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arXiv:1905.08469 · Physical review. E

Equilibration on average in quantum processes with finite temporal resolution

Pedro Figueroa–Romero, Kavan Kishore Modi, Felix A. Pollock

Published Sep 30, 2020• 12 citations
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Abstract

We characterize the conditions under which a multitime quantum process with a finite temporal resolution can be approximately described by an equilibrium one. By providing a generalization of the notion of equilibration on average, where a system remains closed to a fixed equilibrium for most times, to one which can be operationally assessed at multiple times, we place an upper-bound on a new observable distinguishability measure comparing a multitime process with a finite temporal resolution against a fixed equilibrium one. While the same conditions on single-time equilibration, such as a large occupation of energy levels in the initial state remain necessary, we obtain genuine multitime contributions depending on the temporal resolution of the process and the amount of disturbance of the observer's operations on it.

DOI
10.1103/physreve.102.032144

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We characterize the conditions under which a multitime quantum process with a finite temporal resolution can be approximately described by an equilibrium one. By providing a generalization of the notion of equilibration on average, where a system remains closed to a fixed equilibrium for most times, to one which can be operationally assessed at multiple times, we place an upper-bound on a new observable distinguishability measure comparing a multitime process with a finite temporal resolution against a fixed equilibrium one. While the same conditions on single-time equilibration, such as a large occupation of energy levels in the initial state remain necessary, we obtain genuine multitime contributions depending on the temporal resolution of the process and the amount of disturbance of the observer's operations on it.
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.
A generic non-integrable (unitary) out-of-equilibrium quantum process, when interrogated across many times, is shown to yield the same statistics as an (non-unitary) equilibrated process. In particular, using the tools of quantum stochastic processes, we prove that under loose assumptions, quantum processes equilibrate within finite time intervals. Sufficient conditions for this to occur are that multitime observables are coarse grained in both space and time, and that the initial state overlaps with many different energy eigenstates. These results help bridge the gap between (unitary) quantum and (non-unitary) statistical physics, i.e., when all multitime properties and correlations are well approximated by stationary quantities, which includes non-Markovianity and temporal entanglement. We discuss implications of this result for the emergence of classical stochastic processes from multitime measurements of an underlying genuinely quantum system.
Equilibrium statistical mechanics provides powerful tools to understand physics at the macroscale. Yet, the question remains how this can be justified based on a microscopic quantum description. Here, we extend the ideas of pure state quantum statistical mechanics, which focus on single time statistics, to show the equilibration of isolated quantum processes. Namely, we show that most multitime observables for sufficiently large times cannot distinguish a nonequilibrium process from an equilibrium one, unless the system is probed for an extremely large number of times or the observable is particularly fine-grained. A corollary of our results is that the size of non-Markovianity and other multitime characteristics of a nonequilibrium process also equilibrate.
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.
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.
In the dynamics of open quantum systems, information may propagate in time through either the system or the environment, giving rise to Markovian and non-Markovian temporal correlations, respectively. However, despite their notable coexistence in most physical situations, it is not yet clear how these two quantities may limit the existence of one another. Here, we address this issue by deriving several inequalities relating the temporal correlations of general multitime quantum processes. The dynamics are described by process tensors, and the correlations are quantified by the mutual information between subsystems of their Choi states. First, we prove a set of upper bounds to the non-Markovianity of a process given the degree of Markovianity in each of its steps. This immediately implies a nontrivial maximum value for the non-Markovianity of any process, independent of its Markovianity. Finally, we determine how the non-Markovianity limits the amount of total temporal correlations that could be present in a given process. These results show that, although any multitime process must pay a price in total correlations to have a given amount of non-Markovianity, this price vanishes exponentially with the number of steps of the process, while the maximum non-Markovianity grows only linearly. This implies that even a highly non-Markovian process might be arbitrarily close to having the maximum amount of total correlations if it has a sufficiently large number of steps.