Yet Another Proof of the Joint Convexity of Relative Entropy

Abstract

The joint convexity of the map $(X,A)\mapsto X^{\ast }{A}^{-1}X$, an integral representation of operator convex functions, and an observation of Ando are used to obtain a simple proof of both the joint convexity of relative entropy and a trace convexity result of Lieb. The latter was the key ingredient in the original proof of the strong subadditivity of quantum entropy.

arXiv commentAdded dedication to Derek W. Robinson. Added proof of the montonicity of relative entropy under partial traces and strong subadditivity of quantum entropy to v2. Added to v3, a section on generalizations of relative entropy and a remark (not in LMP version) proving the joint convexity of relative entropy using an integral representation for log x instead of operator convex functions

References in corpus (8)

• Monotone Riemannian Metrics and Relative Entropy on Non-Commutative Probability Spaces
• A Matrix Convexity Approach to Some Celebrated Quantum Inequalities
• A Unified Treatment of Convexity of Relative Entropy and Related Trace Functions, with Conditions for Equality
• Another Short and Elementary Proof of Strong Subadditivity of Quantum Entropy
• Families of completely positive maps associated with monotone metrics
• Contraction coefficients for noisy quantum channels
• Operator extension of strong subadditivity of entropy
• Remarks on on Kim's Strong Subadditivity Matrix Inequality: Extensions and Equality Conditions

Cited by in corpus (2)

• Mpemba effects in open nonequilibrium quantum systems
• Log-majorizations between quasi-geometric type means for matrices