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arXiv:2507.05946 · Journal of Physics A Mathematical and Theoretical

A simpler probe of the quantum Mpemba effect in closed systems

Filiberto Ares, Colin Rylands, Pasquale Calabrese

Published Sep 29, 2025• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ae0d18

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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 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.
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 Local relaxation after a quench in 1D quantum many-body systems is a well-known and very active problem with rich phenomenology. Except in pathological cases, the local relaxation is accompanied by the local restoration of the symmetries broken by the initial state that are preserved by unitary evolution. Recently, the entanglement asymmetry has been introduced as a probe to study the interplay between symmetry breaking and relaxation in an extended quantum system. In particular, using the entanglement asymmetry, it has been shown that the more a symmetry is initially broken, the faster it may be restored. This surprising effect, which has also been observed in trapped-ion experiments, can be seen as a quantum version of the Mpemba effect, and is manifested by the crossing at a finite time of the entanglement asymmetry curves of two different initial symmetry-breaking configurations. In this paper we show that, by tuning the initial state, the symmetry dynamics in free fermionic systems can display much richer behavior than seen previously. In particular, for certain classes of initial states, including the ground states of free fermionic models with long-range couplings, the entanglement asymmetry can exhibit multiple crossings. This illustrates that the existence of the quantum Mpemba effect can only be inferred by examining the late-time behavior of the entanglement asymmetry.
Understanding symmetry restoration in isolated quantum many-body systems is an important problem in nonequilibrium many-body quantum physics. Recent studies have shown that the quantum Mpemba effect can be characterized through entanglement asymmetry, where states with stronger initial symmetry breaking restore symmetry faster. However, it remains unclear whether conventional energy-based measures, such as the trace distance, capture the same phenomenon. We investigate this question in closed spin-1/21/21/2 quantum systems with different symmetries by analyzing the dynamics of symmetry-breaking initial states. Combining numerical simulations with an analytical decomposition of the trace distance into symmetry-coherence and residual contributions, we identify the conditions under which trace distance tracks entanglement asymmetry and reproduces the Mpemba--like behavior observed in it. For charge symmetry, the residual contribution is negligible, making the trace distance effectively governed by symmetry-sector coherences. In contrast, for permutation symmetry, a significant residual contribution leads to qualitatively different relaxation dynamics. Our results establish when conventional energy-based diagnostics reliably capture symmetry-restoration dynamics and clarify the distinct physical information encoded by entanglement asymmetry and trace distance.
The essence of the Mpemba effect is that nonequilibrium systems may relax faster the further they are from their equilibrium configuration. In the quantum realm, this phenomenon arises in closed systems dynamics and is witnessed by features of symmetry and entanglement. Here, we study the quantum Mpemba effect in charge-preserving random circuits, combining extensive numerical simulations and analytical arguments. We show that the more asymmetric certain classes of initial states (tilted ferromagnets) are, the faster they restore symmetry and reach the grand-canonical ensemble. Conversely, other classes of states (tilted antiferromagnets) do not show the Mpemba effect. We provide a simple and general mechanism underlying the effect, based on the spreading of nonconserved operators in terms of conserved densities. Grounded only in locality, unitarity, and symmetry, our analysis clarifies the emergence of Mpemba physics in chaotic quantum systems.
Understanding relaxation in isolated quantum many-body systems remains a central challenge. Recently, the quantum Mpemba effect (QME), a counterintuitive relaxation phenomenon, has attracted considerable attention and has been extensively studied in systems with global symmetries. Here, we study the QME in gauge theories with massive local gauge symmetries. In the lattice Schwinger model, we demonstrate that the gauge structure of the reduced density matrix of a subsystem is entirely determined by the initial state and remain unchanged during the time evolution. We then investigate whether gauge symmetry can be dynamically restored following a symmetric quench. Analytical and numerical results show that when the Maxwell term is zero, gauge symmetry restoration fails due to the emergence of a peculiar conservation law. However, for any finite Maxwell term, subsystem gauge symmetry is restored in the thermodynamic limit. Based on these results, we systematically construct a families of initial states exhibiting the QME. We further explore the QME in the quantum link model-a truncated lattice Schwinger model, which has been realized in experiments. Moreover, we propose an experimentally accessible order parameter that correctly captures the QME. Our work demonstrates the generality of the quantum Mpemba effect even in the local gauge symmetries, and are directly relevant to ongoing quantum simulation experiments of gauge theories.