Operator Entropy Inequalities

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 $$S_q^f(\mathbf{A}|\mathbf{B}):=\sum _{j=1}^n{A}_j^{1/2}(A_j^{-1/2}{B}_j{A}_j^{-1/2}{)}^{q}f(A_j^{-1/2}{B}_j{A}_j^{-1/2})A_j^{1/2},$$ 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.

11 pages; to appear in Colloq. Math

References in corpus (1)

• A Matrix Convexity Approach to Some Celebrated Quantum Inequalities