Structure of minimum-error quantum state discrimination

TL;DR

This paper provides a comprehensive analysis of minimum-error quantum state discrimination, offering a general structure, geometric formulation, and explicit solutions for qubit states with equal priors, advancing understanding in quantum information processing.

Contribution

It introduces a general structural framework and geometric approach for minimum-error quantum state discrimination, including explicit solutions for qubit states with equal priors.

Findings

General structure of minimum-error discrimination established

Geometric formulation applicable to quantum states with clear geometry

Explicit analytical solutions for qubit states with equal priors provided

Abstract

Distinguishing different quantum states is a fundamental task having practical applications for information processing. Despite the efforts devoted so far, however, strategies for optimal discrimination are known only for specific examples. We here consider the problem of minimum-error quantum state discrimination where the average error is attempted to be minimized. We show the general structure of minimum-error state discrimination as well as useful properties to derive analytic solutions. Based on the general structure, we present a geometric formulation of the problem, which can be applied to cases where quantum state geometry is clear. We also introduce equivalent classes of sets of quantum states in terms of minimum-error discrimination: sets of quantum states in an equivalence class share the same guessing probability. In particular, for qubit states where the state geometry is found with the Bloch sphere, we illustrate that for an arbitrary set of qubit states, the minimum-error state discrimination with equal prior probabilities can be analytically solved, that is, optimal measurement and the guessing probability are explicitly obtained.