The modified logarithmic Sobolev inequality for quantum spin systems: classical and commuting nearest neighbour interactions

Abstract

Given a uniform, frustration-free family of local Lindbladians defined on a quantum lattice spin system in any spatial dimension, we prove a strong exponential convergence in relative entropy of the system to equilibrium under a condition of spatial mixing of the stationary Gibbs states and the rapid decay of the relative entropy on finite-size blocks. Our result leads to the first examples of the positivity of the modified logarithmic Sobolev inequality for quantum lattice spin systems independently of the system size. Moreover, we show that our notion of spatial mixing is a consequence of the recent quantum generalization of Dobrushin and Shlosman's complete analyticity of the free-energy at equilibrium. The latter typically holds above a critical temperature Tc. Our results have wide-ranging applications in quantum information. As an illustration, we discuss four of them: first, using techniques of quantum optimal transport, we show that a quantum annealer subject to a finite range classical noise will output an energy close to that of the fixed point after constant annealing time. Second, we prove Gaussian concentration inequalities for Lipschitz observables and show that the eigenstate thermalization hypothesis holds for certain high-temperture Gibbs states. Third, we prove a finite blocklength refinement of the quantum Stein lemma for the task of asymmetric discrimination of two Gibbs states of commuting Hamiltonians satisfying our conditions. Fourth, in the same setting, our results imply the existence of a local quantum circuit of logarithmic depth to prepare Gibbs states of a class of commuting Hamiltonians.

69 pages, 11 figures. Added one application, corrected typos and updated results in light of recent advances in the literature

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Cited by in corpus (13)

• Rapid thermalization of spin chain commuting Hamiltonians
• Complete entropic inequalities for quantum Markov chains
• Exponential decay of mutual information for Gibbs states of local Hamiltonians
• Entropy decay for Davies semigroups of a one dimensional quantum lattice
• Quantum concentration inequalities
• Szegedy Walk Unitaries for Quantum Maps
• Continuity of quantum entropic quantities via almost convexity
• Deviation bounds and concentration inequalities for quantum noises
• Weak quasi-factorization for the Belavkin-Staszewski relative entropy
• The topology of data hides in quantum thermal states
• Ricci curvature of quantum channels on non-commutative transportation metric spaces
• Limitations of Noisy Quantum Devices in Computational and Entangling Power
• Geometric Approach Towards Complete Logarithmic Sobolev Inequalities