Minimum error discrimination problem for pure qubit states

TL;DR

This paper reformulates the minimum error discrimination problem for pure qubit states using Bloch vectors, providing algorithms for direct and inverse optimization, and illustrating results with specific state cases.

Contribution

It introduces a reformulation of the discrimination problem in terms of Bloch vectors and offers algorithmic solutions for both direct and inverse optimization problems.

Findings

Derived necessary and sufficient conditions for error minimization.

Provided an algorithmic approach for the direct optimization problem.

Illustrated results with specific cases of pure qubit states.

Abstract

The necessary and sufficient conditions for minimization of the generalized rate error for discriminating among $N$ pure qubit states are reformulated in terms of Bloch vectors representing the states. For the direct optimization problem an algorithmic solution to these conditions is indicated. A solution to the inverse optimization problem is given. General results are widely illustrated by particular cases of equiprobable states and $N=2,3,4$ pure qubit states given with different prior probabilities.