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arXiv:2108.07420 · Quantum

Relaxation of Multitime Statistics in Quantum Systems

Neil Dowling, Pedro Figueroa–Romero, Felix A. Pollock, Philipp Strasberg, Kavan Kishore Modi

Published Jun 1, 2023• 20 citations
arXivHTMLPDF

Abstract

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.

DOI
10.22331/q-2023-06-01-1027

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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.
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.
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 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.
We investigate the role of a statistical complexity measure to assign equilibration in isolated quantum systems. While unitary dynamics preserve global purity, expectation values of observables often exhibit equilibration-like behavior, raising the question of whether a measure of complexity can track this process. In addition to examining observable equilibration, we extend our analysis to study how the complexity of the quantum states evolves, providing insight into the transition from initial coherence to equilibrium. We define a classical statistical complexity measure based on observable entropy and deviation from equilibrium, which captures the dynamical progression towards equilibration and effectively distinguishes between complex and non-complex trajectories. In particular, our measure is sensitive to non-complex dynamics. Such dynamics include the quasi-periodic behavior exhibited by low-dimensional initial states, where the system explores a limited region of Hilbert space while preserving coherence. Numerical simulations of an Ising-like non-integrable Hamiltonian spin-chain model support these findings. Our work provides new insight into the emergence of equilibrium behavior from unitary dynamics and advances complexity as a meaningful tool in the study of the emergence of classicality in microscopic systems.
There has been considerable recent progress in clarifying the conditions under which quantum systems equilibrate. We demonstrate that effective equilibration of an isolated system can occur under much weaker conditions than previously considered, by requiring only an assumption of sufficiently complex dynamics (which encodes a condition on the complexity of the system eigenvectors) and natural information-theoretic constraints resulting from the infeasibility of collecting an astronomically large amount of measurement data. In particular, we prove that, for generic Hamiltonian systems, the measurement statistics for maximally fine-grained measurements on pure states are effectively indistinguishable from the micro-canonical distribution after a finite equilibration time. We demonstrate that our assumptions correspond to physically realistic conditions by numerically confirming their validity for both generic many-body systems limited to two-body interactions and the quantum kicked-top in a globally chaotic regime.
Focusing on isolated macroscopic systems, described in terms of either a quantum mechanical or a classical model, our two key questions are how far does an initial ensemble (usually far from equilibrium and largely unknown in detail) evolve towards a stationary long-time behavior (equilibration) and how far is this steady state in agreement with the microcanonical ensemble as predicted by statistical mechanics (thermalization). A recently developed quantum mechanical treatment of the problem is briefly summarized, putting particular emphasis on the realistic modeling of experimental measurements and nonequilibrium initial conditions. Within this framework, equilibration can be proven under very weak assumptions about those measurements and initial conditions, while thermalization still requires quite strong additional hypotheses. An analogous approach within the framework of classical mechanics is developed and compared with the quantum case. In particular, the assumptions to guarantee classical equilibration are now rather strong, while thermalization then follows under relatively weak additional conditions.