Maximum Entropy and Sufficiency

TL;DR

This paper explores the relationship between Bregman divergence, sufficiency, and spectral sets within convex state spaces, highlighting the unique role of spectral sets like density matrices in quantum information theory.

Contribution

It introduces a general framework for Bregman divergence and sufficiency on convex state spaces, emphasizing the special status of spectral sets and proposing a conjecture linking information theory to Jordan algebras.

Findings

Spectral sets uniquely admit Bregman divergence satisfying sufficiency.

Density matrices are key examples of spectral sets in quantum states.

Conjecture: Information theory naturally leads to Jordan algebra structures.

Abstract

The notion of Bregman divergence and sufficiency will be defined on general convex state spaces. It is demonstrated that only spectral sets can have a Bregman divergence that satisfies a sufficiency condition. Positive elements with trace 1 in a Jordan algebra are examples of spectral sets, and the most important example is the set of density matrices with complex entries. It is conjectured that information theoretic considerations lead directly to the notion of Jordan algebra under some regularity conditions.