Relative information entropy in cosmology: The problem of information entanglement

TL;DR

This paper explores the use of generalized relative entropy measures, like Tsallis and Rénnyi entropies, to quantify information entanglement in inhomogeneous cosmological models, simplifying the complex mutual information calculations.

Contribution

It introduces a parametric extension of the KL measure using Tsallis and Rénnyi entropies to effectively analyze information entanglement in cosmology.

Findings

Tsallis entropy approximates mutual information for small inhomogeneities

Rénnyi entropy measures independent information within the domain

Simplifies complex calculations of information entanglement in cosmological models

Abstract

The necessary information to distinguish a local inhomogeneous mass density field from its spatial average on a compact domain of the universe can be measured by relative information entropy. The Kullback-Leibler (KL) formula arises very naturally in this context, however, it provides a very complicated way to compute the mutual information between spatially separated but causally connected regions of the universe in a realistic, inhomogeneous model. To circumvent this issue, by considering a parametric extension of the KL measure, we develop a simple model to describe the mutual information which is entangled via the gravitational field equations. We show that the Tsallis relative entropy can be a good approximation in the case of small inhomogeneities, and for measuring the independent relative information inside the domain, we propose the R\'enyi relative entropy formula.