.” The payoff function is defined as W := c X µjQ j . (D.31) j=0 We generally regard the payoff function as an observable of the battery’s. We can also consider the W of the system of interest. If the system whose W we refer to is not obvious from context, we will use a subscript. For example, WW denotes the battery’s work function. One might assume that the battery exchanges only finite amounts of charges. Under this assumption, a realistically sized battery can implement the desired protocols with perfect fidelity [19]. Work: We define as average extracted work W the difference in expectation value of the payoff function W:   W := Tr ρ0W W − Tr ( ρW W ) . (D.32) 238 The battery’s initial and final states are denoted by ρW and ρ0W . If the expectation value increases, then W > 0, and work has been extracted from the system of interest. Otherwise, work has been expended. We focus on the average work extracted in the asymptotic limit: We consider processing many copies of the system, then averaging over copies. Alternatively, one could focus on one instance of the transformation. The deterministic or maximal guaranteed work would quantify the protocol’s efficiency better than the average work would [15, 20–22]. Reference frame: Reference frames have appeared in the thermodynamic resource theory for heat exchanges [23–25]. We introduce a non-Abelian reference frame into the thermodynamic resource theory for noncommuting conserved charges. Our agent’s reference frame carries a representation of the G associated with the charges [10, 19]. The reference frame expands the set of allowed operations from a possibly trivial set. A superselection rule restricts the free operations, as detailed below. Every free unitary U conserves (commutes with) each charge. The system charges Q jS might not commute with each other. In the worst case, the Q jS ’s share no multidimensional eigensubspace. The only unitary that conserves all such Q jS ’s is trivial: U ∝ I. A reference frame “frees up” dynamics, enabling the system to evolve nontrivially. A free unitary can fail to commute with a Q jS while preserving Q jtot . This dynamics transfers charges between the system and the reference frame. Our agent’s reference frame doubles as the battery. The reference frame and battery are combined for simplicity, to reduce the number of subsystems under consideration. Ancillas: The agent could manipulate extra subsystems, called “ancillas.” A list ( ρA , HA ,Q1A , . . . ,Q cA ) specifies each ancilla A. Any ancillas evolve cyclically under free operations. That is, NATO preserve the ancillas’ states, ρA . If NATO evolved ancillas acyclically, the agent could “cheat,” extracting work by degrading an ancilla [26]. Example ancillas include catalysts. A catalyst facilitates a transformation that could not occur for free in the catalyst’s absence [26]. Suppose that a state S = ( ρS , HS ,Q1S , . . . ,Q cS ) cannot transform into a state S̃ = ( ρ̃S , H̃S , Q̃1S , . . . , Q̃ cS ) by free operations: S 67→ S̃. Some state X = ( ρX , HX ,Q1X , . . . ,Q cX ) might enable 239 S ⊗ X 7→ S̃ ⊗ X to occur for free. Such a facilitated transformation is called a “catalytic operation.” Non-Abelian Thermal Operations: NATO are the resource theory’s free operations. NATO model exchanges of heat and of charges that might not commute with each other. Definition 3. Every Non-Abelian Thermal Operation (NATO) consists of the following three steps. Every sequence of three such steps forms a NATO: 1. Any number of free states ( ρR , HR ,Q1R , . . . ,Q cR ) can be added. 2. Any unitary U that satisfies the following conditions can be implemented on the whole system: a) U preserves energy: [U, Htot ] = 0. b) U preserves every total charge: [U,Q jtot ] = 0 ∀j = 1, . . . , c. c) Any ancillas return to their original states: Tr\A (U ρtotU † ) = ρA . 