Log-majorization related to R\'enyi divergences

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.