6533b7d3fe1ef96bd1261361

RESEARCH PRODUCT

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

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Abstract In this paper we study the joint convexity/concavity of the trace functions Ψ p , q , s ( A , B ) = Tr ( B q 2 K ⁎ A p K B q 2 ) 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 ) ∈ R 3 for Ψ 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 ( α , z ) for α-z Renyi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of Ψ 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 ) = Tr K ⁎ A p K B 1 − p , − 1 ≤ p ≤ 1 , using a variational method.