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

H. Zhang, Advances in Mathematics (n.d.).

Journal Article | Submitted | English
Author
Department
Abstract
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.
Publishing Year
Date Published
2020-02-17
Journal Title
IST-REx-ID

### Cite this

Zhang H. From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture. Advances in Mathematics. doi:10.1016/j.aim.2020.107053
Zhang, H. (n.d.). From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture. Advances in Mathematics. https://doi.org/10.1016/j.aim.2020.107053
Zhang, Haonan. “From Wigner-Yanase-Dyson Conjecture to Carlen-Frank-Lieb Conjecture.” Advances in Mathematics, n.d. https://doi.org/10.1016/j.aim.2020.107053.
H. Zhang, “From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture,” Advances in Mathematics.
Zhang H. From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture. Advances in Mathematics.
Zhang, Haonan. “From Wigner-Yanase-Dyson Conjecture to Carlen-Frank-Lieb Conjecture.” Advances in Mathematics, doi:10.1016/j.aim.2020.107053.
All files available under the following license(s):
This Item is protected by copyright and/or related rights. [...]

Link(s) to Main File(s)
Access Level
Open Access

### Export

Marked Publications

Open Data IST Research Explorer

arXiv 1811.01205