The minimum-error discrimination via Helstrom family of ensembles and Convex Optimization

TL;DR

This paper develops a convex optimization approach using Helstrom families to solve optimal minimum-error quantum state discrimination problems, providing exact solutions for specific state configurations in qubit systems.

Contribution

It introduces a novel convex optimization method leveraging Helstrom families to find optimal measurements for discriminating multiple quantum states, including special cases with symmetric arrangements.

Findings

Maximum success probability for N equiprobable states was derived.

Exact solutions provided for three arbitrary quantum states.

Method applied to various state configurations including Platonic solid vertices.

Abstract

Using the convex optimization method and Helstrom family of ensembles introduced in Ref. [1], we have discussed optimal ambiguous discrimination in qubit systems. We have analyzed the problem of the optimal discrimination of N known quantum states and have obtained maximum success probability and optimal measurement for N known quantum states with equiprobable prior probabilities and equidistant from center of the Bloch ball, not all of which are on the one half of the Bloch ball and all of the conjugate states are pure. An exact solution has also been given for arbitrary three known quantum states. The given examples which use our method include: 1. Diagonal N mixed states; 2. N equiprobable states and equidistant from center of the Bloch ball which their corresponding Bloch vectors are inclined at the equal angle from z axis; 3. Three mirror-symmetric states; 4. States that have been prepared with equal prior probabilities on vertices of a Platonic solid. Keywords: minimum-error discrimination, success probability, measurement, POVM elements, Helstrom family of ensembles, convex optimization, conjugate states PACS Nos: 03.67.Hk, 03.65.Ta