Differential Entropy on Statistical Spaces

TL;DR

This paper introduces a new way to define differential entropy on fuzzy statistical spaces where points are localized within volumes, impacting integrability and quantum state analysis.

Contribution

It presents a novel formalism for differential entropy on statistical spaces with fuzziness, extending traditional concepts to non-Riemann integrable contexts.

Findings

Differential entropy can be defined on fuzzy statistical spaces.

The formalism accounts for non-Riemann integrability of relevant integrals.

Application to quantum states is discussed.

Abstract

We show that the previously introduced concept of distance on statistical spaces leads to a straightforward definition of differential entropy on these statistical spaces. These spaces are characterized by the fact that their points can only be localized within a certain volume and exhibit thus a feature of fuzziness. This implies that Riemann integrability of relevant integrals is no longer secured. Some discussion on the specialization of this formalism to quantum states concludes the paper.