From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture (New title)

Abstract

In this paper we study the joint convexity/concavity of the trace functions \[ Ψ_{p,q,s}(A,B)=\text{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s,~~p,q,s\in \mathbb{R}, \] where $A$ and $B$ are positive definite matrices and $K$ is any fixed invertible matrix. We will give full range of $(p,q,s)\in {\mathbb{R}}^3$ for ${\Psi }_{p,q,s}$ 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 $(\alpha ,z)$ for $\alpha$-$z$ 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 ${\Psi }_{p,0,1/p}$ 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 \[ Ψ_{p,1-p,1}(A,B)=\text{Tr} K^*A^{p}KB^{1-p},~~-1\le p\le 1, \] using a variational method.

14 pages, 1 figure. Some errors and typos corrected. Title changed. Main results improved: a unified and simple proof of the convexity/concavity of a large family of trace functions ${\Psi }_{p,q,s}$ using a variational method and the convexity/concavity (due to Ando/Lieb) of ${\Psi }_{p,1-p,1}$. To appear in Adv. Math