Sandwiched R\'enyi Convergence for Quantum Evolutions

TL;DR

This paper investigates the exponential convergence rates of quantum evolutions towards their fixed points using sandwiched Rényi divergences, establishing relations with spectral properties and deriving bounds on classical capacities.

Contribution

It introduces a framework connecting sandwiched Rényi divergence convergence rates with spectral and Sobolev constants, enabling direct mixing time bounds and capacity estimates.

Findings

Convergence is typically exponential with rate constants depending on the generator.

Relations between divergence constants, Sobolev constants, and spectral gap are established.

Derived bounds on classical capacity for specific quantum systems, including stabilizer Hamiltonians.

Abstract

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.