3. Any subsystem can be discarded (traced out). Conditions 2a and 2b ensure that the energy and the charges are conserved. The allowed operations are G-invariant, or symmetric with respect to the non-Abelian group G. Conditions 2a and 2b do not significantly restrict the allowed operations, if the agent uses a reference frame. Suppose that the agent wishes to implement, on S, some unitary U that fails to commute with some Q jS . U can be mapped to a whole-system unitary Ũ that conserves Q jtot . The noncommutation represents the transfer of charges to the battery, associated with work. The construction of Ũ from U is described in [19]. (We focus on the subset of free operations analyzed in [19].) Let g, φ ∈ G denote any elements of the symmetry group. Let T denote any subsystem (e.g., T = S,W ). Let VT (g) denote a representation, defined on the Hilbert space of system T, of g. Let |φiT denote a state of S that transforms as the left regular representation of G: VT (g)|φiT = |gφiT . U can be implemented on the system S of interest by the global unitary Z Ũ := dφ |φihφ|W ⊗ [VS (φ) U VS−1 (φ)]. (D.33) The construction (D.33) does not increase the reference frame’s entropy if the reference is initialized to |φ = 1iW . This nonincrease keeps the extracted work 240 “clean” [22, 26, 27]. No entropy is “hidden” in the reference frame W . W allows us to implement the unitary U, providing or storing the charges consumed or outputted by the system of interest. A zeroth law of thermodynamics: Complete passivity of the Non-Abelian Thermal State Which states ρR should the resource-theory agent access for free? The free states are the only states from which work cannot be extracted via free operations. We will ignore S in this section, treating the reservoir R as the system of interest. Free states in the resource theory for heat exchanges: Our argument about noncommuting charges will mirror the argument about extracting work when only the energy is conserved. Consider the thermodynamic resource theory for energy conservation. Let HR denote the Hamiltonian of R. The free state ρR has the form ρR = e− βHR /Z [16, 26]. This form follows from the canonical ensemble’s completely passivity and from the nonexistence of any other completely passive state. Complete passivity was introduced in [7, 8]. Definition 4 (Passivity and complete passivity). Let ρ denote a state governed by a Hamiltonian H. ρ is passive with respect to H if no free unitary U can lower the energy expectation value of ρ:   @ U : Tr U ρU † H < Tr ( ρH) . (D.34) That is, work cannot be extracted from ρ by any free unitary. If work cannot be extracted from any number n of copies of ρ, ρ is completely passive with respect to H:     ∀n = 1, 2, . . . , @ U : Tr U ρ⊗nU † H < Tr ρ⊗n H . (D.35) A free U could lower the energy expectation value only if the energy expectation value of a work-storage system increased. This transfer of energy would amount to work extraction. Conditions under which ρ is passive have been derived [7, 8]: Let {pi } and {Ei } denote the eigenvalues of ρ and H. ρ is passive if 1. [ρ, H] = 0 and 2. Ei > E j implies that pi ≤ p j for all i, j. 241 One can check that e− βHR /Z is completely passive with respect to HR . No other states are completely passive (apart from the ground state). Suppose that the agent could access copies of some ρ0 , e− βHR /Z. The agent could extract work via thermal operations [26]. Free (worthless) states could be transformed into a (valuable) resource for free. Such a transformation would be unphysical, rendering the resource theory trivial, in a sense. (As noted in [28], if a reference frame is not allowed, the theory might be nontrivial in that creating superpositions of energy eigenstates would not be possible). Free states in the resource theory of Non-Abelian Thermal Operations: We have reviewed the free states in the resource theory for heat exchanges. Similar considerations characterize the resource theory for noncommuting charges Q j . The free states, we show, have the NATS’s form. If any other state were free, the agent could extract work for free. Theorem 9. There exists an m > 0 such that a NATO can extract a nonzero amount P of chemical work from ( ρR ) ⊗m if and only if ρR , e− β (HR + j µ j Q jR ) /Z for some β ∈ R. Proof. We borrow from [7, 8] the proof that canonical-type states, and only canonicaltype states, are completely passive. We generalize complete passivity with respect to a Hamiltonian H to complete passivity with respect to the work function W. Every free unitary preserves every global charge. Hence the lowering of the expectation value of the work function W of a system amounts to transferring work from the system to the battery: ∆Tr(WW ρW ) = −∆Tr(WR ρR ). (D.36) Just as e− βH /Z is completely passive with respect to H [7, 8], the NATS is completely passive with respect to WR for some β. Conversely, if ρR is not of the NATS form, it is not completely passive with respect to WR . Some unitary UR ⊗m lowers the energy expectation value of ρR⊗m , Tr(UR ⊗m [ρR⊗m ]UR† ⊗m WR ⊗m ) < Tr( ρR⊗m WR ⊗m ), for some great-enough m. A joint unitary defined on R ⊗m and W approximates UR ⊗m well and uses the system W as a reference frame [Eq. (D.33)]. This joint unitary conserves every global charge. Because the expectation value of WR ⊗m decreases, chemical work is transferred to the battery.  