Quantum state discrimination: a geometric approach

TL;DR

This paper presents a geometric approach to quantum state discrimination, analyzing the problem through the embedding of maximal-distance simplexes and deriving bounds for distances between quantum states, simplifying the problem to diagonal states.

Contribution

It introduces a geometric framework for quantum state discrimination, deriving bounds for distances and showing the sufficiency of considering diagonal states under unitary transformations.

Findings

Derived bounds for trace distance and fidelity between quantum states.

Showed that optimal discrimination can be analyzed using diagonal states only.

Illustrated the geometric approach using Weyl chambers.

Abstract

We analyse the problem of finding sets of quantum states that can be deterministically discriminated. From a geometric point of view this problem is equivalent to that of embedding a simplex of points whose distances are maximal with respect to the Bures distance (or trace distance). We derive upper and lower bounds for the trace distance and for the fidelity between two quantum states, which imply bounds for the Bures distance between the unitary orbits of both states. We thus show that when analysing minimal and maximal distances between states of fixed spectra it is sufficient to consider diagonal states only. Hence considering optimal discrimination, given freedom up to unitary orbits, it is sufficient to consider diagonal states. This is illustrated geometrically in terms of Weyl chambers.