Yet Another Proof of the Joint Convexity of Relative Entropy

Abstract

The joint convexity of the map $(X,A)\mapsto X^{\ast }{A}^{-1}X$, an integral representation of operator convex functions, and an observation of Ando are used to obtain a simple proof of both the joint convexity of relative entropy and a trace convexity result of Lieb. The latter was the key ingredient in the original proof of the strong subadditivity of quantum entropy.

Added dedication to Derek W. Robinson. Added proof of the montonicity of relative entropy under partial traces and strong subadditivity of quantum entropy to v2. Added to v3, a section on generalizations of relative entropy and a remark (not in LMP version) proving the joint convexity of relative entropy using an integral representation for log x instead of operator convex functions

References in corpus (1)

• A Matrix Convexity Approach to Some Celebrated Quantum Inequalities

Cited by in corpus (2)

• Mpemba effects in open nonequilibrium quantum systems
• Log-majorizations between quasi-geometric type means for matrices