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Ruminations on Matrix Convexity and the Strong Subadditivity of Quantum Entropy

Michael Aizenman, Giorgio Cipolloni

TL;DR

The paper develops a local second-derivative criterion for matrix convexity and leverages resolvent calculus to obtain joint concavity results for matrix functions, including parallel sums, Lieb concavity, and Kubo–Ando operator means. These tools are then applied to quantum entropy, yielding a streamlined proof of the Lieb–Ruskai concavity of the conditional entropy and the strong subadditivity (SSA) of quantum entropy, via unitary averaging and Jensen’s inequality. The approach unifies matrix-analytic convexity with quantum information, providing concise, resolvent-based proofs that illuminate how operator convexity governs entropy inequalities. The results have both foundational significance in mathematical physics and practical implications for analyses of quantum systems and entropy-related functionals.

Abstract

The familiar second derivative test for convexity, combined with resolvent calculus, is shown to yield a useful tool for the study of convex matrix-valued functions. We demonstrate the applicability of this approach on a number of theorems in this field. These include convexity principles which play an essential role in the Lieb-Ruskai proof of the strong subadditivity of quantum entropy.

Ruminations on Matrix Convexity and the Strong Subadditivity of Quantum Entropy

TL;DR

The paper develops a local second-derivative criterion for matrix convexity and leverages resolvent calculus to obtain joint concavity results for matrix functions, including parallel sums, Lieb concavity, and Kubo–Ando operator means. These tools are then applied to quantum entropy, yielding a streamlined proof of the Lieb–Ruskai concavity of the conditional entropy and the strong subadditivity (SSA) of quantum entropy, via unitary averaging and Jensen’s inequality. The approach unifies matrix-analytic convexity with quantum information, providing concise, resolvent-based proofs that illuminate how operator convexity governs entropy inequalities. The results have both foundational significance in mathematical physics and practical implications for analyses of quantum systems and entropy-related functionals.

Abstract

The familiar second derivative test for convexity, combined with resolvent calculus, is shown to yield a useful tool for the study of convex matrix-valued functions. We demonstrate the applicability of this approach on a number of theorems in this field. These include convexity principles which play an essential role in the Lieb-Ruskai proof of the strong subadditivity of quantum entropy.
Paper Structure (13 sections, 10 theorems, 56 equations)

This paper contains 13 sections, 10 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. A bit of background
  3. Outline of the paper's contents
  4. Local test of convexity
  5. Computing with resolvents
  6. Resolvent based proofs of joint concavity in multiple matrix variables
  7. Joint concavity of the parallel sums
  8. Lieb concavity
  9. The Kubo-Ando Theorem
  10. Applications to quantum entropy
  11. Basics notions
  12. The Lieb-Ruskai concavity
  13. The strong subadditivity of quantum entropy

Key Result

Proposition 2.1

A sufficient condition for $f:(a,b)\to \mathbb{R}$ to be in $\mathcal{C}_n(a,b)$ is that for any matrix $M\in {\mathcal{B}}_n(a,b)$ and any bounded self adjoint $Q$ of equal rank the matrix valued function $f(M+tQ)$ is twice differentiable at $t=0$ and satisfies in the sense of quadratic forms.

Theorems & Definitions (20)

  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • proof
  • ...and 10 more