Search results for "Dyson conjecture"

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

2020

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…