Log-majorization related to R\'enyi divergences

TL;DR

This paper investigates log-majorization relations between two matrix functions related to Re9nyi divergences, providing precise conditions on parameters for these inequalities to hold in quantum information theory.

Contribution

It precisely characterizes the parameter ranges where log-majorization holds between two matrix functions associated with Re9nyi divergences, advancing understanding in quantum information theory.

Findings

Identifies parameter conditions for log-majorization between matrix functions

Clarifies the relation between different Re9nyi divergence measures

Enhances mathematical understanding of quantum information measures

Abstract

For $\alpha,z>0$ with $\alpha\ne1$, motivated by comparison between different kinds of R\'enyi divergences in quantum information, we consider log-majorization between the matrix functions \begin{align*} P_\alpha(A,B)&:=B^{1/2}(B^{-1/2}AB^{-1/2})^\alpha B^{1/2}, \\ Q_{\alpha,z}(A,B)&:=(B^{1-\alpha\over2z}A^{\alpha\over z}B^{1-\alpha\over2z})^z \end{align*} of two positive (semi)definite matrices $A,B$. We precisely determine the parameter $\alpha,z$ for which $P_\alpha(A,B)\prec_{\log}Q_{\alpha,z}(A,B)$ and $Q_{\alpha,z}(A,B)\prec_{\log}P_\alpha(A,B)$ holds, respectively.