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arXiv:2610.03683

Modified logarithmic Sobolev inequality for 1D non-commuting Hamiltonians

Ángela Capel, David Pérez-García, Matteo Scandi

quant-phcond-mat.stat-mechmath-phSubmitted Oct 2, 2026• Announced Oct 5, 2026• 0 citations
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Abstract

Given a finite-range, non-commuting Hamiltonian on a 1D chain of finite spins, we prove a modified logarithmic Sobolev inequality with constant $Ω(1/\log n)$ for the heat-bath dynamics, a quasi-local Gibbs sampler, and for the regularised heat-bath dynamics. This implies rapid mixing of both samplers towards the Gibbs state in relative entropy and trace norm, with mixing times of order $\mathcal{O}(\log n\cdot \log(n/ε))$. The interactions may be spatially inhomogeneous. The proof combines weak quasi-factorisation of relative entropy, decay-of-correlation estimates, a uniform conditional local gap, and a buffered comparison between conditional entropy and quadratic energy. The resulting bounded entropy defect is removed using the global gap. To the best of our knowledge, this is the first proof of MLSI in the non-commuting regime.

Comments
56 pages, 10 figures

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We propose a new method for proving L2L^2L2 and modified logarithmic Sobolev inequalities with constants independent of system size for quantum Gibbs samplers whose Lindblad generators satisfy Kubo-Martin-Schwinger (KMS) detailed balance. We first establish the L2L^2L2 logarithmic Sobolev inequality by transporting it along a temperature-dependent path from the infinite-temperature product state to the target Gibbs state while maintaining a size-independent decay rate for the sandwiched Rényi-2 divergence. A tensor-power argument, combined with the quantum asymptotic equipartition property, then transfers this decay bound to relative entropy, yielding the modified logarithmic Sobolev inequality. This approach provides a general framework for proving size-independent logarithmic Sobolev inequalities for quantum Gibbs samplers. We apply it to two regimes: one-dimensional quantum spin chains at every finite temperature, and finite-range quantum spin systems at sufficiently high temperature in any fixed dimension.
We prove rapid mixing of a quantum Gibbs sampler for general one-dimensional non-commuting local spin systems at every fixed positive temperature. In particular, for a system of nnn sites, we show that the Kubo--Martin--Schwinger (KMS) detailed-balanced Gibbs sampler of~\cite{CKG2023} converges from any initial state to trace-norm error εεε in time O(log⁡4(n)+log⁡(1/ε))O(\log^4(n)+\log(1/ε))O(log4(n)+log(1/ε)). This improves the previous O(n+log⁡(1/ε))O(n+\log(1/ε))O(n+log(1/ε)) bound and establishes rapid mixing beyond high-temperature and weak-interaction regimes. Our proof explains why the slowest decay rate, captured by the spectral gap, can substantially underestimate the speed of equilibration. The key insight is that a state far from equilibrium must have a substantial component outside the span of low-weight Pauli operators, where stronger dissipation estimates apply. This yields two stages of relaxation: a double-exponential convergence from a bad initial state, followed by exponential convergence near equilibrium. We turn this mechanism into a framework for proving rapid mixing through spectral analysis beyond the gap. Our decay estimate is equivalent to exponential convergence in sandwiched Rényi-2 divergence. Together with the spectral gap, it also implies a modified logarithmic Sobolev inequality with rate Ω(log⁡−4(n))Ω(\log^{-4}(n))Ω(log−4(n)).
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.
We prove that a quasi-local quantum Gibbs sampler rapidly mixes to the Gibbs state of any 1D finite-range Hamiltonian at all finite temperatures in O(log⁡(n/ε))\mathcal{O}\left(\log (n/\varepsilon)\right)O(log(n/ε)) time. Each update traces out a contiguous block of constant length and reconstructs it using the Petz recovery map, following early proposals for quantum heat-bath Gibbs samplers by Kastoryano and Brandão~\cite{kastoryano2016quantum}. Our proof directly bounds the distance between the updated state and the target Gibbs state using quantum Wasserstein distance of order 111. We further provide O(polylog(n/ε))\mathcal{O}(\text{polylog}(n/\varepsilon))O(polylog(n/ε)) depth of quantum circuits to prepare these Gibbs states. We hope these techniques will facilitate the analysis and design of other Gibbs samplers.