Finite-time quantum equilibration from local correlations
Abstract
An isolated quantum system evolves reversibly, yet a small subsystem can lose nearly all information about its initial state. Conserved quantities may retain some of this information, depending on how strongly they overlap with local observables. Here we consider a subsystem in an arbitrary initial state coupled to a maximally mixed bath. We show that the worst-input mean-square equilibration error, maximised over window origins, is bounded from above and below by the mean square of a complete local autocorrelation, with the stationary part subtracted. The two mean-square quantities are linearly comparable. A correlation record over a finite interval therefore controls the equilibration error in every interval of the same length, and the result extends to nonnegative observation weights with nonnegative Fourier transform. The result allows arbitrary degeneracies of energies and gaps. We apply it to two static Hamiltonian families. For fixed spin subsystems in the one-particle sector of a bounded-range periodic clock, fixed-accuracy thermalisation is guaranteed on windows that grow only slightly faster than linearly with system size, despite exponentially small positive energies. A random commuting model with logarithmic interaction range retains an extensive set of conserved observables. Its expected mean-square error is bounded by a finite-size term and an inverse-time term. The inverse-time dependence is unavoidable because of the initial transient. These examples show how local relaxation can coexist with exact conservation laws and a finely spaced spectrum.
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BibTeX
@misc{cwiklinski2026finitetime,
title = {{Finite-time quantum equilibration from local correlations}},
author = {Piotr Ćwikliński},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16351},
eprint = {2610.16351},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16351},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Finite-time quantum equilibration from local correlations
AU - Piotr Ćwikliński
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.16351
UR - https://arxiv.org/abs/2610.16351
AB - An isolated quantum system evolves reversibly, yet a small subsystem can lose nearly all information about its initial state. Conserved quantities may retain some of this information, depending on how strongly they overlap with local observables. Here we consider a subsystem in an arbitrary initial state coupled to a maximally mixed bath. We show that the worst-input mean-square equilibration error, maximised over window origins, is bounded from above and below by the mean square of a complete local autocorrelation, with the stationary part subtracted. The two mean-square quantities are linearly comparable. A correlation record over a finite interval therefore controls the equilibration error in every interval of the same length, and the result extends to nonnegative observation weights with nonnegative Fourier transform. The result allows arbitrary degeneracies of energies and gaps. We apply it to two static Hamiltonian families. For fixed spin subsystems in the one-particle sector of a bounded-range periodic clock, fixed-accuracy thermalisation is guaranteed on windows that grow only slightly faster than linearly with system size, despite exponentially small positive energies. A random commuting model with logarithmic interaction range retains an extensive set of conserved observables. Its expected mean-square error is bounded by a finite-size term and an inverse-time term. The inverse-time dependence is unavoidable because of the initial transient. These examples show how local relaxation can coexist with exact conservation laws and a finely spaced spectrum.
ER -