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.

中文翻译：

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从 Wigner-Yanase-Dyson 猜想到 Carlen-Frank-Lieb 猜想

摘要 在本文中，我们研究了迹函数 Ψ p , q , s ( A , B ) = Tr ( B q 2 K ⁎ A p KB q 2 ) s , p , q , s ∈ R 的联合凸度/凹度，其中 A 和 B 是正定矩阵，K 是任何固定可逆矩阵。我们将给出全范围的 ( p , q , s ) ∈ R 3 ，因为 Ψ p , q , s 是所有 K 的联合凸/凹。因此，我们证实了 Carlen、Frank 和 Lieb 的猜想。特别是，我们证实了 Audenaert 和 Datta 的较弱猜想，并获得了在完全正迹保留映射下 α-z Renyi 相对熵是单调的 (α, z) 的全范围。我们还给出了许多已知结果的更简单的证明，包括 Ψ p , 0 , 1 / p 对于 0 p 1 的凹度，这是爱泼斯坦首先使用复分析证明的。