A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II: Convexity and Concavity

TL;DR

This paper proves new convexity inequalities for trace functions of positive operators, extending previous results, and applies these to define operator norms and provide a novel proof of strong subadditivity of quantum entropy.

Contribution

It establishes the joint convexity of a class of trace functionals for a broader parameter range, including q ≥ 1, and introduces an operator L^q(L^p) norm with Minkowski inequality.

Findings

Proved convexity of _{p,q} for 1 q p q 2 and q q 1.

Defined an L^q(L^p) operator norm for 1 q p q 2.

Provided a new proof of strong subadditivity of quantum entropy.

Abstract

We revisit and prove some convexity inequalities for trace functions conjectured in the earlier part I. The main functional considered is \Phi_{p,q}(A_1,A_2,...,A_m) = (trace((\sum_{j=1}^m A_j^p)^{q/p}))^{1/q} for m positive definite operators A_j. In part I we only considered the case q=1 and proved the concavity of \Phi_{p,1} for 0 < p \leq 1 and the convexity for p=2. We conjectured the convexity of \Phi_{p,1} for 1< p < 2. Here we not only settle the unresolved case of joint convexity for 1 \leq p \leq 2, we are also able to include the parameter q\geq 1 and still retain the convexity. Among other things this leads to a definition of an L^q(L^p) norm for operators when 1 \leq p \leq 2 and a Minkowski inequality for operators on a tensor product of three Hilbert spaces -- which leads to another proof of strong subadditivity of entropy. We also prove convexity/concavity properties of some other, related functionals.