Bounds of the Pinsker and Fannes Types on the Tsallis Relative Entropy

TL;DR

This paper derives bounds on Tsallis relative entropy using quantum f-divergence properties, extending classical inequalities like Pinsker, Fannes, and Fano to the Tsallis and Rényi entropy frameworks, with implications for quantum information theory.

Contribution

It introduces new bounds for Tsallis relative entropy, including Pinsker and Fannes type inequalities, and extends classical inequalities to the Tsallis and Rényi entropy contexts.

Findings

Derived Pinsker-type lower bounds for Tsallis relative entropy.

Established upper continuity bounds for Tsallis relative entropy in the commutative case.

Extended Fano inequality to all positive alpha for Tsallis entropies.

Abstract

Pinsker's and Fannes' type bounds on the Tsallis relative entropy are derived. The monotonicity property of the quantum $f$-divergence is used for its estimating from below. For order $\alpha\in(0,1)$, a family of lower bounds of Pinsker type is obtained. For $\alpha>1$ and the commutative case, upper continuity bounds on the relative entropy in terms of the minimal probability in its second argument are derived. Both the lower and upper bounds presented are reformulated for the case of R\'{e}nyi's entropies. The Fano inequality is extended to Tsallis' entropies for all $\alpha>0$. The deduced bounds on the Tsallis conditional entropy are used for obtaining inequalities of Fannes' type.