242 The NATS is completely passive with respect to WR but not necessarily with respect to each charge Q j . The latter lack of passivity was viewed as problematic in [18]. The lowering of the NATS’s hQ j i’s creates no problems in our framework, because free operations cannot lower the NATS’s hWi. The possibility of extracting charge of a desired type Q j , rather than energy, is investigated also in [29]. For example, let the Q j ’s be the components J j of the spin operator J. Let the z-axis point in the direction of µ, and let µ z > 0: 3 X µ j J j ≡ µ z Jz . (D.37) j=1 The NATS has the form ρR = e− β(HR −µz JzR ) /Z. This ρR shares an eigenbasis with JzR . Hence the expectation value of the battery’s Jx charge vanishes: Tr( ρR Jx R ) = 0. A free unitary, defined on R and W , can rotate the spin operator that appears in the exponential of ρR . Under this unitary, the eigenstates of ρR become eigenstates of Jx R . Tr(Jx ρR ) becomes negative; work appears appears to be extracted “along the Jx -direction” from ρR . Hence the NATS appears to lack completely passivity. The unitary, however, extracts no chemical work: The decrease in Tr( ρR Jx R ) is compensated for by an increase in Tr( ρR JzR ). Another example concerns the charges Ji and ρR = e− β(HR −µz JzR ) /Z. No amount of the charge Jz can be extracted from ρR . But the eigenstates of −Jz are inversely populated: The eigenstate |zi associated with the low eigenvalue − ~2 of −Jz has the small population e− β~/2 . The eigenstate |−zi associated with the large eigenvalue ~ β~/2 . Hence the charge −J can be extracted z 2 of −Jz has the large population e from ρR . This extractability does not prevent ρR from being completely passive, according our definition. Only the extraction of W corresponds to chemical work. The extraction of just one charge does not. The interconvertibility of types of free energy associated with commuting charges was noted in [17]. Let Q1 and Q2 denote commuting charges, and let ρR = e− β(HR −µ1 Q1R −µ2 Q2R ) . One can extract Q1 work at the expense of Q2 work, by swapping Q1 and Q2 (if an allowed unitary implements the swap). Non-Abelian Thermal Operations preserve the Non-Abelian Thermal State. The NATS, we have shown, is the only completely passive state. It is also the only state preserved by NATO. 243 Theorem 10. Consider the resource theory, defined by NATO, associated with a fixed β. Let each free state be specified by ( ρR , HR ,Q1R , . . . ,Q cR ), wherein ρR := Pc e− β (HR − j=1 µ j Q jR ) /Z. Suppose that the agent has access to the battery, associated with the payoff function (D.31). The agent cannot, at a cost of hWi ≤ 0, transform any number of copies of free states into any other state. In particular, the agent cannot change the state’s β or µ j ’s. Proof. Drawing on Theorem 9, we prove Theorem 10 by contradiction. Imagine that some free operation could transform some number m of copies of γv := P e− β (HR − j µ j Q jR ) /Z into some other state γv0 : γv⊗m 7→ γv0 . (γv0 could have a different form from the NATS’s. Alternatively, γv0 could have the same form but have different µ j ’s or a different β.) γv0 is not completely passive. Work could be extracted from some number n of copies of γv0 , by Theorem 9. By converting copies of γv into copies of γv0 , and extracting work from copies of γv0 , the agent could extract work from γv for free. But work cannot be extracted from γv , by Theorem 9. Hence γv⊗m must not be convertible into any γv0 , γv , for all m = 1, 2, . . ..  