A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, with Conditions for Equality

TL;DR

This paper presents a unified framework for understanding the convexity properties of relative entropy and related trace functions, providing new conditions for equality and extending classical inequalities in quantum information theory.

Contribution

It introduces a generalized relative entropy based on Wigner-Yanase-Dyson entropy and derives unified convexity and equality conditions for various trace functions and inequalities.

Findings

Proves convexity of a generalized relative entropy.

Derives conditions for equality in key trace inequalities.

Unifies and extends classical results like strong subadditivity.

Abstract

We introduce a generalization of relative entropy derived from the Wigner-Yanase-Dyson entropy and give a simple, self-contained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for the map (A,B) --> Tr K^* A^p K B^{1-p} Lieb's joint concavity for 0 < p < 1 and Ando's joint convexity for 1 < p < 2. This approach allows us to obtain conditions for equality in these cases, as well as conditions for equality in a number of inequalities which follow from them. These include the monotonicity under partial traces, and some Minkowski type matrix inequalities proved by Lieb and Carlen for mixed (p,q) norms. In all cases the equality conditions are independent of p; for extensions to three spaces they are identical to the conditions for equality in the strong subadditivity of relative entropy.