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

We consider 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 Tr K* A p K B 1-p Lieb's joint concavity in (A, B) 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 Carlen and Lieb for Formula: see text. In all cases, the equality conditions are independent of p; for extensions to three spaces they are identical to the conditions for equality in the strong subadditivity of relative entropy.