Optimal self-concordant barriers for quantum relative entropies

Abstract

Quantum relative entropies are jointly convex functions of two positive definite matrices that generalize the Kullback-Leibler divergence and arise naturally in quantum information theory. In this paper, we prove self-concordance of natural barrier functions for the epigraphs of various quantum relative entropies and divergences. Furthermore we show that these barriers have optimal barrier parameter. These barriers allow convex optimization problems involving quantum relative entropies to be directly solved using interior point methods for non-symmetric cones, avoiding the approximations and lifting techniques used in previous approaches. More generally, we establish the self-concordance of natural barriers for various closed convex cones related to the noncommutative perspectives of operator concave functions, and show that the resulting barrier parameters are optimal.