Ruminations on Matrix Convexity and the Strong Subadditivity of Quantum Entropy

TL;DR

This paper introduces a new method combining second derivative tests and resolvent calculus to analyze convex matrix functions, with applications to quantum entropy and its fundamental properties.

Contribution

It presents a novel approach for studying convex matrix functions and applies it to key theorems like the strong subadditivity of quantum entropy.

Findings

New convexity analysis tool for matrix functions

Application to Lieb-Ruskai proof of quantum entropy subadditivity

Enhanced understanding of convexity principles in quantum information

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.