Continuity and Stability of Partial Entropic Sums

TL;DR

This paper extends Fannes' inequality to partial sums of Tsallis entropy for classical and quantum systems, providing new measures and characterizations of continuity and stability, with applications and reformulations in terms of fidelities.

Contribution

It introduces a new framework for analyzing partial entropic sums, enhancing understanding of their continuity and stability in quantum and classical information theory.

Findings

Derived estimates characterize continuity and stability properties.

Introduced partial sums of Tsallis entropy with basic properties.

Reformulated results using Uhlmann's partial fidelities.

Abstract

Extensions of Fannes' inequality with partial sums of the Tsallis entropy are obtained for both the classical and quantum cases. The definition of kth partial sum under the prescribed order of terms is given. Basic properties of introduced entropic measures and some applications are discussed. The derived estimates provide a complete characterization of the continuity and stability properties in the refined scale. The results are also reformulated in terms of Uhlmann's partial fidelities.