Log-majorization related to Rényi divergences

2018/08/11 by Hiai, Fumio · 2 citations
#15A42 #15A45 #47A64 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1808.03866

Abstract

For α,z>0 with α≠1, motivated by comparison between different kinds of Rényi divergences in quantum information, we consider log-majorization between the matrix functions P_α(A,B)amp;:=B^1/2(B^-1/2AB^-1/2)^αB^1/2,
Q_α,z(A,B)amp;:=(B^1-α\over2zA^{α\over z}B^1-α\over2z)^z of two positive (semi)definite matrices A,B. We precisely determine the parameter α,z for which P_α(A,B)\prec_logQ_α,z(A,B) and Q_α,z(A,B)\prec_logP_α(A,B) holds, respectively.

Cited by

• Log-majorizations between quasi-geometric type means for matrices
• Various inequalities between quasi-arithmetic mean and quasi-geometric type means for matrices

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