Entropy Distance: New Quantum Phenomena

TL;DR

This paper explores novel quantum phenomena such as discontinuities in maximum-entropy inference and entropy distance in complex 3x3 matrices, revealing features absent in classical systems through geometric and information-theoretic analysis.

Contribution

It introduces new quantum phenomena related to entropy and mean value sets, providing a detailed geometric and topological analysis of these features.

Findings

Discontinuous maximum-entropy inference identified

Discontinuous entropy distance observed

Non-exposed faces of the mean value set discovered

Abstract

We study a curve of Gibbsian families of complex 3x3-matrices and point out new features, absent in commutative finite-dimensional algebras: a discontinuous maximum-entropy inference, a discontinuous entropy distance and non-exposed faces of the mean value set. We analyze these problems from various aspects including convex geometry, topology and information geometry. This research is motivated by a theory of info-max principles, where we contribute by computing first order optimality conditions of the entropy distance.