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This paper advances the mathematical foundations of quantum information theory by establishing generalized Poincaré inequalities for non-tracial von Neumann algebras, a critical step toward understanding convergence rates in complex quantum systems. By extending semi-commutative results to the noncommutative regime using spectral gap assumptions, the work provides rigorous tools for estimating concentration inequalities and Lipschitz diameters essential for analyzing quantum Markov processes.
Generalized Poincar\'e inequality for quantum Markov semigroups
arXiv:2601.06005v2 Announce Type: replace-cross 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.
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
Operator Algebras (math.OA); Probability (math.PR); Quantum Physics (quant-ph)
arXiv:2601.06005 [math.OA]
(or arXiv:2601.06005v2 [math.OA] for this version)
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