Second laws: Consider any resource theory defined by operations that preserve Pc some state, e.g., states of the form e− β (HR − j=1 µ j Q jR ) /Z. Consider any distance measure on states that is contractive under the free operations. Every state’s distance from the preserved state ρR decreases monotonically under the operations. NATO can be characterized with any distance measure from ρR that is contractive under completely positive trace-preserving maps. We focus on the Rényi divergences, extending the second laws developed in [26] for the resource theory for heat exchanges. To avoid excessive subscripting, we alter our notation for the NATS. For any subsystem T, we denote by γT the NATS relative to the fixed β, to the fixed µ j ’s, and to the Hamiltonian HT and the charges Q1T , . . . ,Q cT associated with T. For examPc ple, γSW := e− β[(HS +HW )+ j=1 µ j (Q jS +Q jW )] /Z denotes the NATS associated with the system-and-battery composite. We define the generalized free energies Fα ( ρS , γS ) := k BT Dα ( ρS kγS ) − k BT log(Z ). The classical Rényi divergences Dα ( ρS kγS ) are defined as   X sgn(α) log  pkα qk1−α  , Dα ( ρS kγS ) := α−1 k (D.38) (D.39) 244 wherein pk and qk denote the probabilities of the possible outcomes of measurements of the work function W associated with ρS and with γS . The state ρS of S is compared with the NATS associated with HS and with the Q jS ’s. The Fα ’s generalize the thermodynamic free energy. To see how, we consider transforming n copies ( ρS ) ⊗n of a state ρS . Consider the asymptotic limit, similar to the thermodynamic limit, in which n → ∞. Suppose that the agent has some arbitrarily small, nonzero probability ε of failing to achieve the transformation. ε can be incorporated into any Fα via “smoothing” [26]. The smoothed Fαε per copy of ρS approaches F1 in the asymptotic limit [26]:  1 ε Fα ( ρS ) ⊗n , (γS ) ⊗n = F1 ( ρS ) n→∞ n c X = hHS i ρS − T S( ρS ) + µ j hQ jS i. lim (D.310) (D.311) j=1 P This expression resembles the definition F := E − T S + cj=1 µ j Q j of a thermodynamic free energy F. In terms of these generalized free energies, we formulate second laws. Proposition 2. In the presence of a heat bath of inverse temperature β and chemical potentials µ j , the free energies Fα ( ρS , γS ) decrease monotonically: Fα ( ρS , γS ) ≥ Fα ( ρ0S , γS ) ∀α ≥ 0, (D.312) wherein ρS and ρ0S denote the system’s initial and final states. If [WS , ρ0S ] = 0 and Fα ( ρS , γS ) ≥ Fα ( ρ0S , γS ) ∀α ≥ 0, (D.313) some catalytic NATO maps ρS to ρ0S . The Fα ( ρS , γS )’s are called “monotones.” Under NATO, the functions cannot increase. The transformed state approaches the NATS or retains its distance. Two remarks about extraneous systems are in order. First, the second laws clearly govern operations during which no work is performed on the system S. But the second laws also govern work performance: Let SW denote the system-and-battery composite. The second laws govern the transformations of SW . During such transformations, work can be transferred from W to S. 245 Second, the second laws govern transformations that change the system’s Hamiltonian. An ancilla facilitates such transformations [15]. Let us model the change, via external control, of an initial Hamiltonian HS into HS0 . Let γS and γS0 denote the NATSs relative to HS and to HS0 . The second laws become Fα ( ρS , γS ) ≥ Fα ( ρ0S , γS0 ) ∀α ≥ 0. (D.314) Extractable work: In terms of the free energies, we can bound the work extractable from a resource state via NATO. Unlike in the previous section, we consider the battery W separately from the system S of interest. We assume that W and S initially occupy a product state. (This assumption is reasonable for the idealised, infinite-dimensional battery we have been considering. As we will show, the assumption can be dropped when we focus on average work.) Let ρW and ρ0W denote the battery’s initial and final states. For all α, Fα ( ρS ⊗ ρW , γSW ) ≥ Fα ( ρ0S ⊗ ρ0W , γSW ). (D.315) Since Fα ( ρS ⊗ ρW , γSW ) = Fα ( ρS , γS ) + Fα ( ρW , γW ),   Fα ρ0W , γW − Fα ( ρW , γW ) ≤ Fα ( ρS , γS ) − Fα ( ρ0S , γS ). (D.316) If the battery states ρW and ρ0W are energy eigenstates, the left-hand side of Ineq. (D.316) represents the work extractable during one implementation of the protocol. Hence the right-hand side bounds the work extractable during the transition ρS 7→ ρ0S . This bound is a necessary condition under which work can be extracted. When α = 1, we need not