On a 2-relative entropy

TL;DR

This paper extends the concept of relative entropy into a 2-categorical framework, providing a functorial construction that captures information measures between channels with properties analogous to classical relative entropy.

Contribution

It introduces a 2-categorical extension of relative entropy, establishing its functoriality and key properties like convex linearity and lower semicontinuity in this higher categorical context.

Findings

Constructed a 2-categorical relative entropy functor

Proved functoriality with respect to vertical morphisms

Established properties analogous to classical relative entropy

Abstract

We construct a 2-categorical extension of the relative entropy functor of Baez and Fritz, and show that our construction is functorial with respect to vertical morphisms. Moreover, we show such a `2-relative entropy' satisfies natural 2-categorial analogues of convex linearity, vanishing under optimal hypotheses, and lower semicontinuity. While relative entropy is a relative measure of information between probability distributions, we view our construction as a relative measure of information between channels.