  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.
