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arXiv:2512.11522 · Physical Review A

Equilibration and the eigenstate thermalization hypothesis as limits to observing macroscopic quantum superpositions

Gabriel Dias Carvalho, Pedro S. Correia, Thiago R. de Oliveira

quant-phSubmitted Dec 12, 2025• Announced Dec 15, 2025• 0 citations• Revised Jun 5, 2026 (v2)
v1 · Dec 12, 2025v2 · Jun 5, 2026
arXivHTMLPDF

Abstract

Macroscopic quantum superpositions are widely believed to be unobservable because large systems cannot be perfectly isolated from their environments. Here, we show that even under perfect isolation, intrinsic unitary dynamics in generic many-body systems, as analyzed through the eigenstate thermalization hypothesis and random matrix theory, can suppress the observable signatures of macroscopic coherence. Using the Greenberger-Horne-Zeilinger (GHZ) state as a representative example, we demonstrate that while fully correlated measurements can initially distinguish a macroscopic superposition from its corresponding classical mixture, generic many-body evolution renders them operationally indistinguishable for most times. By analyzing both distinguishability measures and established quantifiers of macroscopic quantumness, we find that equilibration not only hides coherence from accessible observables but also suppresses the corresponding signatures of macroscopic quantumness, in particular within the additive-local framework considered here. These results identify unitary thermalization, independent of environmental decoherence, as a fundamental mechanism that limits the observation of macroscopic quantum effects.

