# Sandwiched R\'enyi Convergence for Quantum Evolutions. (arXiv:1607.00041v2 [quant-ph] UPDATED)

We study the speed of convergence of a primitive quantum time evolution

towards its fixed point in the distance of sandwiched R\'enyi 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.