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.
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.
Quantum systems governed by unitary and reversible microscopic dynamics may nevertheless exhibit equilibration, in the sense that some effective description becomes time-independent. Standard equilibration results usually consider two separate situations: system-environment structures, in which the composite system evolves unitarily while the system of interest equilibrates, and restricted measurements, such as coarse-grained POVMs and observables, in which the measurement statistics equilibrate. Here, we bring these descriptions into a common state-level framework using the concept of generalized subsystems, where the accessible effective state appears as the output of a quantum channel acting on the microscopic state. We derive bounds showing that generalized subsystems equilibrate when their dimension is small compared with the effective dimension of the discarded microscopic information. We further show that this condition is met for typical initial states in large subspaces and that the resulting equilibrium description is largely insensitive to microscopic initial details. The framework recovers the usual equilibration bounds for ordinary subsystems and finite families of POVMs. As an example, we also introduce a finite-resolution energy channel that maps unresolved microscopic energy levels into effective energy levels, thereby making residual effective coherences explicit and showing how spectral multiplicities constrain those coherences while strengthening equilibration. Our results provide a unified state-level formulation of quantum equilibration under general forms of limited accessible information.
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.
Understanding how classical physics emerges from quantum mechanics remains a central problem in the foundations of physics. Here we derive a classical limit from finite-resolution measurements, modeled by continuous coarse-grained POVMs. When the resolved phase-space area is large compared with Planck's constant, the accessible statistics of any quantum state admit an effective classical description: coarse-grained observables become approximately jointly measurable and the induced probability density is positive. We derive the exact evolution equation for this density and show that, in the strong coarse-graining regime, its non-Liouville corrections are suppressed up to an Ehrenfest time, resulting in classical Hamiltonian flow generated by a Hamiltonian smoothed over the measurement cell. When the Hamiltonian varies negligibly across such a cell, the smoothed Hamiltonian reduces to the classical Hamiltonian whose quantization produced the quantum dynamics, thereby closing the quantization--classical-limit loop. Repeated finite-resolution measurements then generate stochastic records confined, with high probability, to tubes around classical trajectories. Our results provide a unified operational framework for the quantum-to-classical transition in microscopic and macroscopic systems, and establish the consistency of the quantization--classical-limit cycle.
We develop a unified dynamical-spectral framework for equilibration in isolated quantum systems based on a subspace coarse-graining approach. Central to our formulation is the Leakage Fidelity Function (LFF), defined as the probability that a unitarily evolving state escapes the support of its initial subspace. This quantity provides a direct, operational measure of information flow and memory loss without invoking ensemble assumptions or perturbative arguments. We derive universal bounds on temporal fluctuations of the LFF, in terms of the spectral gap structure and the square of the effective dimension, evincing that large spectral delocalization suppresses fluctuations and guarantees equilibration on average. By introducing spectral power distributions and associated entropic measures, we establish a quantitative link between phase mixing, gap participation, and dynamical stability. We further investigate the equilibration timescale by connecting the LFF to quantum speed limits, thereby revealing the average time required for equilibration. Our results provide a state-dependent, geometrically transparent perspective on how spectral complexity and subspace information leakage jointly govern irreversibility in closed quantum many-body systems.
Quantum stochastic processes are shown, within finite time intervals and for finite dimensional time-independent Hamiltonians, to be close to a corresponding stationary equilibrium process. This occurs for multitime observables that are coarse grained in both space and time, and for systems which have many significantly interacting energy eigenstates. This dictates conditions on time scales at when the multitime statistics of any arbitrary non-equilibrium quantum process look equilibrium on average. We show that this leads to the equilibration of general geometric measures of quantum processes, such as the degree of non-Markovianity or classicality. The implications of this result for the emergence of classical stochastic processes from underlying quantum mechanics are also discussed.
We identify a general mechanism for the finite-time thermalization of macroscopic observables, such as coarse-grained charge densities, in terms of elementary forms of the quantum dynamics of initial states: (1) aperiodicity, which provides a computable measure of (2) a dynamical partially ergodic exploration of the Hilbert space. Specifically, this mechanism predicts the equilibration of all (concentrated) macroscopic observables, in almost all states in an initial ensemble and almost all times within finite and longer intervals, given only the observable-independent information that the return probability of the ensemble of initial states is small over a finite time range. As a special case, it also accesses standard results on equilibration over infinitely long times in terms of (stronger versions of) the effective dimension of initial state delocalization in the energy eigenbasis. Our results incorporate macroscopic thermalization into the domain of operational quantum statistical mechanics, recently developed to provide finitely computable criteria for microscopic thermalization. We discuss an overall characterization of this approach as establishing connections between (1) the decay of a (theoretically or experimentally) computable probe indicating memorylessness, (2) a fundamental invariant mechanism in terms of the alignment of observables or states in the Hilbert space, and (3) predicting different natural forms of (classical and) quantum thermalization, most of which rigorously recover conventional eigenstate-based descriptions of infinite-time thermalization as a special case but provide stronger accessible predictions over finite observation times in the thermodynamic limit.