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Quantum relative entropies are jointly convex functions of two positivedefinite matrices that generalize the Kullback-Leibler divergence and arisenaturally in quantum information theory. In this paper, we proveself-concordance of natural barrier functions for the epigraphs of variousquantum relative entropies and divergences. Furthermore we show that thesebarriers have optimal barrier parameter. These barriers allow convexoptimization problems involving quantum relative entropies to be directlysolved using interior point methods for non-symmetric cones, avoiding theapproximations and lifting techniques used in previous approaches. Moregenerally, we establish the self-concordance of natural barriers for variousclosed convex cones related to the noncommutative perspectives of operatorconcave functions, and show that the resulting barrier parameters are optimal.

Optimal self-concordant barriers for quantum relative entropies