Generalized Poincar\'e inequality for quantum Markov semigroups

TL;DR

This paper establishes a generalized noncommutative Poincaré inequality for quantum Markov semigroups, extending classical results to the quantum setting under spectral gap assumptions, with applications to concentration inequalities.

Contribution

It introduces a noncommutative $(p,p)$-Poincaré inequality for quantum Markov semigroups, extending prior semi-commutative results to broader non-tracial von Neumann algebras.

Findings

Proves a noncommutative Poincaré inequality under spectral gap conditions.

Recovers noncommutative Khintchine and concentration inequalities.

Extends results to non-tracial von Neumann algebras with GNS-detailed balance.

Abstract

We prove a noncommutative $(p,p)$-Poincar\'e inequality for trace-symmetric quantum Markov semigroups on tracial von Neumann algebras, assuming only the existence of a spectral gap. Extending semi-commutative results of Huang and Tropp, our argument uses Markov dilations to obtain chain-rule estimates for Dirichlet forms and employs amalgamated free products to define an appropriate noncommutative derivation. We further generalize the argument to non-tracial $\sigma$-finite von Neumann algebras under the weaker assumption of GNS-detailed balance, using Haagerup's reduction and Kosaki's interpolation theorem. As applications, we recover noncommutative Khintchine and sub-exponential concentration inequalities.