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Log-majorization related to Rényi divergences (1808.03866v1)

Published 11 Aug 2018 in math.FA

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.

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