Generalized Poincaré inequality for quantum Markov semigroups

• URL: http://arxiv.org/abs/2601.06005v1
• Date: Fri, 09 Jan 2026 18:41:12 GMT
• Title: Generalized Poincaré inequality for quantum Markov semigroups
• Authors: Marius Junge, Jia Wang,
• Abstract summary: We prove a noncommutative $(p,p)$-Poincaré inequality for trace-symmetric quantum Markov semigroups on tracial von Neumann algebras.
• Score: 5.0858152916077595
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We prove a noncommutative $(p,p)$-Poincaré 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 $σ$-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.

Related papers

• Exponential Relative Entropy Decay Along Quantum Markov Semigroups [0.0]
  We establish the equivalence between exponential decay of the relative entropy along a quantum Markov semigroup and the modified logarithmic Sobolev inequality for general von Neumann algebras.<n>We extend an intertwining criterion for the modified logarithmic Sobolev inequality to GNS-symmetric quantum Markov semigroups on infinite-dimensional von Neumann algebras.
  arXiv  Detail & Related papers  (2025-05-12T13:29:34Z)
• Quantum and classical low-degree learning via a dimension-free Remez inequality [52.12931955662553]
  We show a new way to relate functions on the hypergrid to their harmonic extensions over the polytorus. We show the supremum of a function $f$ over products of the cyclic group $exp(2pi i k/K)_k=1K$. We extend to new spaces a recent line of work citeEI22, CHP, VZ22 that gave similarly efficient methods for learning low-degrees on hypercubes and observables on qubits.
  arXiv  Detail & Related papers  (2023-01-04T04:15:40Z)
• Relative entropy decay and complete positivity mixing time [11.225649178057697]
  We prove that the complete modified logarithmic Sobolev constant of a quantum Markov semigroup is bounded by the inverse of its complete positivity mixing time. Our results apply to GNS-symmetric semigroups on general von Neumanns.
  arXiv  Detail & Related papers  (2022-09-22T17:40:22Z)
• Quantum teleportation in the commuting operator framework [63.69764116066747]
  We present unbiased teleportation schemes for relative commutants $N'cap M$ of a large class of finite-index inclusions $Nsubseteq M$ of tracial von Neumann algebras. We show that any tight teleportation scheme for $N$ necessarily arises from an orthonormal unitary Pimsner-Popa basis of $M_n(mathbbC)$ over $N'$.
  arXiv  Detail & Related papers  (2022-08-02T00:20:46Z)
• Annihilating Entanglement Between Cones [77.34726150561087]
  We show that Lorentz cones are the only cones with a symmetric base for which a certain stronger version of the resilience property is satisfied. Our proof exploits the symmetries of the Lorentz cones and applies two constructions resembling protocols for entanglement distillation.
  arXiv  Detail & Related papers  (2021-10-22T15:02:39Z)
• Optimal policy evaluation using kernel-based temporal difference methods [78.83926562536791]
  We use kernel Hilbert spaces for estimating the value function of an infinite-horizon discounted Markov reward process. We derive a non-asymptotic upper bound on the error with explicit dependence on the eigenvalues of the associated kernel operator. We prove minimax lower bounds over sub-classes of MRPs.
  arXiv  Detail & Related papers  (2021-09-24T14:48:20Z)
• Spectral clustering under degree heterogeneity: a case for the random walk Laplacian [83.79286663107845]
  This paper shows that graph spectral embedding using the random walk Laplacian produces vector representations which are completely corrected for node degree. In the special case of a degree-corrected block model, the embedding concentrates about K distinct points, representing communities.
  arXiv  Detail & Related papers  (2021-05-03T16:36:27Z)
• Some Hoeffding- and Bernstein-type Concentration Inequalities [47.24550702683417]
  We prove concentration inequalities for functions of independent random variables under sub-gaussian and sub-exponential conditions. The utility of the inequalities is demonstrated by an extension of the now classical method of Rademacher complexities to Lipschitz function classes and unbounded sub-exponential distribution.
  arXiv  Detail & Related papers  (2021-02-11T23:09:13Z)
• Concentration inequality for U-statistics of order two for uniformly ergodic Markov chains [0.0]
  We prove a concentration inequality for U-statistics of order two for uniformly ergodic Markov chains. We show that we can recover the convergence rate of Arcones and Gin'e who proved a concentration result for U-statistics of independent random variables and canonical kernels.
  arXiv  Detail & Related papers  (2020-11-20T15:14:34Z)
• Multivariate Trace Inequalities, p-Fidelity, and Universal Recovery Beyond Tracial Settings [4.56877715768796]
  We show that the physics of quantum field theory and holography motivate entropy inequalities in type III von Neumann algebras that lack a semifinite trace. The Haagerup and Kosaki $L_p$ spaces enable re-expressing trace inequalities in non-tracial von Neumann algebras.
  arXiv  Detail & Related papers  (2020-09-24T21:05:13Z)
• The semiring of dichotomies and asymptotic relative submajorization [0.0]
  We study quantum dichotomies and the resource theory of asymmetric distinguishability using a generalization of Strassen's theorem on preordered semirings. We find that an variant of relative submajorization, defined on unnormalized dichotomies, is characterized by real-valued monotones that are multiplicative under the tensor product and additive under the direct sum.
  arXiv  Detail & Related papers  (2020-04-22T14:13:26Z)
• Joint measurability meets Birkhoff-von Neumann's theorem [77.34726150561087]
  We prove that joint measurability arises as a mathematical feature of DNTs in this context, needed to establish a characterisation similar to Birkhoff-von Neumann's. We also show that DNTs emerge naturally from a particular instance of a joint measurability problem, remarking its relevance in general operator theory.
  arXiv  Detail & Related papers  (2018-09-19T18:57:45Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.