DOI
10.1103/qlz3-4tfx
Comments
Accepted to PRA

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Macroscopic quantum superpositions are widely believed to be unobservable because large systems cannot be perfectly isolated from their environments. Here, we show that even under perfect isolation, intrinsic unitary dynamics in generic many-body systems, as analyzed through the eigenstate thermalization hypothesis and random matrix theory, can suppress the observable signatures of macroscopic coherence. Using the Greenberger-Horne-Zeilinger (GHZ) state as a representative example, we demonstrate that while fully correlated measurements can initially distinguish a macroscopic superposition from its corresponding classical mixture, generic many-body evolution renders them operationally indistinguishable for most times. By analyzing both distinguishability measures and established quantifiers of macroscopic quantumness, we find that equilibration not only hides coherence from accessible observables but also suppresses the corresponding signatures of macroscopic quantumness, in particular within the additive-local framework considered here. These results identify unitary thermalization, independent of environmental decoherence, as a fundamental mechanism that limits the observation of macroscopic quantum effects.
The emergence of classical behavior and the origin of thermal equilibrium are two central problems in the foundations of physics. In the standard accounts, both phenomena are typically explained through interactions with an external environment: decoherence suppresses quantum interference, while coupling to a thermal bath drives relaxation toward equilibrium. Over the last decades, however, it has become clear that equilibration and thermalization can arise even in fully isolated quantum systems, in the operational sense that the expectation values of relevant observables remain close to equilibrium values for most of the time. Here, we ask whether the same intrinsic equilibration mechanism can also account for the emergence of classical behavior. Using rigorous bounds on equilibration in closed systems, we derive sufficient conditions under which a time-evolved pure state becomes, for most times, operationally indistinguishable from a classical mixture associated with a chosen physical property. We identify two complementary routes to such operational classicality: either the chosen property almost commutes with the system Hamiltonian or the observables used to probe the system lose access to the remaining coherence after equilibration. Our results show that classical behavior need not be confined to the energy basis and may emerge even when substantial coherence remains present in the equilibrium state. This establishes a direct connection between two foundational questions: the origin of thermalization in isolated quantum systems and the quantum-to-classical transition
The eigenstate thermalization hypothesis explains the emergence of the thermodynamic equilibrium in isolated quantum many-body systems by assuming a particular structure of the observable's matrix elements in the energy eigenbasis. Schematically, it postulates that off-diagonal matrix elements are random numbers and the observables can be described by random matrix theory (RMT). To what extent a RMT description applies, more precisely at which energy scale matrix elements of physical operators become truly uncorrelated, is, however, not fully understood. We study this issue by introducing a novel numerical approach to probe correlations between matrix elements for Hilbert-space dimensions beyond those accessible by exact diagonalization. Our analysis is based on the evaluation of higher moments of operator submatrices, defined within energy windows of varying width. Considering nonintegrable quantum spin chains, we observe that matrix elements remain correlated even for narrow energy windows corresponding to timescales of the order of thermalization time of the respective observables. We also demonstrate that such residual correlations between matrix elements are reflected in the dynamics of out-of-time-ordered correlation functions.
The eigenstate thermalization hypothesis (ETH), which asserts that every eigenstate of a many-body quantum system is indistinguishable from a thermal ensemble, plays a pivotal role in understanding thermalization of isolated quantum systems. Yet, no evidence has been obtained as to whether the ETH holds for all few-body operators in a chaotic system; such few-body operators include key quantities in statistical mechanics, such as the total magnetization, the momentum distributions, and their low-order thermal and quantum fluctuations. Here, we formulate a conjecture that for a generic nonintegrable system the ETH holds simultaneously for all mmm-body operators with m<α∗Nm < α_{\ast} Nm<α∗​N in the thermodynamic limit for some nonzero constant α∗>0α_{\ast} > 0α∗​>0. We first show the existence of such nontrivial constants for idealized (pseudo) random-matrix descriptions of many-body eigenstates. We then verify the conjecture for generic spin, Bose, and Fermi systems with local and few-body interactions by large-scale numerical calculations. Our results imply that generic systems satisfy the ETH simultaneously for all few-body operators, including their thermal and quantum fluctuations.
The Eigenstate Thermalization Hypothesis (ETH) explains how isolated quantum many-body systems thermalize by proposing that each energy eigenstate is already thermal. The hypothesis is believed to be essential to the understanding of quantum chaos and implies various important thermodynamic relations. While by now there are many numerical validations of the ETH, only few analytical arguments supporting the ETH have been found (M. Rigol et al., Nature 452, 2008). One of them is based on a semiclassical limit (M. Srednicki, Phys. Rev. E 50, 1994). Another important reasoning was given by Deutsch, who showed how a small interaction, which is modeled by a random matrix, leads to ETH (J.M. Deutsch, Phys. Rev. A 43, 1991). This thesis aims to find justification for the assumptions made by Deutsch from a microscopic point of view. The idea is to analyze whether a generic quantum system can be brought to a form that satisfies Deutsch's ansatz. The method is to use flow equations to define unitary transformations to map the initial Hamiltonian to an effective one, which can be compared to a random matrix. The test system is a one-dimensional hardcore boson model, which is set up using exact diagonalization and numerous symmetries. We analyze the statistical properties of the numerically obtained Hamiltonian matrices and test for normality and compare them to a random transformation. In a second step we verify that the continuous unitary transformations do not destroy the structure of typical few-body observables by studying the scaling of the inverse participation ratio. The obtained results support our hypothesis that it is possible to extend Deutsch's argument to generic microscopic many-body Hamiltonians.
Abstract Identifying which quantum states qualify as macroscopic superpositions remains a fundamental challenge in quantum physics, as no universally accepted operational criterion currently exists. In this work we introduce a dynamical criterion for macroscopic quantum superposition based on the continuous-measurement-based measure of quantum superposition together with the principle of macrorealism. The key idea is to characterize macroscopicity through the rate of purity decay of the quantum state under continuous monitoring, which quantifies the susceptibility of quantum coherence to classicalizing mechanisms. Two classes of macrorealistic dynamics are considered: environmental decoherence described by a Lindblad master equation (class A) and objective collapse dynamics in the continuous spontaneous localization model (class B). Applying this framework to an optomechanical oscillator, we show that the emergence of macroscopic superposition depends on the competition between measurement-induced decoherence and the macrorealistic mechanism. In class A the relevant contributions increase with excitation number, whereas in class B large excitation numbers can suppress macroscopicity. Unlike many existing macroscopicity measures based solely on phase-space structure or decoherence times, the present criterion directly relates macroscopicity to experimentally accessible parameters in continuous quantum measurement.
We study the equilibration properties of isolated ergodic quantum systems initially prepared in a cat state, i.e a macroscopic quantum superposition of states. Our main result consists in showing that, even though decoherence is at work in the mean, there exists a remnant of the initial quantum coherences visible in the strength of the fluctuations of the steady state. We back-up our analysis with numerical results obtained on the XXX spin chain with a random field along the z-axis in the ergodic regime and find good qualitative and quantitative agreement with the theory. We also present and discuss a framework where equilibrium quantities can be computed from general statistical ensembles without relying on microscopic details about the initial state, akin to the eigenstate thermalization hypothesis.