assume that W and S occupy a product state. The reason is that subadditivity implies F1 ( ρSW , γSW ) ≤ F1 ( ρS , γS ) + F1 ( ρW , γW ). F1 is the relevant free energy if only the average work is important. Quantum second laws: As in [26], additional laws can be derived in terms of quantum Rényi divergences [30–33]. These laws provide extra constraints if ρS (and/or ρ0S ) has coherences relative to the WS eigenbasis. Such coherences would prevent ρS from commuting with the work function. Such noncommutation is a signature of truly quantum behavior. Two quantum analogues of Fα ( ρS , γS ) are defined as F̃α ( ρS , γS ) := k BT    sgn(α) log Tr ραS (γS ) 1−α − k BT log(Z ) α−1 (D.317) 246 and     1−α α 1−α 1 2α 2α F̂α ( ρS , γS ) := k BT log Tr (γS ) ρS (γS ) α−1 − kBT log(Z ). (D.318) The additional second laws have the following form. Proposition 3. NATO can transform ρS into ρ0S only if F̂α ( ρS , γS ) ≥ F̂α ( ρ0S , γS ) F̂α (γS , ρS ) ≥ F̂α (γS , ρS ) F̃α ( ρS , γS ) ≥ F̃α ( ρ0S , γS ) 1 , "2 # 1 ∀α ∈ , 1 , 2 ∀α ≥ ∀α ∈ [0, 2]. (D.319) and (D.320) (D.321) These laws govern transitions during which the Hamiltonian changes via an ancilla, as in [15]. 247 BIBLIOGRAPHY [1] Y. Ogata, Journal of Functional Analysis 264, 2005 (2013). [2] F. Hiai, M. Ohya, and M. Tsukada, Pacific J. Math. 96, 99 (1981). [3] W. Hoeffding, Journal of the American Statistical Association 58, 13 (1963). [4] A. Winter, Coding Theorems of Quantum Information Theory, PhD thesis, Universität Bielefeld, 1999. [5] M. Hayashi, Journal of Physics A Mathematical General 35, 10759 (2002). [6] A. Aleksandrov and V. Peller, Advances in Mathematics 224, 910 (2010). [7] W. Pusz and S. L. Woronowicz, Comm. Math. Phys. 58, 273 (1978). [8] A. Lenard, J. Stat. Phys. 19, 575 (1978). [9] R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Rev. Mod. Phys. 81, 865 (2009). [10] S. D. Bartlett, T. Rudolph, and R. W. Spekkens, Reviews of Modern Physics 79, 555 (2007). [11] V. Veitch, S. A. Hamed Mousavian, D. Gottesman, and J. Emerson, New Journal of Physics 16, 013009 (2014). [12] A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016). [13] I. Devetak, A. W. Harrow, and A. Winter, IEEE Trans. Inf. Theor. 54, 4587 (2008). [14] D. Janzing, P. Wocjan, R. Zeier, R. Geiss, and T. Beth, Int. J. Theor. Phys. 39, 2717 (2000). [15] M. Horodecki and J. Oppenheim, Nature Communications 4, 2059 (2013). [16] N. Yunger Halpern and J. M. Renes, Phys. Rev. E 93, 022126 (2016). [17] N. Y. Halpern, 094001 (2018). Journal of Physics A: Mathematical and Theoretical 51, [18] M. Lostaglio, D. Jennings, and T. Rudolph, 1511.04420. ArXiv e-prints (2015), [19] A. Kitaev, D. Mayers, and J. Preskill, Phys. Rev. A 69, 052326 (2004). [20] O. C. O. Dahlsten, R. Renner, E. Rieper, and V. Vedral, New Journal of Physics 13, 053015 (2011). 248 [21] L. Del Rio, J. Åberg, R. Renner, O. Dahlsten, and V. Vedral, Nature 474, 61 (2011). [22] J. Åberg, Nature Communications 4, 1925 (2013). [23] F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Physical Review Letters 111, 250404 (2013). [24] J. Åberg, Phys. Rev. Lett. 113, 150402 (2014). [25] K. Korzekwa, M. Lostaglio, J. Oppenheim, and D. Jennings, New Journal of Physics 18 (2016). [26] F. G. S. L. Brandao, M. Horodecki, N. H. Y. Ng, J. Oppenheim, and S. Wehner, Proc. Natl. Acad. Sci. 112, 3275 (2015). [27] P. Skrzypczyk, A. J. Short, and S. Popescu, Nature Communications 5, 4185 (2014). [28] M. Lostaglio, K. Korzekwa, D. Jennings, and T. Rudolph, Physical Review X 5, 021001 (2015). [29] Y. Guryanova, S. Popescu, A. J. Short, R. Silva, and P. Skrzypczyk, Nature Communications 7, 12049 (2016), 1512.01190. [30] F. Hiai, M. Mosonyi, D. Petz, and C. Bény, Rev. Math. Phys. 23, 691 (2011). [31] M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, Journal of Mathematical Physics 54 (2013). [32] M. M. Wilde, A. Winter, and D. Yang, Communications in Mathematical Physics 331, 593 (2014). [33] V. Jaksic, Y. Ogata, Y. Pautrat, and C.-A. Pillet, Entropic fluctuations in quantum statistical mechanics. an introduction, in Quantum Theory from Small to Large Scales: Lecture Notes of the Les Houches Summer School, edited by J. Fröhlich, S. Manfred, M. Vieri, W. De Roeck, and L. F. Cugliandolo, , Lecture Notes of the Les Houches Summer School Vol. 95, Oxford University Press, 2012.