Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2412.00982

Finite-time quantum equilibration for continuous variables

Alberto Acevedo, Antonio Falcó

Published Dec 1, 2024• 0 citations
arXivHTMLPDF

Abstract

Leveraging the techniques found in the literature on Quantum Equilibration for finite dimensional systems, we develop the theory of Quantum Equilibration for the case of infinite-dimensional systems, particularly the cases where the dynamics-generating Hamiltonians have continuous spectrum. The main goal of this paper will be to propose a framework to extend the results obtained by Short in, where estimates for the equilibration-on-average and effective equilibration for the case of Hamiltonians with continuous spectrum are derived. We will show that in the latter setting, it is compulsory to constrain ourselves to finite time equilibration; we then develop estimates analogous to the main results in the proposed setting.

DOI
10.48550/arxiv.2412.00982

Related papers

Finite-time equilibration bounds for continuous-spectrum quantum systems via spectral coarse graining

Alberto Acevedo, Antonio Falcó

We investigate equilibration-on-average and effective equilibration for quantum systems evolving on infinite-dimensional Hilbert spaces under Hamiltonians with purely continuous spectrum. In this regime, the standard infinite-time dephasing picture underlying finite-dimensional equilibration theory ceases to be informative: the corresponding ergodic average becomes trivial in the relevant sense and therefore cannot serve as a physically meaningful equilibrium state. This makes equilibration intrinsically a finite-time phenomenon. Motivated by this obstruction, we introduce a framework based on finite-time ergodic averages and spectral coarse graining at finite resolution. Within this setting, we derive explicit upper bounds for equilibration-on-average of bounded observables under purely continuous-spectrum dynamics. Unlike their finite-dimensional counterparts, these bounds depend not only on the Hamiltonian and the initial state, but also on the observation time horizon and on the resolution scale of the spectral partition. We also obtain corresponding bounds for effective equilibration with respect to finite families of physically realistic POVM (Positive Operator-Valued Measure) measurements. Our results provide a quantitative formulation of equilibration for continuous-spectrum quantum systems in a regime where the usual infinite-time dephasing paradigm breaks down. Within the coarse-grained framework developed here, the resulting bounds depend explicitly on both the finite observation time and the chosen spectral resolution. An explicit example based on a free-particle Hamiltonian illustrates the mechanism and interpretation of the general bounds.

Journal of Mathematical Physics•Aug 1, 2026•0 citations

Continuous discretization of infinite-dimensional Hilbert spaces

Pierre Vernaz-Gris, Andreas Ketterer, Arne Keller +3 more

Observables with a continuous spectrum are known to fundamentally differ from those with a discrete and finite spectrum. While some fundamental tests and applications of quantum mechanics originally formulated for discrete variables have been translated to continuous ones, this is not the case in general. Here we show that it is possible to manipulate, detect, and classify continuous-variable states using observables with a continuous spectrum revealing properties and symmetries which are analogous to finite discrete systems. Our approach leads to an operational way to define, and adapt, to arbitrary continuous quantum systems, quantum protocols, and algorithms developed for discrete systems.

arXiv:1310.4452•Physical Review A•May 12, 2014•25 citations

Quantum equilibration in finite time

Anthony J Short, Terry Farrelly

It has recently been shown that small quantum subsystems generically equilibrate, in the sense that they spend most of the time close to a fixed equilibrium state. This relies on just two assumptions: that the state is spread over many different energies, and that the Hamiltonian has non-degenerate energy gaps. Given the same assumptions, it has also been shown that closed systems equilibrate with respect to realistic measurements. We extend these results in two important ways. Firstly, we prove equilibration over a finite (rather than infinite) time-interval, allowing us to bound the equilibration time. Secondly, we weaken the non-degenerate energy gaps condition, showing that equilibration occurs provided that no energy gap is hugely degenerate.

arXiv:1110.5759•New Journal of Physics•Jan 31, 2012•237 citations

Quantum equilibration under extended experimentally realistic conditions

M. R. Passos, Thiago Rodrigues de Oliveira

Understanding how macroscopic systems exhibit irreversible thermal behavior has been a longstanding challenge, first brought to prominence by Boltzmann. Recent advances have established rigorous conditions for isolated quantum systems to equilibrate to a maximum entropy state, contingent upon weak assumptions. These theorems, while powerful, applied for a sudden quench. However, natural processes involve finite-time perturbations or quenches, which raises a crucial question: Can these systems still equilibrate under more realistic, finite-time dynamics? In this work, we extend the established results to account for finite-time quenches, demonstrating that, even under finite-time perturbations, the system will equilibrate provided it populates many significant energy levels. While the mathematical proof is more intricate than in the instantaneous case, the physical conclusion remains the same: sufficient perturbation leads to equilibration. Our results provide a broader and more physically realistic framework for understanding thermalization in isolated quantum systems.

arXiv:2410.07429•Physical Review A•Feb 14, 2025•1 citation

Equilibration of isolated macroscopic quantum systems

Peter Reimann, Michael Kästner

We investigate the equilibration of an isolated macroscopic quantum system in the sense that deviations from a steady state become unmeasurably small for the overwhelming majority of times within any sufficiently large time interval. The main requirements are that the initial state, possibly far from equilibrium, exhibits a macroscopic population of at most one energy level and that degeneracies of energy eigenvalues and of energy gaps (differences of energy eigenvalues) are not of exceedingly large multiplicities. Our approach closely follows and extends recent works by Short and Farrelly (2012 New. J. Phys. 14 013063), in particular going beyond the realm of finite-dimensional systems and large effective dimensions.

