Generalized Poincaré inequality for quantum Markov semigroups

Abstract

We prove a noncommutative $p$-Poincaré inequality for GNS-detailed-balance quantum Markov semigroups (QMSs) on non-tracial $\sigma$-finite von Neumann algebras, assuming only the existence of a spectral gap. Extending the semi-commutative results of Huang and Tropp, we first establish the inequality for tracial von Neumann algebras. We use Markov dilations to obtain chain-rule estimates for Dirichlet forms and amalgamated free products to define an appropriate noncommutative derivation. We then extend the result to QMSs satisfying GNS detailed balance on non-tracial $\sigma$-finite von Neumann algebras, using Haagerup's reduction and Kosaki's interpolation theorem. As applications, we recover sub-exponential concentration inequalities and estimate the Lipschitz and completely bounded Lipschitz diameters of the QMSs.

Added a completely bounded Lipschitz-diameter estimate; Updated the proof of the sub-exponential concentration inequalities in Section 5.2; Reframed the abstract and introduction to emphasize the non-tracial results; Clarified the domain and approximation convention for the $L^p$ gradient seminorm