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Journal of Statistical Mechanics Theory and Experiment

The quantum Mpemba effect in closed systems: from theory to experiment

Pasquale Calabrese

Published Mar 1, 2026• 3 citations
Publisher

Abstract

Abstract The Mpemba effect refers to a counterintuitive phenomenon whereby a system initially prepared further from equilibrium may relax faster than one prepared closer to equilibrium. While extensively studied in classical nonequilibrium physics, its extension to isolated quantum systems only started in the last few years. In this contribution we review recent progress on the quantum Mpemba effect in closed many-body systems, emphasizing the role of reduced density matrix, entanglement and symmetry restoration. We discuss why and how the entanglement asymmetry provides a natural and experimentally accessible framework to characterize Mpemba-like behavior in unitary quantum evolution.

DOI
10.1088/1742-5468/ae4bb6

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Abstract The Mpemba effect refers to a counterintuitive phenomenon whereby a system initially prepared further from equilibrium may relax faster than one prepared closer to equilibrium. While extensively studied in classical nonequilibrium physics, its extension to isolated quantum systems only started in the last few years. In this contribution we review recent progress on the quantum Mpemba effect in closed many-body systems, emphasizing the role of reduced density matrix, entanglement and symmetry restoration. We discuss why and how the entanglement asymmetry provides a natural and experimentally accessible framework to characterize Mpemba-like behavior in unitary quantum evolution.
Abstract Non-equilibrium dynamics have become a central research focus, exemplified by the counterintuitive Mpemba effect where initially hotter systems can cool faster than colder ones. Studied extensively in both classical and quantum regimes, this phenomenon reveals diverse and complex behaviors across different systems. This review provides a concise overview of the quantum Mpemba effect (QME), specifically emphasizing its connection to symmetry breaking and restoration in closed quantum many-body systems. We begin by outlining the classical Mpemba effect and its quantum counterparts, summarizing key findings. Subsequently, we introduce entanglement asymmetry and charge variance as key metrics for probing the QME from symmetry perspectives. Leveraging these tools, we analyze the early- and late-time dynamics of these quantities under Hamiltonian evolution and random unitary circuits. We conclude by discussing significant challenges and promising avenues for future research.
Abstract We study the local relaxation of closed quantum systems through the relative entropy between the reduced density matrix and its long time limit. We show, using analytic arguments combined with numerical checks, that this relative entropy can be very well approximated by an entropy difference, affording a significant computational advantage. We go on to relate this to the entanglement asymmetry of the subsystem with respect to time translation invariance. In doing this, we obtain a simple probe of the relaxation dynamics of closed many-body systems and use it to re-examine the quantum Mpemba effect, wherein states can relax faster if they are initially further from equilibrium. We reproduce earlier instances of the effect related to symmetry restoration as well as uncover new cases in the absence of such symmetries. For integrable models, we obtain the criteria for this to occur using the quasiparticle picture. Lastly, we show that, in models obeying the entanglement membrane picture, the quantum Mpemba effect cannot occur for a large class of initial states.
The quantum Mpemba effect is the counterintuitive phenomenon whereby a quantum state initially farther from equilibrium relaxes faster than one initially closer to it. We demonstrate this effect in random Clifford circuits with a conserved U(1)\mathrm{U}(1)U(1) charge, using the entanglement asymmetry to characterize relaxation through dynamical symmetry restoration. Exact numerical simulations show that an initially more asymmetric state can become locally more symmetric than an initially less asymmetric one. We explain this behavior by mapping the dynamics onto a charge-conserving quantum automaton, in which the decay of the entanglement asymmetry is controlled by the statistics of encounters between two particle species evolving according to a symmetric simple exclusion process. This mapping yields an analytic expression for the entanglement asymmetry and provides a simple microscopic mechanism for the quantum Mpemba effect: stronger initial symmetry breaking corresponds to a denser particle configuration, which enhances the frequency of particle encounters and thereby accelerates symmetry restoration.
The highly complicated nature of far from equilibrium systems can lead to a complete breakdown of the physical intuition developed in equilibrium. A famous example of this is the Mpemba effect, which states that nonequilibrium states may relax faster when they are further from equilibrium or, put another way, hot water can freeze faster than warm water. Despite possessing a storied history, the precise criteria and mechanisms underpinning this phenomenon are still not known. Here, we study a quantum version of the Mpemba effect that takes place in closed many-body systems with a U(1) conserved charge: in certain cases a more asymmetric initial configuration relaxes and restores the symmetry faster than a more symmetric one. In contrast to the classical case, we establish the criteria for this to occur in arbitrary integrable quantum systems using the recently introduced entanglement asymmetry. We describe the quantum Mpemba effect in such systems and relate the properties of the initial state, specifically its charge fluctuations, to the criteria for its occurrence. These criteria are expounded using exact analytic and numerical techniques in several examples, a free fermion model, the Rule 54 cellular automaton, and the Lieb-Liniger model.
The Mpemba effect, where a system initially farther from equilibrium relaxes faster than one closer to equilibrium, has been extensively studied in classical systems and recently explored in quantum settings. While previous studies of the quantum Mpemba effect (QME) have largely focused on isolated systems with global symmetries, we argue that the QME is ubiquitous in generic, nonintegrable many-body systems lacking such symmetries, including U(1) charge conservation, spatial symmetries, and even energy conservation. Using paradigmatic models such as the quantum Ising model with transverse and longitudinal fields, we show that the QME can be understood through the energy density of initial states and their inverse participation ratio in the energy eigenbasis. Our findings provide a unified framework for the QME, linking it with classical thermal relaxation.
In classical settings, the Mpemba effect occurs when a hotter system cools faster than an initially colder one. In quantum systems, this effect can be reinterpreted exploiting the concept of symmetries, with the asymmetry of a subsystem playing the role of temperature. A quantum Mpemba effect arises when a more asymmetric state restores the symmetry faster than a less asymmetric one. Previous work mainly focuses on closed systems characterized by thermal equilibration and Hamiltonian symmetries. In this paper, we analyze the dynamics of asymmetry in an open quantum many-body system featuring symmetry breaking and uncover dynamical behavior that appears to be unique to these settings. In the symmetric phase, we demonstrate the existence of a quantum Mpemba effect, which emerges as a direct consequence of a non-monotonic evolution of the asymmetry. In the broken-symmetry phase, we analyze the imbalance between the system's ability to increase or to decrease its asymmetry. Our results extend the notion of quantum Mpemba effects to open quantum many-body systems exhibiting symmetry-breaking phase transitions and establish them as a platform for observing and controlling anomalous relaxation phenomena.