MSc Defence: Mutually Unbiased Bases and Weyl Commutation Relations in Quantum State Distinguishability

CANDIDATE: Mukesh Taank

ABSTRACT: This thesis investigates the relationship between mutually unbiased bases, the Weyl commutation relation and one-way local operations and classical communication (LOCC) distinguishability in 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 to prove this, as well as provide our own proof, which uses the argument of dimension. We use the properties of the Weyl commutation relation of matrices, the generalized Pauli matrices and mutually unbiased bases to find methods of distinguishing quantum states by one-way LOCC. Further, we take a deep look at established results that involve the notion of common unbiased bases, which provides 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.