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Operator and Graph-Theoretic Techniques for Distinguishing Quantum States in the Local Operations and Classical Communication Framework

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Title: Operator and Graph-Theoretic Techniques for Distinguishing Quantum States in the Local Operations and Classical Communication Framework
Author: Comfort Mintah
Department: Department of Mathematics and Statistics
Program: Mathematics and Statistics
Advisor: Kribs, DavidPereira, Rajesh
Abstract: The quantum communication framework called quantum local operations and classical communication (LOCC) plays a fundamental role in near-term efforts to develop hybrid classical and quantum technologies. Recent advances in the subject include expanded investigations of the mathematical foundations of LOCC. We begin this thesis with an introduction to quantum information theory and the structure theory of finite-dimensional $C^{*}$-algebras to help build the mathematical framework for quantum LOCC. We focus on an important scheme of LOCC called one-way LOCC, where classical communication is in a predetermined direction, and we consider its mathematical description. We apply algebraic techniques to one-way LOCC, such as the existence of a separating vector for certain operator algebras. We explore the intersection between quantum error correction and quantum one-way LOCC. We consider the protocol of one-way LOCC measurement as a quantum channel and derive new methods to distinguish sets of states via one-way LOCC. We establish conditions for one-way LOCC state distinguishability for the set of maximally entangled states that arise from the stabilizer formalism for quantum error correction. We apply the results from quantum error correction and from one-way LOCC to build small sets of states that are indistinguishable under one-way LOCC. We extend these constructions to the generalized Pauli states and compare our results with other recent works. We next introduce a graph-theoretic approach to distinguish sets of product states via one-way LOCC. We see that one-way LOCC state distinguishability is equivalent to the existence of an edge graph clique cover of the graph with nice properties. We establish a number of results for which a set of states can be distinguished with one-way LOCC using the properties of the underlying graphs associated with the set of states. We present some illustrative examples.
Date: 2021-06
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Related Publications: Kribs, D. W., Mintah, C., Nathanson, M., & Pereira, R. (2020). Vector representations of graphs and distinguishing quantum product states with one-way LOCC. Linear Algebra and Its Applications, 602, 223–239.

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