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A sharp complete modified logarithmic Sobolev inequality for Hamiltonian dephasing

Christoph Hartmann

Published Sep 25, 2026• 1 citation
Publisher

Abstract

We consider the pure dephasing semigroup on n x n matrices whose generator is one half of the double commutator with a fixed self-adjoint matrix H. Let delta be the smallest distance between two distinct eigenvalues of H. We show that the relative entropy of any state with respect to its pinching onto the commutant of H decays at least like exp(-delta^2 t), and that the same holds, with the same constant, when the system is coupled to an arbitrary finite-dimensional reference system. Equivalently, the modified logarithmic Sobolev constant and its complete version are both equal to delta^2 / 2. This is the spectral gap of the generator and hence the largest possible value. The spectrum of H is arbitrary; it may be degenerate and need not lie in a lattice. The proof adapts the argument of L. Zhao for the heat semigroup on the circle (arXiv:2608.23482). For lattice spectra, transference from the circle gives a weaker rate unless the lattice spacing equals delta. We also note that equality of this constant with the spectral gap can fail for two commuting dephasing operators. Status: checked by three AI-based passes and numerically, but not by a human referee. Novelty has not yet been independently confirmed.

DOI
10.5281/zenodo.22967383

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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.