Yet Another Proof of the Joint Convexity of Relative Entropy

TL;DR

This paper provides a simple proof of the joint convexity of quantum relative entropy and related trace convexity results, which are fundamental in quantum information theory and the proof of strong subadditivity.

Contribution

It offers a new, simplified proof of the joint convexity of relative entropy using operator convex functions and Ando's observation.

Findings

Proves joint convexity of relative entropy.

Establishes trace convexity result of Lieb.

Simplifies understanding of quantum entropy properties.

Abstract

The joint convexity of the map $(X,A) \mapsto X^* A^{-1} X$, 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.