Relative Entropy Decay via BKM coercivity for Quantum Markov Semigroups

• URL: http://arxiv.org/abs/2609.15902v1
• Date: Mon, 14 Sep 2026 17:19:43 GMT
• Title: Relative Entropy Decay via BKM coercivity for Quantum Markov Semigroups
• Abstract summary: We show that the positivity of the BKM coercivity constant governs the exponential relative entropy decay of the semigroup.<n>We prove that the modified log-Sobolev constant is equivalent to the BKM coercivity constant up to a constant depending on the conditional expectation.
• Score: 12.387975598917158
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We introduce the notion of the BKM coercivity constant and show that the positivity of this constant governs the exponential relative entropy decay of the semigroup. Indeed, we prove that the modified log-Sobolev constant is equivalent to the BKM coercivity constant up to a constant depending on the asymptotic conditional expectation. We also discover that the BKM coercivity constants for matrix amplifications are already attained with a qubit auxiliary system, which also coincides with the GNS spectral gap of the generating Lindbladian. As a corollary, we establish a criterion that the complete modified log-Sobolev inequality holds if and only if the GNS gap of the semigroup is strictly positive. All the discussion above applies to quantum Markov semigroups that admit a faithful asymptotic conditional expectation as long-time equilibration, with no detailed balance condition assumed. As an application, we show the finite-dimensional Chen-Kastoryano-Gilyén Gibbs samplers always satisfy the complete modified log-Sobolev inequality.

Related papers

• Entropy Moduli and BKM Coercivity for Rank-Deficient Non-Commutative Markov Semigroups [0.0]
  entropy-based certification of convergence can exhibit a slowdown near the boundary regime, with a rate of order e-alpha t/sqrt(alpha t)<n>We show that, under explicit local conditions, entropy-based certification of convergence can exhibit a slowdown near the boundary regime, with a rate of order e-alpha t/sqrt(alpha t)<n>The mechanism is identified as a coherence-population effect and disappears for incoherent initial states, where classical Pinsker-type bounds are recovered.
  arXiv  Detail & Related papers  (2026-04-17T18:13:15Z)
• Typical entanglement in anyon chains: Page curves beyond Lie group symmetries [41.99844472131922]
  We study bipartite entanglement statistics in one-dimensional anyon chains constrained by fusion rules of unitary pre-modular categories.<n>Surprisingly, despite the constrained Hilbert space structure, the large $L$ expansion has no universal $O(sqrtL)$ or $O(1)$ symmetry-type corrections.<n> Numerical simulations of the integrable and quantum-chaotic golden chain Hamiltonian show that chaotic mid-spectrum eigenstates match the Haar-random predictions.
  arXiv  Detail & Related papers  (2026-03-26T18:00:03Z)
• Entanglement and Bell inequality violation in vector diboson systems produced in decays of spin-0 particles [44.99833362998488]
  We discuss conserving and the violation of the entanglementMP inequality in a system of two vector bosons produced in the decay of a spin-0 particle.<n>As an exemplary process of this kind we use the decay $Hto ZZ$ with anomalous coupling.
  arXiv  Detail & Related papers  (2024-05-26T11:21:06Z)
• Measurement phase transitions in the no-click limit as quantum phase transitions of a non-hermitean vacuum [77.34726150561087]
  We study phase transitions occurring in the stationary state of the dynamics of integrable many-body non-Hermitian Hamiltonians. We observe that the entanglement phase transitions occurring in the stationary state have the same nature as that occurring in the vacuum of the non-hermitian Hamiltonian.
  arXiv  Detail & Related papers  (2023-01-18T09:26:02Z)
• 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)
• Role of boundary conditions in the full counting statistics of topological defects after crossing a continuous phase transition [62.997667081978825]
  We analyze the role of boundary conditions in the statistics of topological defects. We show that for fast and moderate quenches, the cumulants of the kink number distribution present a universal scaling with the quench rate.
  arXiv  Detail & Related papers  (2022-07-08T09:55:05Z)
• Self-restricting Noise and Exponential Relative Entropy Decay Under Unital Quantum Markov Semigroups [0.8702432681310401]
  We analyze continuous processes that combine dissipative with Hamiltonian time-evolution.<n>We show that when dissipation is much stronger than Hamiltonian time-evolution, exponential decay toward the semigroup's decoherence-free subspace is bounded inversely in the decay rate of the dissipative part alone.
  arXiv  Detail & Related papers  (2022-03-07T22:13:40Z)
• Quantum concentration inequalities [12.56413718364189]
  We establish transportation cost inequalities (TCI) with respect to the quantum Wasserstein distance. We prove Gibbs states of commuting Hamiltonians on arbitrary hypergraphs $H=(V,E)$ satisfy a TCI with constant scaling as $O(|V|)$. We argue that the temperature range for which the TCI holds can be enlarged by relating it to recently established modified logarithmic Sobolev inequalities.
  arXiv  Detail & Related papers  (2021-06-30T05:44:12Z)
• 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)
• 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)

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.