Unambiguous State Discrimination of two density matrices in Quantum Information Theory

TL;DR

This thesis develops theoretical methods for unambiguously discriminating two mixed quantum states, providing bounds, conditions, and explicit solutions, with applications to quantum communication protocols.

Contribution

It introduces reduction theorems simplifying the USD problem, derives tighter bounds and conditions for optimal discrimination, and provides analytical solutions for specific classes of mixed states.

Findings

Derived tighter bounds on failure probability using fidelity.

Constructed optimal measurements for certain mixed states.

Applied results to quantum key distribution protocols.

Abstract

In this thesis we study the problem of unambiguously discriminating two mixed quantum states. We first present reduction theorems for optimal unambiguous discrimination of two generic density matrices. We show that this problem can be reduced to that of two density matrices that have the same rank $r$ in a 2$r$-dimensional Hilbert space. These reduction theorems also allow us to reduce USD problems to simpler ones for which the solution might be known. As an application, we consider the unambiguous comparison of $n$ linearly independent pure states with a simple symmetry. Moreover, lower bounds on the optimal failure probability have been derived. For two mixed states they are given in terms of the fidelity. Here we give tighter bounds as well as necessary and sufficient conditions for two mixed states to reach these bounds. We also construct the corresponding optimal measurement. With this result, we provide analytical solutions for unambiguously discriminating a class of generic mixed states. This goes beyond known results which are all reducible to some pure state case. We however show that examples exist where the bounds cannot be reached. Next, we derive properties on the rank and the spectrum of an optimal USD measurement. This finally leads to a second class of exact solutions. Indeed we present the optimal failure probability as well as the optimal measurement for unambiguously discriminating any pair of geometrically uniform mixed states in four dimensions. This class of problems includes for example the discrimination of both the basis and the bit value mixed states in the BB84 QKD protocol with coherent states.