On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems

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.

arXiv comment28 pages, 4 figures. Included some additional comments and updated results in light of recent advances in the literature

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• Continuity of quantum entropic quantities via almost convexity
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• Limitations of optimization algorithms on noisy quantum devices
• Approximate tensorization of the relative entropy for noncommuting conditional expectations
• Stability of logarithmic Sobolev inequalities under a noncommutative change of measure
• Rapid thermalization of dissipative many-body dynamics of commuting Hamiltonians
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• Mixing Time of Open Quantum Systems via Hypocoercivity
• Improved Quantum Hypercontractivity Inequality for the Qubit Depolarizing Channel
• Limitations of Noisy Quantum Devices in Computational and Entangling Power
• Strong decay of correlations for Gibbs states in any dimension
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• Efficient and simple Gibbs state preparation of the 2D toric code via duality to classical Ising chains