[Paper Review] Lieb's simple proof of concavity of Tr A^p K^* B^(1-p) K and remarks on related inequalities

Mary Beth Ruskai|arXiv (Cornell University)|Apr 21, 2004

Analytic Number Theory Research10 references4 citations

TL;DR

This paper presents a simplified, self-contained proof of Lieb's concavity theorem for the trace functional $(A,B)\mapsto \mathrm{Tr}\,A^p K^\dagger B^{1-p}K$ using the maximum modulus principle, establishing joint concavity for $p \in (0,1)$. The proof is elementary and accessible, with implications to quantum entropy inequalities including strong subadditivity and relative entropy convexity.

ABSTRACT

A simple, self-contained proof is presented for the concavity of the map (A,B) --> Tr(A^p K^* B^(1-p) K). The author makes no claim to originality; this note gives Lieb's original argument in its simplest, rather than its most general, form. A sketch of the chain of implications from this result to concavity of A --> Tr e^[K + log(A)] is then presented. An independent elementary proof is given for the joint convexity of the map (A,B,X) --> Tr \int X^* (A+ uI)^{-1} X (B+ uI)^{-1} du which plays a key role in entropy inequalities.

Motivation & Objective

• To provide a clear, elementary proof of the joint concavity of the trace functional $\mathrm{Tr}\,A^p K^\dagger B^{1-p}K$ for $p \in (0,1)$, using only the maximum modulus principle.
• To clarify and correct minor errors and omissions in the original presentation of Lieb's proof, particularly regarding trace inequalities and operator bounds.
• To demonstrate the implications of this concavity result to key quantum entropy inequalities, including strong subadditivity and joint convexity of relative entropy.
• To offer an independent, elementary proof of the joint convexity of the integral trace functional $\mathrm{Tr}\int_0^\infty X^\dagger \frac{1}{A+uI} X \frac{1}{B+uI} du$.
• To make Lieb's original argument more accessible to researchers and students by presenting it in its simplest, most transparent form without generalizations.

Proposed method

• Apply the maximum modulus principle to a complex analytic function $f(z)$ constructed from the trace functional, defined on the strip $0 \leq \mathrm{Re}\,z \leq 1$.
• Define $f_k(z) = \mathrm{Tr}\,A_k^z C^{-z/2} M^\dagger C^{-(1-z)/2} A_k^{1-z} C^{-(1-z)/2} M C^{-z/2}$ with $M = C^{(1-p)/2} K C^{p/2}$, to express the concavity inequality.
• Use the Cauchy-Schwarz inequality for the trace: $|\mathrm{Tr}\,X^\dagger Y| \leq \left(\mathrm{Tr}\,X^\dagger X\right)^{1/2} \left(\mathrm{Tr}\,Y^\dagger Y\right)^{1/2}$, to bound the modulus of $f_k(z)$.
• Establish uniform boundedness of $|f_k(z)|$ on the boundary of the strip ($\mathrm{Re}\,z = 0$ and $\mathrm{Re}\,z = 1$) using trace norms and operator bounds.
• Leverage the fact that the supremum of $|f(z)|$ on the strip is attained on the boundary, and use this to prove the concavity inequality via the maximum modulus principle.
• Provide a corrected and pedagogically clearer version of the proof, addressing issues in trace cyclicity and operator norm estimates, particularly for $z = iy$.

Experimental results

Research questions

• RQ1 How can Lieb's original proof of the joint concavity of $\mathrm{Tr}\,A^p K^\dagger B^{1-p}K$ be simplified and made fully self-contained?
• RQ2 What is the precise role of the maximum modulus principle in establishing concavity of the trace functional?
• RQ3 How does the concavity of $\mathrm{Tr}\,A^p K^\dagger B^{1-p}K$ imply the joint convexity of the relative entropy and strong subadditivity of quantum entropy?
• RQ4 Can the joint convexity of the integral functional $\mathrm{Tr}\int_0^\infty X^\dagger \frac{1}{A+uI} X \frac{1}{B+uI} du$ be proven independently and elementarily?
• RQ5 What are the technical corrections needed in the original proof to ensure rigor in trace and norm estimates, especially for purely imaginary $z$?

Key findings

• The map $(A,B)\mapsto \mathrm{Tr}\,A^p K^\dagger B^{1-p}K$ is jointly concave on the cone of positive semi-definite matrices for $p \in (0,1)$, as proven via the maximum modulus principle.
• The proof is simplified and corrected, resolving issues in trace cyclicity and norm estimates, particularly for $z = iy$ on the boundary of the strip.
• The concavity result implies the joint convexity of relative entropy, as shown by taking the limit $p \to 0$ in the trace functional.
• An independent, elementary proof is given for the joint convexity of the integral functional $\mathrm{Tr}\int_0^\infty X^\dagger \frac{1}{A+uI} X \frac{1}{B+uI} du$, which is key to entropy inequalities.
• The concavity of $A \mapsto \mathrm{Tr}\,e^{K + \log A}$ follows from the main result, providing a short route to strong subadditivity of quantum entropy.
• The homogeneity and superadditivity of the trace functionals are used to derive derivative inequalities, such as $\lim_{x \to 0} \frac{g(A + xB) - g(A)}{x} \leq g(B)$, which are useful in entropy analysis.