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Finite-time quantum equilibration for continuous variables

Alberto Acevedo, Antonio Falco

TL;DR

This work extends quantum equilibration theory from finite-dimensional systems to infinite-dimensional Hilbert spaces with continuous-spectrum Hamiltonians, addressing the breakdown of infinite-time equilibration and proposing finite-time, spectrum-aware bounds. By developing a framework based on $c_{0}$-semigroups, spectral measures, and Bochner integration, it derives explicit bounds on equilibration-on-average and effective equilibration for continuous variables, incorporating spectrum-resolution scale $oldsymbol{ riangle}$, partition distance $D$, and finite time $T$. The main contributions include a continuous-spectrum analogue of Short’s equilibration bounds, a detailed decomposition via spectral projections, and a bound on the time-averaged variance of observables that accounts for coherence between spectral blocks; a toy example illustrates the parameter trade-offs. The results advance understanding of how quantum systems with continuous spectra approach equilibrium within finite times and under realistic measurement constraints, with implications for quantum thermodynamics and dynamics in infinite-dimensional settings.

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.

Finite-time quantum equilibration for continuous variables

TL;DR

This work extends quantum equilibration theory from finite-dimensional systems to infinite-dimensional Hilbert spaces with continuous-spectrum Hamiltonians, addressing the breakdown of infinite-time equilibration and proposing finite-time, spectrum-aware bounds. By developing a framework based on -semigroups, spectral measures, and Bochner integration, it derives explicit bounds on equilibration-on-average and effective equilibration for continuous variables, incorporating spectrum-resolution scale , partition distance , and finite time . The main contributions include a continuous-spectrum analogue of Short’s equilibration bounds, a detailed decomposition via spectral projections, and a bound on the time-averaged variance of observables that accounts for coherence between spectral blocks; a toy example illustrates the parameter trade-offs. The results advance understanding of how quantum systems with continuous spectra approach equilibrium within finite times and under realistic measurement constraints, with implications for quantum thermodynamics and dynamics in infinite-dimensional settings.

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.

Paper Structure

This paper contains 21 sections, 12 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bounded operators and its limits with respect to the strong operator topology
  4. Bochner integration
  5. $c_{0}$-semigroups
  6. The spectrum of an operator
  7. Trace-class and Hilbert-Schmidt operators
  8. The Density Operators
  9. Ergodic averages and more on semigroups
  10. Equilibration-on-Average
  11. A first attempt at studying equilibration for the case of continuous variables
  12. Spectral families of projection operators
  13. Preliminary estimates
  14. A bound for estimating $\mathscr{K}(T, \Delta)$ and $\mathscr{D}_{\mathbf{\hat{A}}}(T,\Delta)$
  15. Estimates for $\mathscr{D}_{\mathbf{\hat{A}}}(T,\Delta)$ and $\mathscr{R}_{\mathbf{\hat{A}}}(T,\Delta)$
  16. ...and 6 more sections

Key Result

Theorem 1

Consider a strongly continuous unitary one-parameter semigroup on a Hilbert space $\mathscr{H}$. Namely, $\mathbf{\hat{U}}_{t} :\mathbb{R}_{+}\rightarrow L(\mathscr{H})$. Then there exists a unique (possibly unbounded) operator $\mathbf{\hat{A}} : \mathrm{Dom}(\mathbf{\hat{A}})\rightarrow \mathscr{H Conversely, let $\mathbf{\hat{A}}:\mathrm{Dom}(\mathbf{\hat{A}})\rightarrow \mathscr{H}$ be a (poss

Theorems & Definitions (38)

  • Definition 1: Operator Norm
  • Definition 2: Operator limit with respect to the strong operator topology
  • Definition 3: Bochner integrable
  • Remark 1: Proving Bochner integrability
  • Definition 4: $c_{0}$-semigroup
  • Theorem 1: Stone's Theorem
  • Definition 5: The Spectrum of an operator
  • Definition 6: Trace of an operator
  • Definition 7: Hilbert-Schmidt and Trace-Class operators
  • Definition 8: Density Operator
  • ...and 28 more