Generalised Brègman relative entropies: a brief introduction
Ryszard Paweł Kostecki
TL;DR
The paper develops a unified framework for Brègman type relative entropies that extends beyond reflexive Banach spaces to arbitrary state spaces, including probabilistic, quantum, and postquantum settings. It achieves this by combining Euler‑Legendre functions with nonlinear embeddings to define generalized Brègman information $D_{ell,Psi}$, which pulls back classical reflexive theory to nonreflexive contexts. The work proves that $D_{ell,Psi}$ is an information on $Z$, derives nonlinear pythagorean projection properties, and establishes conditions for continuity of projections; it also provides several concrete instantiations in operator algebras and Orlicz spaces that recover standard quantum divergences and their postquantum analogues. The results unify reflexive Banach space methods and information geometric approaches, offering a versatile geometric toolkit for optimization and state-space geometry across classical, quantum, and postquantum theories.
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.