From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture

In this paper we study the joint convexity/concavity of the trace functions Ψp,q,s(A,B)=Tr(Bq2K∗ApKBq2)s, p,q,s∈R, where A and B are positive definite matrices and K is any fixed invertible matrix. We will give full range of (p,q,s)∈R3 for Ψp,q,s to be jointly convex/concave for all K. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of (α,z) for α-z Rényi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of Ψp,0,1/p for 0<p<1 which was first proved by Epstein using complex analysis. The key is to reduce the problem to the joint convexity/concavity of the trace functions Ψp,1−p,1(A,B)=TrK∗ApKB1−p, −1≤p≤1, using a variational method.