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arXiv:2601.06005

Generalized Poincaré inequality for quantum Markov semigroups

Marius Junge, Jia Wang

math.OAmath.PRquant-phSubmitted Jan 9, 2026• Announced Jan 12, 2026• Revised Aug 31, 2026 (v2)

v1 · Jan 9, 2026 v2 · Aug 31, 2026

Abstract

We prove a noncommutative $p$-Poincaré inequality for GNS-detailed-balance quantum Markov semigroups (QMSs) on non-tracial $σ$-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 $σ$-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.

Comments: 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

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We prove a noncommutative

p

-Poincaré inequality for GNS-detailed-balance quantum Markov semigroups (QMSs) on non-tracial

σ

-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

σ

-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.

We generalize the notions of the non-commutative Poincaré and modified log-Sobolev inequalities for primitive quantum Markov semigroups (QMS) to not necessarily primitive ones. These two inequalities provide estimates on the decoherence time of the evolution. More precisely, we focus on an algebraic definition of environment-induced decoherence in open quantum systems which happens to be generic on finite dimensional systems and describes the asymptotic behavior of any QMS. An essential tool in our analysis is the explicit structure of the decoherence-free algebra generated by the QMS, a central object in the study of passive quantum error correction schemes. The Poincaré constant corresponds to the spectral gap of the QMS, which implies its positivity, while we prove that the modified log-Sobolev constant is positive under the

\mathbb L_1

-regularity of the Dirichlet form, a condition that also appears in the primitive case. We furthermore prove that strong

σ

-regularity holds for QMS that satisfy a strong form of detailed balance condition for

p\geq1

. The latter condition includes all known cases where this strong regularity was proved. Finally and to emphasize the mathematical interest of this study compared to the classical case, we focus on two truly quantum scenarios, one exhibiting quantum coherence, and the other, quantum correlations.

Abstract We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional matrix algebra satisfies a modified log-Sobolev inequality. In the discrete time setting, we prove that every finite dimensional GNS-symmetric quantum channel satisfies a strong data processing inequality with respect to its decoherence free part. Moreover, we establish the first general approximate tensorization property of the relative entropy. This extends the famous strong subadditivity of the quantum entropy (SSA) of two subsystems to the general setting of two subalgebras. All three results are independent of the size of the environment and hence satisfy the tensorization property. They are obtained via a common, conceptually simple method for proving entropic inequalities via spectral or

L_2

L 2 -estimates. As an application, we combine our results on the modified log-Sobolev inequality and approximate tensorization to derive tight bounds for local generators.

We establish a relation between the exponential decay rates of quantum Markov semigroups with respect to different inner products. More precisely, it was conjectured by Fagnola, Poletti, Sasso and Umanità that for a Gaussian quantum Markov semigroup, the exponential decay rate with respect to the KMS inner product is bounded below by the exponential decay rate for the GNS inner product. We show that this is indeed the case and not limited to Gaussian quantum Markov semigroups, but holds for quantum Markov semigroups with a faithful normal invariant state on arbitrary von Neumann algebras. Additionally, the KMS inner product can be replaced by a whole class of inner products induced by operator monotone functions.

We investigate the mixing properties of primitive Markovian Lindblad dynamics (i.e., quantum Markov semigroups), where the detailed balance is disrupted by a coherent drift term. It is known that the sharp

L^2

-exponential convergence rate of Lindblad dynamics is determined by the spectral gap of the generator. We show that incorporating a Hamiltonian component into a detailed balanced Lindbladian can generically enhance its spectral gap, thereby accelerating the mixing. In addition, we analyze the asymptotic behavior of the spectral gap for Lindblad dynamics with a large coherent contribution. However, estimating the spectral gap, particularly for a non-detailed balanced Lindbladian, presents a significant challenge. In the case of hypocoercive Lindblad dynamics, we extend the variational framework originally developed for underdamped Langevin dynamics to derive fully explicit and constructive exponential decay estimates for convergence in the noncommutative

L^2

-norm. This analysis relies on establishing a quantum analog of space-time Poincaré inequality. Furthermore, we provide several examples with connections to quantum noise and quantum Gibbs samplers as applications of our theoretical results.