Sandwiched R\'enyi Convergence for Quantum Evolutions

Understand

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.