Entanglement of Multipartite Quantum States and the Generalized Quantum Search

Author: Gingrich, Robert Michael

Year: 2002

Degree: Dissertation (Ph.D.)

Advisor: Preskill, John P.

Committee Members: Preskill, John P.; Adami, Christoph Carl; Frautschi, Steven C.; Mabuchi, Hideo

Option: Physics; Computer Science

DOI: 10.7907/0M1P-NS54

Abstract

In chapter 2 various parameterizations for the orbits under local unitary transformations of three-qubit pure states are analyzed. It is shown that the entanglement monotones of any multipartite pure state uniquely determine the orbit of that state. It follows that there must be an entanglement monotone for three-qubit pure states which depends on the Kempe invariant defined in [1]. A form for such an entanglement monotone is proposed. A theorem is proved that significantly reduces the number of entanglement monotones that must be looked at to find the maximal probability of transforming one multipartite state to another.

In chapter 3 Grover's unstructured quantum search algorithm is generalized to use an arbitrary starting superposition and an arbitrary unitary matrix. A formula for the probability of the generalized Grover's algorithm succeeding after n iterations is derived. This formula is used to determine the optimal strategy for using the unstructured quantum search algorithm. The speedup obtained illustrates that a hybrid use of quantum computing and classical computing techniques can yield a performance that is better than either alone. The analysis is extended to the case of a society of k quantum searches acting in parallel.

In chapter 4 the positive map Г : p → (Trρ) - ρ is introduced as a separability criterion. Any separable state is mapped by the tensor product of Г and the identity in to a non-negative operator, which provides a necessary condition for separability. If Г acts on a two-dimensional subsystem, then it is equivalent to partial transposition and therefore also sufficient for 2 x 2 and 2 x 3 systems. Finally, a connection between this map for two qubits and complex conjugation in the "magic" basis [2] is displayed.

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