•Jan 1, 2012•107 citations

Equilibration of isolated macroscopic quantum systems

Peter Reimann, Michael Kastner

We investigate the equilibration of an isolated macroscopic quantum system in the sense that deviations from a steady state become unmeasurably small for the overwhelming majority of times within any sufficiently large time interval.The main requirements are that the initial state, possibly far from equilibrium, exhibits a macroscopic population of at most one energy level and that degeneracies of energy eigenvalues and of energy gaps (differences of energy eigenvalues) are not of exceedingly large multiplicities.Our approach closely follows and extends recent works by Short and Farrelly (2012 New.J. Phys.14 013063), in particular going beyond the realm of finite-dimensional systems and large effective dimensions.

arXiv:1202.2768•New Journal of Physics•Apr 18, 2012•116 citations
Leveraging the techniques found in the literature on Quantum Equilibration for finite dimensional systems, we develop the theory of Quantum Equilibration for the case of infinite-dimensional systems, particularly the cases where the dynamics-generating Hamiltonians have continuous spectrum. The main goal of this paper will be to propose a framework to extend the results obtained by Short in, where estimates for the equilibration-on-average and effective equilibration for the case of Hamiltonians with continuous spectrum are derived. We will show that in the latter setting, it is compulsory to constrain ourselves to finite time equilibration; we then develop estimates analogous to the main results in the proposed setting.
We investigate equilibration-on-average and effective equilibration for quantum systems evolving on infinite-dimensional Hilbert spaces under Hamiltonians with purely continuous spectrum. In this regime, the standard infinite-time dephasing picture underlying finite-dimensional equilibration theory ceases to be informative: the corresponding ergodic average becomes trivial in the relevant sense and therefore cannot serve as a physically meaningful equilibrium state. This makes equilibration intrinsically a finite-time phenomenon. Motivated by this obstruction, we introduce a framework based on finite-time ergodic averages and spectral coarse graining at finite resolution. Within this setting, we derive explicit upper bounds for equilibration-on-average of bounded observables under purely continuous-spectrum dynamics. Unlike their finite-dimensional counterparts, these bounds depend not only on the Hamiltonian and the initial state, but also on the observation time horizon and on the resolution scale of the spectral partition. We also obtain corresponding bounds for effective equilibration with respect to finite families of physically realistic POVM (Positive Operator-Valued Measure) measurements. Our results provide a quantitative formulation of equilibration for continuous-spectrum quantum systems in a regime where the usual infinite-time dephasing paradigm breaks down. Within the coarse-grained framework developed here, the resulting bounds depend explicitly on both the finite observation time and the chosen spectral resolution. An explicit example based on a free-particle Hamiltonian illustrates the mechanism and interpretation of the general bounds.
Observables with a continuous spectrum are known to fundamentally differ from those with a discrete and finite spectrum. While some fundamental tests and applications of quantum mechanics originally formulated for discrete variables have been translated to continuous ones, this is not the case in general. Here we show that it is possible to manipulate, detect, and classify continuous-variable states using observables with a continuous spectrum revealing properties and symmetries which are analogous to finite discrete systems. Our approach leads to an operational way to define, and adapt, to arbitrary continuous quantum systems, quantum protocols, and algorithms developed for discrete systems.
It has recently been shown that small quantum subsystems generically equilibrate, in the sense that they spend most of the time close to a fixed equilibrium state. This relies on just two assumptions: that the state is spread over many different energies, and that the Hamiltonian has non-degenerate energy gaps. Given the same assumptions, it has also been shown that closed systems equilibrate with respect to realistic measurements. We extend these results in two important ways. Firstly, we prove equilibration over a finite (rather than infinite) time-interval, allowing us to bound the equilibration time. Secondly, we weaken the non-degenerate energy gaps condition, showing that equilibration occurs provided that no energy gap is hugely degenerate.
Understanding how macroscopic systems exhibit irreversible thermal behavior has been a longstanding challenge, first brought to prominence by Boltzmann. Recent advances have established rigorous conditions for isolated quantum systems to equilibrate to a maximum entropy state, contingent upon weak assumptions. These theorems, while powerful, applied for a sudden quench. However, natural processes involve finite-time perturbations or quenches, which raises a crucial question: Can these systems still equilibrate under more realistic, finite-time dynamics? In this work, we extend the established results to account for finite-time quenches, demonstrating that, even under finite-time perturbations, the system will equilibrate provided it populates many significant energy levels. While the mathematical proof is more intricate than in the instantaneous case, the physical conclusion remains the same: sufficient perturbation leads to equilibration. Our results provide a broader and more physically realistic framework for understanding thermalization in isolated quantum systems.
We investigate the equilibration of an isolated macroscopic quantum system in the sense that deviations from a steady state become unmeasurably small for the overwhelming majority of times within any sufficiently large time interval. The main requirements are that the initial state, possibly far from equilibrium, exhibits a macroscopic population of at most one energy level and that degeneracies of energy eigenvalues and of energy gaps (differences of energy eigenvalues) are not of exceedingly large multiplicities. Our approach closely follows and extends recent works by Short and Farrelly (2012 New. J. Phys. 14 013063), in particular going beyond the realm of finite-dimensional systems and large effective dimensions.
We investigate the equilibration of an isolated macroscopic quantum system in the sense that deviations from a steady state become unmeasurably small for the overwhelming majority of times within any sufficiently large time interval.The main requirements are that the initial state, possibly far from equilibrium, exhibits a macroscopic population of at most one energy level and that degeneracies of energy eigenvalues and of energy gaps (differences of energy eigenvalues) are not of exceedingly large multiplicities.Our approach closely follows and extends recent works by Short and Farrelly (2012 New.J. Phys.14 013063), in particular going beyond the realm of finite-dimensional systems and large effective dimensions.