Log-majorization related to Rényi divergences

Abstract

For $\alpha ,z>0$ with $\alpha \ne 1$, motivated by comparison between different kinds of Rényi divergences in quantum information, we consider log-majorization between the matrix functions \begin{align*} P_α(A,B)&:=B^{1/2}(B^{-1/2}AB^{-1/2})^αB^{1/2}, \\ Q_{α,z}(A,B)&:=(B^{1-α\over2z}A^{α\over z}B^{1-α\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.

21 pages

References in corpus (2)

• Strong converse for the classical capacity of entanglement-breaking and Hadamard channels via a sandwiched Renyi relative entropy
• A new quantum version of f-divergence