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Communications Physics

Typical thermalization of low-entanglement states

Christian Bertoni, Clara Wassner, Giacomo Guarnieri, Jens Eisert

Published Jul 17, 2025• 5 citations
Publisher

Abstract

Proving thermalization from the unitary evolution of closed quantum systems is one of the oldest questions that is still only partially resolved. Efforts led to various versions of the eigenstate thermalization hypothesis (ETH), which implies thermalization under certain conditions. Whether the ETH holds in specific systems is however difficult to verify from the microscopic description of the system. In this work, we focus on thermalization under local Hamiltonians of low-entanglement initial states, which are operationally accessible in many natural physical settings, including schemes for testing thermalization in experiments and quantum simulators. We prove thermalization of these states under precise conditions that have operational significance. More specifically, motivated by arguments of unavoidable finite resolution, we define a random energy smoothing on local Hamiltonians that leads to local thermalization when the initial state has low entanglement. Finally we show that this transformation affects neither the Gibbs state locally nor, under generic smoothness conditions on the spectrum, the short-time dynamics.

DOI
10.1038/s42005-025-02161-7

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•Jan 1, 2023•1 citation
Proving thermalization from the unitary evolution of closed quantum systems is one of the oldest questions that is still only partially resolved. Efforts led to various versions of the eigenstate thermalization hypothesis (ETH), which implies thermalization under certain conditions. Whether the ETH holds in specific systems is however difficult to verify from the microscopic description of the system. In this work, we focus on thermalization under local Hamiltonians of low-entanglement initial states, which are operationally accessible in many natural physical settings, including schemes for testing thermalization in experiments and quantum simulators. We prove thermalization of these states under precise conditions that have operational significance. More specifically, motivated by arguments of unavoidable finite resolution, we define a random energy smoothing on local Hamiltonians that leads to local thermalization when the initial state has low entanglement. Finally we show that this transformation affects neither the Gibbs state locally nor, under generic smoothness conditions on the spectrum, the short-time dynamics.
Proving thermalization from the unitary evolution of a closed quantum system is one of the oldest questions that is still nowadays only partially resolved. Several efforts have led to various formulations of what is called the eigenstate thermalization hypothesis, which leads to thermalization under certain conditions on the initial states. These conditions, however, are sensitive to the precise formulation of the hypothesis. In this work, we focus on the important case of low entanglement initial states, which are operationally accessible in many natural physical settings, including experimental schemes for testing thermalization and for quantum simulation. We prove thermalization of these states under precise conditions that have operational significance. More specifically, motivated by arguments of unavoidable finite resolution, we define a random energy smoothing on local Hamiltonians that leads to local thermalization when the initial state has low entanglement. Finally we show that such a transformation affects neither the Gibbs state locally nor, under generic smoothness conditions on the spectrum, the short-time dynamics.
Deriving conditions under which a macroscopic system thermalizes directly from the underlying quantum many-body dynamics of its microscopic constituents is a long-standing challenge in theoretical physics. The well-known eigenstate thermalization hypothesis (ETH) is presumed to be a key mechanism, but has defied rigorous verification for generic systems thus far. A weaker variant (weak ETH), by contrast, is provably true for a large variety of systems, including even many integrable models, but its implications with respect to the problem of thermalization are still largely unexplored. Here we analytically demonstrate that systems satisfying the weak ETH exhibit thermalization for two very natural classes of far-from-equilibrium initial conditions: the overwhelming majority of all pure states with a preset nonequilibrium expectation value of some given local observable, and the Gibbs states of a Hamiltonian which subsequently is subject to a quantum quench in the form of a sudden change of some local system properties.
Quantum thermalization describes how closed quantum systems can effectively reach thermal equilibrium, resolving the apparent incongruity between the reversibility of Schrödinger's equation and the second law of thermodynamics. Despite its ubiquity and conceptual significance, the precise conditions that give rise to quantum thermalization are still not well understood. After nearly a century of efforts, we have yet to find a complete mathematical proof that an effective statistical description naturally emerges the underlying quantum dynamics in generic settings. Here, we prove that quantum thermalization must occur in any qubit system with local interactions under three conditions: (i) high effective temperature, (ii) translation invariance, and (iii) no perfect resonances in the energy spectrum. Specifically, we show that a typical, low-complexity pure state drawn from any ensemble with large entropy and well-defined effective temperature becomes locally indistinguishable from a Gibbs state upon unitary evolution. In this setting, our rigorous results prove the widely anticipated notion that statistical physics should be understood as an emergent phenomenon, explicitly derived from the first principles of quantum mechanics.
The Eigenstate Thermalization Hypothesis (ETH) has played a major role in understanding thermodynamic phenomena in closed quantum systems. However, its connection to the timescale of thermalization for open system dynamics has remained elusive. This paper establishes a rigorous link between ETH and fast thermalization to the global Gibbs state. Specifically, we demonstrate fast thermalization for a system coupled weakly to a bath of quasi-free Fermions that we refresh periodically. To describe the joint evolution, we derive a finite-time version of Davies' generator with explicit error bounds and resource estimates. Our approach exploits a critical feature of ETH: operators in the energy basis can be modeled by independent random matrices in a near-diagonal band. This gives quantum expanders at nearby eigenstates of the Hamiltonian and reduces the problem to a one-dimensional classical random walk on the energy eigenstates. Our results explain finite-time thermalization in chaotic open quantum systems.
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.