Sandwiched Rényi Convergence for Quantum Evolutions

Alexander Müller-Hermes1 and Daniel Stilck Franca2

1Department of Mathematical Sciences, University of Copenhagen, 2100 Copenhagen, Denmark
2Department of Mathematics, Technische Universität München, 85748 Garching, Germany

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We study the speed of convergence of a primitive quantum time evolution towards its fixed point in the distance of sandwiched Rényi divergences. For each of these distance measures the convergence is typically exponentially fast and the best exponent is given by a constant (similar to a logarithmic Sobolev constant) depending only on the generator of the time evolution. We establish relations between these constants and the logarithmic Sobolev constants as well as the spectral gap. An important consequence of these relations is the derivation of mixing time bounds for time evolutions directly from logarithmic Sobolev inequalities without relying on notions like lp-regularity. We also derive strong converse bounds for the classical capacity of a quantum time evolution and apply these to obtain bounds on the classical capacity of some examples, including stabilizer Hamiltonians under thermal noise.

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