Some trace inequalities for exponential and logarithmic functions

TL;DR

This paper establishes new trace inequalities involving exponential and logarithmic functions of positive matrices, providing simplified proofs and generalizations of known quantum entropy inequalities, with implications for matrix analysis and quantum information theory.

Contribution

The paper introduces conditions on functions of positive matrices to derive trace inequalities, generalizes known quantum entropy inequalities, and refines the Golden-Thompson inequality.

Findings

Established conditions for trace inequalities involving matrix functions.

Provided simplified proofs of Hiai and Petz inequalities.

Derived new bounds and refinements for quantum relative entropies.

Abstract

Consider a function $F(X,Y)$ of pairs of positive matrices with values in the positive matrices such that whenever $X$ and $Y$ commute $F(X,Y)= X^pY^q.$ Our first main result gives conditions on $F$ such that ${\rm Tr}[ X \log (F(Z,Y))] \leq {\rm Tr}[X(p\log X + q \log Y)]$ for all $X,Y,Z$ such that ${\rm Tr} Z = {\rm Tr} X$. (Note that $Z$ is absent from the right side of the inequality.) We give several examples of functions $F$ to which the theorem applies. Our theorem allows us to give simple proofs of the well known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables $X,Y,Z$ instead of just $X,Y$ alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy $D(X||Y) = {\rm Tr} [X(\log X-\log Y])$, and two others, the Donald relative entropy $D_D(X||Y)$, and the Belavkin-Stasewski relative entropy $D_{BS}(X||Y)$. They are known to satisfy $D_D(X||Y) \leq D(X||Y)\leq D_{BS}(X||Y)$. We prove that the Donald relative entropy provides the sharp upper bound, independent of $Z$, on ${\rm Tr}[ X \log (F(Z,Y))]$ in a number of cases in which $(Z,Y)$ is homogeneous of degree $1$ in $Z$ and $-1$ in $Y$. We also investigate the Legendre transforms in $X$ of $D_D(X||Y)$ and $D_{BS}(X||Y)$, and show how our results for these lead to new refinements of the Golden-Thompson inequality.