Generalised Brègman relative entropies: a brief introduction

Abstract

We present some basic elements of the theory of generalised Brègman relative entropies over nonreflexive Banach spaces. Using nonlinear embeddings of Banach spaces together with the Euler--Legendre functions, this approach unifies two former approaches to Brègman relative entropy: one based on reflexive Banach spaces, another based on differential geometry. This construction allows to extend Brègman relative entropies, and related geometric and operator structures, to arbitrary-dimensional state spaces of probability, quantum, and postquantum theory. We give several examples, not considered previously in the literature.

arXiv admin note: text overlap with arXiv:1710.01837 (author's note: this overlap will be removed in the upcoming v5 of arXiv:1710.01837 and the upcoming v4 of this paper); v3: minor change (grant numbers)