Yet Another Proof of the Joint Convexity of Relative Entropy

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