Optimal Observables for Minimum-Error State Discrimination in General Probabilistic Theories

TL;DR

This paper develops a general framework for optimal state discrimination in probabilistic theories, proving the existence of optimal observables, extending geometric methods, and revealing operational meanings of intrinsic metrics.

Contribution

It establishes the existence of optimal observables in general probabilistic theories and extends geometric discrimination methods to arbitrary dimensions.

Findings

Optimal observables always exist for finite state discrimination.

The geometric approach is effective in arbitrary dimensions for two-state discrimination.

The success probability generalizes the trace distance and operationally relates to Gudder's intrinsic metric.

Abstract

General Probabilistic Theories provide the most general mathematical framework for the theory of probability in an operationally natural manner, and generalize classical and quantum theories. In this article, we study state-discrimination problems in general probabilistic theories using a Bayesian strategy. After re-formulation of the theories with mathematical rigor, we first prove that an optimal observable to discriminate any (finite) number of states always exists in the most general setting. Next, we revisit our recently proposed geometric approach for the problem and show that, for two-state discrimination, this approach is indeed effective in arbitrary dimensional cases. Moreover, our method reveals an operational meaning of Gudder's ``intrinsic metric'' by means of the optimal success probability, which turns out to be a generalization of the trace distance for quantum systems. As its by-product, an information-disturbance theorem in general probabilistic theories is derived, generalizing its well known quantum version.