Convexity of quasi-entropy type functions: Lieb's and Ando's convexity theorems revisited

TL;DR

This paper revisits Lieb's and Ando's convexity theorems by analyzing a generalized function related to quantum information measures, providing unified characterizations of convexity, concavity, and monotonicity properties.

Contribution

It unifies and extends existing results on quantum information functions by characterizing their convexity, concavity, and monotonicity properties for a broad class of functions.

Findings

Characterizes joint convexity and concavity of the function $I_f^ heta$.

Establishes conditions for monotonicity of the function.

Unifies various quantum information quantities under a common framework.

Abstract

Given a positive function $f$ on $(0,\infty)$ and a non-zero real parameter $\theta$, we consider a function $I_f^\theta(A,B,X)=Tr X^*(f(L_AR_B^{-1})R_B)^\theta(X)$ in three matrices $A,B>0$ and $X$. In the literature $\theta=\pm1$ has been typical. The concept unifies various quantum information quantities such as quasi-entropy, monotone metrics, etc. We characterize joint convexity/concavity and monotonicity properties of the function $I_f^\theta$, thus unifying some known results for various quantum quantities.