Optimal State Discrimination in General Probabilistic Theories

TL;DR

This paper develops a geometric method for optimal state discrimination applicable across classical, quantum, and generalized probabilistic theories, using a concept called Helstrom families of ensembles.

Contribution

It introduces a unified geometric framework for optimal state discrimination in general probabilistic theories, extending beyond quantum mechanics.

Findings

Method successfully applied to 2-level quantum systems.

Reproduces optimal success probabilities for symmetric quantum states.

Shows existence of ensemble families in classical and quantum binary cases.

Abstract

We investigate a state discrimination problem in operationally the most general framework to use a probability, including both classical, quantum theories, and more. In this wide framework, introducing closely related family of ensembles (which we call a {\it Helstrom family of ensembles}) with the problem, we provide a geometrical method to find an optimal measurement for state discrimination by means of Bayesian strategy. We illustrate our method in 2-level quantum systems and in a probabilistic model with square-state space to reproduce e.g., the optimal success probabilities for binary state discrimination and $N$ numbers of symmetric quantum states. The existences of families of ensembles in binary cases are shown both in classical and quantum theories in any generic cases.