Entropy on Spin Factors

TL;DR

This paper investigates the properties of Bregman divergences on rank 2 convex bodies, such as spin factors, revealing conditions under which these bodies exhibit spectral or ball-shaped geometries related to entropy functions.

Contribution

It characterizes Bregman divergences on rank 2 convex bodies, especially spin factors, and establishes conditions for spectrality and shape, linking geometry with entropy properties.

Findings

Convex bodies of rank 2 with Bregman divergence satisfying sufficiency are spectral.

If the Bregman divergence is monotone, the convex body is ball-shaped.

Results impose restrictions on possible state spaces of physical systems with well-behaved entropy.

Abstract

Recently it has been demonstrated that the Shannon entropy or the von Neuman entropy are the only entropy functions that generate a local Bregman divergences as long as the state space has rank 3 or higher. In this paper we will study the properties of Bregman divergences for convex bodies of rank 2. The two most important convex bodies of rank 2 can be identified with the bit and the qubit. We demonstrate that if a convex body of rank 2 has a Bregman divergence that satisfies sufficiency then the convex body is spectral and if the Bregman divergence is monotone then the convex body has the shape of a ball. A ball can be represented as the state space of a spin factor, which is the most simple type of Jordan algebra. We also study the existence of recovery maps for Bregman divergences on spin factors. In general the convex bodies of rank 2 appear as faces of state spaces of higher rank. Therefore our results give strong restrictions on which convex bodies could be the state space of a physical system with a well-behaved entropy function.