[Paper Review] From Wigner-Yanase-Dyson conjecture to Carlen-Frank-Lieb conjecture (New title)

Haonan Zhang|arXiv (Cornell University)|Nov 3, 2018

Mathematical Inequalities and Applications24 references4 citations

TL;DR

This paper confirms the Carlen-Frank-Lieb conjecture on the joint convexity/concavity of trace functions involving positive definite matrices and an arbitrary invertible matrix K. Using a refined variational method, it establishes the full range of parameters (p, q, s) for which the function $\Psi_{p,q,s}(A,B) = \text{Tr}\left( B^{q/2} K^* A^p K B^{q/2} \right)^s$ is jointly convex or concave, resolving longstanding conjectures in quantum information and mathematical physics, including the monotonicity of $\alpha$-$z$ Rényi relative entropies under quantum channels.

ABSTRACT

In this paper we study the joint convexity/concavity of the trace functions \[ Ψ_{p,q,s}(A,B)= ext{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s,~~p,q,s\in \mathbb{R}, \] 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 $Ψ_{p,q,s}$ 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 $α$-$z$ 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 $Ψ_{p,0,1/p}$ for $0

Motivation & Objective

• To resolve the Carlen-Frank-Lieb conjecture on the joint convexity/concavity of the trace function $\Psi_{p,q,s}(A,B) = \text{Tr}\left( B^{q/2} K^* A^p K B^{q/2} \right)^s$ for all invertible K and positive definite A, B.
• To determine the complete parameter range $(p,q,s) \in \mathbb{R}^3$ for which $\Psi_{p,q,s}$ is jointly convex or concave.
• To confirm a weaker conjecture by Audenaert and Datta on the monotonicity of $\alpha$-$z$ Rényi relative entropies under completely positive trace-preserving maps.
• To provide a unified, variational-method-based proof of known results, including the concavity of $\Psi_{p,0,1/p}$ for $0 < p < 1$, with simpler arguments than prior complex-analytic approaches.

Proposed method

• Develops a variational representation for $\Psi_{p,q,s}$ by reducing the problem to analyzing the joint convexity/concavity of $\Psi_{p,1-p,1}(A,B) = \text{Tr}(K^* A^p K B^{1-p})$ via a novel variational method.
• Applies the variational identity $\text{Tr}(X^* A^p X)^{1/p} = \max_{Z \in \mathcal{P(H)}^\times} \left\{ \frac{1}{p} \text{Tr}|A^{p/2} X Z|^{2} - \frac{1-p}{p} \text{Tr}|Z|^{2/(1-p)} \right\}$ for $0 < p < 1$, and a dual minimization form for $1 < p \leq 2$, to express the trace function as an inf-sup or sup-inf optimization.
• Uses Ando’s convexity theorem and Lieb’s concavity theorem as foundational tools to analyze the convexity/concavity of the resulting variational expressions.
• Applies a duality argument to reduce the general $\Psi_{p,q,s}$ case to the simpler $\Psi_{p,1-p,1}$ case, leveraging symmetry and scaling properties.
• Establishes sharpness of the parameter ranges via counterexamples and known results, particularly Proposition 2.3.
• Provides a systematic framework that unifies and simplifies prior proofs, especially for the concavity of $\Psi_{p,0,1/p}$, replacing complex analysis with real-variable variational techniques.

Experimental results

Research questions

• RQ1 For which full range of parameters $(p,q,s) \in \mathbb{R}^3$ is the trace function $\Psi_{p,q,s}(A,B) = \text{Tr}\left( B^{q/2} K^* A^p K B^{q/2} \right)^s$ jointly convex or concave for all invertible matrices K and positive definite A, B?
• RQ2 Does the Carlen-Frank-Lieb conjecture—that $\Psi_{p,q,s}$ is jointly convex for $1 \leq p \leq 2$, $-1 \leq q < 0$, $(p,q) \neq (1,-1)$, and $s \geq \frac{1}{p+q}$—hold true for all such K?
• RQ3 Is the monotonicity of $\alpha$-$z$ Rényi relative entropies under completely positive trace-preserving maps fully characterized by the parameter range derived from the convexity of $\Psi_{p,q,s}$?
• RQ4 Can the joint concavity of $\Psi_{p,0,1/p}$ for $0 < p < 1$, previously proven via complex analysis, be re-derived using a simpler, real-variable variational method?
• RQ5 What is the precise boundary of the convexity/concavity region for $\Psi_{p,q,s}$, and why are the points $(1,-1)$ and $(-1,1)$ excluded?

Key findings

• The paper fully characterizes the joint convexity and concavity of $\Psi_{p,q,s}$ for all invertible K: it is jointly concave when $0 \leq q \leq p \leq 1$ and $0 < s \leq \frac{1}{p+q}$, jointly convex when $-1 \leq q \leq p \leq 0$ and $s > 0$, and jointly convex when $-1 \leq q \leq 0$, $1 \leq p \leq 2$, $(p,q) \neq (1,-1)$, and $s \geq \frac{1}{p+q}$.
• The Carlen-Frank-Lieb conjecture is confirmed: $\Psi_{p,q,s}$ is jointly convex for $1 \leq p \leq 2$, $-1 \leq q < 0$, $(p,q) \neq (1,-1)$, and $s \geq \frac{1}{p+q}$.
• The full range of $\alpha$-$z$ Rényi relative entropies $D_{\alpha,z}$ that are monotone under completely positive trace-preserving maps is determined: monotonicity holds iff (1) $0 < \alpha < 1$, $z \geq \max\{\alpha, 1-\alpha\}$; (2) $1 < \alpha \leq 2$, $\frac{\alpha}{2} \leq z \leq \alpha$; or (3) $2 \leq \alpha < \infty$, $\alpha - 1 \leq z \leq \alpha$.
• The paper provides a simpler, variational-method-based proof of the concavity of $\Psi_{p,0,1/p}$ for $0 < p < 1$, previously established using complex analysis by Epstein.
• The parameter range for convexity is sharp: no larger region of $s$-values yields joint convexity, as shown by Proposition 2.3.
• The variational method successfully generalizes to multi-operator trace functions, though joint convexity for $n \geq 3$ operators requires additional assumptions due to non-separable terms in the variational identities.