Exponential Relative Entropy Decay Along Quantum Markov Semigroups

• URL: http://arxiv.org/abs/2505.07549v2
• Date: Thu, 26 Jun 2025 14:00:49 GMT
• Title: Exponential Relative Entropy Decay Along Quantum Markov Semigroups
• Authors: Melchior Wirth,
• Abstract summary: 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.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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. We also extend an intertwining criterion for the modified logarithmic Sobolev inequality to GNS-symmetric quantum Markov semigroups on infinite-dimensional von Neumann algebras.

Related papers

• Quantum Random Walks and Quantum Oscillator in an Infinite-Dimensional Phase Space [45.9982965995401]
  We consider quantum random walks in an infinite-dimensional phase space constructed using Weyl representation of the coordinate and momentum operators. We find conditions for their strong continuity and establish properties of their generators.
  arXiv  Detail & Related papers  (2024-06-15T17:39:32Z)
• 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)
• Thermal equilibrium in Gaussian dynamical semigroups [77.34726150561087]
  We characterize all Gaussian dynamical semigroups in continuous variables quantum systems of n-bosonic modes which have a thermal Gibbs state as a stationary solution. We also show that Alicki's quantum detailed-balance condition, based on a Gelfand-Naimark-Segal inner product, allows the determination of the temperature dependence of the diffusion and dissipation matrices.
  arXiv  Detail & Related papers  (2022-07-11T19:32:17Z)
• Trace- and improved data processing inequalities for von Neumann algebras [0.0]
  We prove a version of the data-processing inequality for the relative entropy for general von Neumann algebras with an explicit lower bound involving the measured relative entropy. The natural applications of our results are in quantum field theory where the von Neumann algebras are known to be of type III.
  arXiv  Detail & Related papers  (2021-02-15T11:36:44Z)
• 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)
• Complete entropic inequalities for quantum Markov chains [17.21921346541951]
  We prove that every GNS-symmetric quantum Markov semigroup on a finite dimensional algebra satisfies a modified log-Sobolev inequality. We also establish the first general approximateization property of relative entropy.
  arXiv  Detail & Related papers  (2021-02-08T11:47:37Z)
• 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)
• Unraveling the topology of dissipative quantum systems [58.720142291102135]
  We discuss topology in dissipative quantum systems from the perspective of quantum trajectories. We show for a broad family of translation-invariant collapse models that the set of dark state-inducing Hamiltonians imposes a nontrivial topological structure on the space of Hamiltonians.
  arXiv  Detail & Related papers  (2020-07-12T11:26:02Z)
• Lagrangian description of Heisenberg and Landau-von Neumann equations of motion [55.41644538483948]
  An explicit Lagrangian description is given for the Heisenberg equation on the algebra of operators of a quantum system, and for the Landau-von Neumann equation on the manifold of quantum states which are isospectral with respect to a fixed reference quantum state.
  arXiv  Detail & Related papers  (2020-05-04T22:46:37Z)
• Approximate tensorization of the relative entropy for noncommuting conditional expectations [3.4376560669160385]
  We derive a new generalisation of the strong subadditivity of the entropy to the setting of general conditional expectations onto finite-dimensional von Neumann algebras. We show the equivalence between conditional expectations arising from Petz recovery maps and those of general Davies semigroups.
  arXiv  Detail & Related papers  (2020-01-22T12:20:53Z)
• 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.