Generalized Poincar\'e inequality for quantum Markov semigroups

Marius Junge; Jia Wang

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arXiv · 2601.06005

Generalized Poincar\'e inequality for quantum Markov semigroups

Marius Junge·Jia Wang

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Abstract

We prove a noncommutative $p$-Poincar\'e 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.

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Marius Junge, Jia Wang. 2026-01-09. Generalized Poincar\'e inequality for quantum Markov semigroups. https://arxiv.org/abs/2601.06005

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