  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.
