Operator Entropy Inequalities

TL;DR

This paper extends the concept of relative operator entropy for sequences of positive operators using operator monotone functions and explores bounds and inequalities, including applications to Shannon entropy.

Contribution

It introduces a generalized entropy notion for operator sequences and derives new bounds and inequalities, extending previous results by Furuta.

Findings

Established bounds for the extended operator entropy

Derived inequalities related to Shannon entropy

Extended Furuta's inequality to new operator entropy context

Abstract

In this paper we investigate a notion of relative operator entropy, which develops the theory started by J.I. Fujii and E. Kamei [Math. Japonica 34 (1989), 341--348]. For two finite sequences $\mathbf{A}=(A_1,...,A_n)$ and $\mathbf{B}=(B_1,...,B_n)$ of positive operators acting on a Hilbert space, a real number $q$ and an operator monotone function $f$ we extend the concept of entropy by $$ S_q^f(\mathbf{A}|\mathbf{B}):=\sum_{j=1}^nA_j^{1/2}(A_j^{-1/2}B_jA_j^{-1/2})^qf(A_j^{-1/2}B_jA_j^{-1/2})A_j^{1/2}\,, $$ and then give upper and lower bounds for $S_q^f(\mathbf{A}|\mathbf{B})$ as an extension of an inequality due to T. Furuta [Linear Algebra Appl. 381 (2004), 219--235] under certain conditions. Afterwards, some inequalities concerning the classical Shannon entropy are drawn from it.