1. On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems
- Author
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Ivan Bardet, Cambyse Rouzé, David Pérez-García, Angelo Lucia, Angela Capel, Cryptologie symétrique, cryptologie fondée sur les codes et information quantique (COSMIQ), Inria de Paris, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Instituto de Ciencias Matemàticas [Madrid] (ICMAT), Universidad Carlos III de Madrid [Madrid] (UC3M)-Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM)-Universidad Autónoma de Madrid (UAM)-Consejo Superior de Investigaciones Científicas [Madrid] (CSIC), California Institute of Technology (CALTECH), Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM), Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), Ivan Bardet is supported by French A.N.R. grant: ANR-14-CE25-0003 'StoQ'. Angela Capel is partially supported by a La Caixa-Severo Ochoa grant (ICMAT Severo Ochoa project SEV-2011-0087, MINECO) and Angela Capel and David Perez Garcia acknowledge support from MINECO (grant MTM2017-88385-P) and from Comunidad de Madrid (grantQUITEMAD-CM, ref. P2018/TCS-4342). Angelo Lucia acknowledges support from the Walter Burke Institute for Theoretical Physics in the form of the Sherman Fairchild Fellowship as well as support from the Institute for Quantum Information and Matter (IQIM), an NSF Physics Frontiers Center (NFS Grant PHY-1733907), and from the BBVA Fundation.This project has received funding from the European Research Council (ERC) under theEuropean Union’s Horizon 2020 research and innovation programme (grant agreementNo 648913). Cambyse Rouzé acknowledges financial support from the TUM university Foundation Fellowship and by the DFG cluster of excellence 2111 (Munich Center for Quantum Science and Technology)., Universidad Autonoma de Madrid (UAM)-Consejo Superior de Investigaciones Científicas [Madrid] (CSIC)-Universidad Complutense de Madrid = Complutense University of Madrid [Madrid] (UCM)-Universidad Carlos III de Madrid [Madrid] (UC3M), Security, Cryptology and Transmissions (SECRET), Instituto de Ciencias Matemàticas [Madrid], Consejo Superior de Investigaciones Científicas [Spain] (CSIC), Universidad Complutense de Madrid [Madrid] (UCM), and Technische Universität München [München] (TUM)
- Subjects
[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,FOS: Physical sciences ,Gibbs state ,[MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA] ,01 natural sciences ,Entropy (classical thermodynamics) ,symbols.namesake ,[PHYS.QPHY]Physics [physics]/Quantum Physics [quant-ph] ,0103 physical sciences ,0101 mathematics ,Gibbs measure ,Mixing (physics) ,Condensed Matter - Statistical Mechanics ,Mathematical Physics ,Mathematical physics ,Mathematics ,Quantum Physics ,Statistical Mechanics (cond-mat.stat-mech) ,010102 general mathematics ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,Sobolev space ,symbols ,Dissipative system ,010307 mathematical physics ,Constant (mathematics) ,Quantum Physics (quant-ph) ,Hamiltonian (control theory) - Abstract
The mixing time of Markovian dissipative evolutions of open quantum many-body systems can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure, via quasi-factorization results for the entropy. Inspired by the classical case, we present a strategy to derive the positivity of the modified logarithmic Sobolev constant associated to the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. In particular we show that for the heat-bath dynamics for 1D systems, the modified logarithmic Sobolev constant is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi-factorization of the relative entropy., 28 pages, 4 figures. Included some additional comments and updated results in light of recent advances in the literature
- Published
- 2021
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