A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, with Conditions for Equality

Understand

We introduce a generalization of relative entropy derived from the Wigner-Yanase-Dyson entropy and give a simple, self-contained proof that it is convex.

• Moreover, special cases yield the joint convexity of relative entropy, and for the map (A,B) --> Tr K^* A^p K B^{1-p} Lieb's joint concavity for 0 < p < 1 and Ando's joint convexity for 1 < p < 2.
• This approach allows us to obtain conditions for equality in these cases, as well as conditions for equality in a number of inequalities which follow from them.
• These include the monotonicity under partial traces, and some Minkowski type matrix inequalities proved by Lieb and Carlen for mixed (p,q) norms.