[Paper Review] Optimal 2-uniform convexity of Schatten classes revisited

Haonan Zhang|arXiv (Cornell University)|Oct 31, 2020

Mathematical Inequalities and Applications7 references4 citations

TL;DR

This paper revisits the optimal 2-uniform convexity of Schatten classes $S_p$ for $1 < p \leq 2$ using multiple operator integrals and generalized monotone metrics from quantum information theory. It provides a new proof of the inequality $\|A+B\|_p^2 + \|A-B\|_p^2 \geq 2\|A\|_p^2 + 2(p-1)\|B\|_p^2$, recovering the sharp constant $(p-1)^{-1/2}$ via operator-theoretic techniques and conditional expectations, with applications to noncommutative $L_p$-spaces and hypercontractivity.

ABSTRACT

The optimal 2-uniform convexity of Schatten classes $S_p, 1

Motivation & Objective

• To re-derive the optimal 2-uniform convexity constant for Schatten classes $S_p$ with $1 < p \leq 2$ using modern operator-theoretic tools.
• To establish a new proof of the inequality $\|A+B\|_p^2 + \|A-B\|_p^2 \geq 2\|A\|_p^2 + 2(p-1)\|B\|_p^2$ via multiple operator integrals and generalized monotone metrics.
• To extend the applicability of the result to noncommutative $L_p$-spaces by leveraging techniques from quantum information theory.
• To provide a detailed proof of the monotonicity property of trace functionals involving $f_{\alpha}^{[1]}$-type kernels, filling a gap in prior work.

Proposed method

• Uses the second derivative of the trace functional $\operatorname{Tr}(f_p(A + tB))$ at $t=0$ via multiple operator integrals, expressed as a triple sum over spectral projections of $A$.
• Applies the formula $\left.\frac{d^2}{dt^2}\right|_{t=0} \operatorname{Tr}(f_p(A + tB)) = 2\sum_{i,j,k} f_p^{[2]}(\lambda_i, \lambda_j, \lambda_k) E^A_i B E^A_j B E^A_k$ for self-adjoint $A,B$.
• Employs the generalized monotonicity property of the functional $\langle X, Q_{f_{\alpha}^{[1]}}^{A,B}(X) \rangle$ under unital completely positive trace-preserving maps $\beta$, proven via integral representations of $x^{\alpha-1}$.
• Uses the conditional expectation $\mathcal{E}$ to reduce the problem to commuting operators, enabling the use of standard Hölder and trace inequalities.
• Relies on the integral representation $x^{\alpha-1} = \frac{\pi}{\sin(\alpha\pi)} \int_0^\infty \frac{t^{\alpha-1}}{t+x} dt$ to express $f_{\alpha}^{[1]}$ as an integral of $g_t^{[1]}(x,y) = \log(x+t) - \log(y+t)$.
• Establishes the key inequality $\beta^* Q_{g_t^{[1]}}^{\beta(A),\beta(B)} \beta \leq Q_{g_t^{[1]}}^{A,B}$ using known results on operator monotonicity of $h(x) = \frac{x-1}{\log x}$.

Experimental results

Research questions

• RQ1 Can the optimal 2-uniform convexity constant for Schatten classes $S_p$, $1 < p \leq 2$, be re-derived using multiple operator integrals and quantum information-theoretic tools?
• RQ2 What is the role of generalized monotone metrics in proving trace inequalities for noncommutative $L_p$-spaces?
• RQ3 How can the second derivative of $\operatorname{Tr}(f_p(A + tB))$ be expressed and bounded using spectral projections and divided differences?
• RQ4 Does the monotonicity of the functional $\langle X, Q_{f_{\alpha}^{[1]}}^{A,B}(X) \rangle$ under unital completely positive maps imply stronger convexity or smoothness properties?
• RQ5 Can the proof of Ball–Carlen–Lieb’s inequality be made more transparent by reducing to commuting operators via conditional expectations?

Key findings

• The paper establishes the inequality $\|A+B\|_p^2 + \|A-B\|_p^2 \geq 2\|A\|_p^2 + 2(p-1)\|B\|_p^2$ for all $1 < p \leq 2$ and all matrices $A,B$, recovering the sharp constant $(p-1)^{-1/2}$.
• The second derivative of $\operatorname{Tr}(f_p(A + tB))$ at $t=0$ is bounded below by $p(p-1)\|A\|_p^{p-2}\|B\|_p^2$ when $A$ is positive definite.
• The use of conditional expectations $\mathcal{E}$ ensures $\|\mathcal{E}(A)\|_p \leq \|A\|_p$, which is crucial for preserving the lower bound in the trace inequality.
• The monotonicity property $\left\langle \beta(X), Q_{f_{\alpha}^{[1]}}^{\beta(A),\beta(B)}(\beta(X)) \right\rangle \leq \left\langle X, Q_{f_{\alpha}^{[1]}}^{A,B}(X) \right\rangle$ holds for all unital completely positive trace-preserving maps $\beta$, with a complete proof provided in Appendix A.
• The function $h(x) = \frac{x-1}{\log x}$ is operator monotone, which underpins the monotonicity of the metric $Q_{g^{[1]}}^{A,B}$ and enables the key inequality $\beta^* Q_{g_t^{[1]}}^{\beta(A),\beta(B)} \beta \leq Q_{g_t^{[1]}}^{A,B}$.
• The proof technique via multiple operator integrals and integral representations of $x^{\alpha-1}$ provides a transparent and generalizable framework for proving convexity and smoothness inequalities in noncommutative $L_p$-spaces.