Hypothesis testing and Stein's lemma in general probability theories with Euclidean Jordan algebra and its quantum realization

TL;DR

This paper explores the mathematical foundations of Stein's Lemma within general probabilistic theories modeled by Euclidean Jordan Algebras, revealing structural similarities between classical and quantum information theories.

Contribution

It generalizes Stein's Lemma to models of GPTs generated by EJAs, extending the mathematical understanding of information-theoretic principles beyond quantum theory.

Findings

Stein's Lemma holds in models of GPTs generated by EJAs

Established a generalization of information-theoretic tools for EJAs

Unified classical and quantum information structures under a common framework

Abstract

Even though quantum information theory gives advantage over classical information theory, these two information theories have a structural similarity that many exponet rates of information tasks asymptotically equal to entropic quantities. A typical example is Stein's Lemma, which many researchers still keep interested in. In this paper, in order to analyze the mathemtaical roots of the structural similarity, we investigate mathematically minimum structure where Stein's Lemma holds. We focus on the structure of Euclidean Jordan Algebras (EJAs), which is a generalization of the algebraic structure in quantum theory, and we investigate the properties of general models of General Probabilistic Theories (GPTs) generated by EJAs. As a result, we prove Stein's Lemma in any model of GPTs generated by EJAs by establishing a generalization of information theoretical tools from the mathematical properties of EJAs.