Mutually Unbiased Bases and Weyl Commutation Relations in Quantum State Distinguishability

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University of Guelph

This thesis explores the interplay between mutually unbiased bases, the Weyl commutation relation, and one-way local operations and classical communication (LOCC) distinguishability within quantum systems. We focus on three main concepts in this thesis: the Weyl commutation relation of matrices, the generalized Pauli matrices and mutually unbiased bases. We detail the well-known proof that in any dimension d, the number of mutually unbiased bases is at most d + 1. Specifically, we look at a proof that uses the argument of rank, as well as provide our own proof, which uses the argument of dimension. Leveraging these concepts, we develop an approach for distinguishing quantum states through one-way LOCC. Furthermore, we delve into established results related to common unbiased bases, which provide a framework for one-way LOCC distinguishability. The results and exposition contribute to a deeper understanding of quantum systems, with potential applications in secure communication protocols and quantum privacy.

Mutually unbiased bases, LOCC, Quantum states, State distinguishability