  Quantum information quantities play a substantial role in characterizing
operational quantities in various quantum information-theoretic problems. We
consider numerical computation of four quantum information quantities:
Petz-Augustin information, sandwiched Augustin information, conditional
sandwiched Renyi entropy and sandwiched Renyi information. To compute these
quantities requires minimizing some order-$\alpha$ quantum Renyi divergences
over the set of quantum states. Whereas the optimization problems are obviously
convex, they violate standard bounded gradient/Hessian conditions in
literature, so existing convex optimization methods and their convergence
guarantees do not directly apply. In this paper, we propose a new class of
convex optimization methods called mirror descent with the Polyak step size. We
prove their convergence under a weak condition, showing that they provably
converge for minimizing quantum Renyi divergences. Numerical experiment results
show that entropic mirror descent with the Polyak step size converges fast in
minimizing quantum Renyi divergences.
