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arXiv:2501.12459 · Physical Review Research

Entanglement asymmetry dynamics in random quantum circuits

Filiberto Ares, Sara Murciano, Pasquale Calabrese, Lorenzo Piroli

Published Jun 10, 2025• 19 citations
arXivHTMLPDF

Abstract

We study the dynamics of entanglement asymmetry in random unitary circuits (RUCs). Focusing on a local U ( 1 ) charge, we consider symmetric initial states evolved by both local one-dimensional circuits and geometrically nonlocal RUCs made of two-qudit gates. We compute the entanglement asymmetry of subsystems of arbitrary size, analyzing the relaxation timescales. We show that the entanglement asymmetry of the whole system approaches its stationary value in a time independent of the system size for both local and nonlocal circuits. For subsystems, we find qualitative differences depending on their size. When the subsystem is larger than half of the full system, the equilibration timescales are again independent of the system size for both local and nonlocal circuits and the entanglement asymmetry grows monotonically in time. Conversely, when the subsystems are smaller than half of the full system, we show that the entanglement asymmetry is nonmonotonic in time and that it equilibrates in a time proportional to the quantum-information scrambling time, providing a physical intuition. As a consequence, the subsystem-equilibration time depends on the locality of interactions, scaling linearly and logarithmically in the system size, respectively, for local and nonlocal RUCs. Our work confirms the entanglement asymmetry as a versatile and computable probe of symmetry in many-body physics and yields a phenomenological overview of entanglement-asymmetry evolution in typical nonintegrable dynamics.

DOI
10.1103/m3np-p5xj

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We study the dynamics of entanglement asymmetry in random unitary circuits (RUCs). Focusing on a local U ( 1 ) charge, we consider symmetric initial states evolved by both local one-dimensional circuits and geometrically nonlocal RUCs made of two-qudit gates. We compute the entanglement asymmetry of subsystems of arbitrary size, analyzing the relaxation timescales. We show that the entanglement asymmetry of the whole system approaches its stationary value in a time independent of the system size for both local and nonlocal circuits. For subsystems, we find qualitative differences depending on their size. When the subsystem is larger than half of the full system, the equilibration timescales are again independent of the system size for both local and nonlocal circuits and the entanglement asymmetry grows monotonically in time. Conversely, when the subsystems are smaller than half of the full system, we show that the entanglement asymmetry is nonmonotonic in time and that it equilibrates in a time proportional to the quantum-information scrambling time, providing a physical intuition. As a consequence, the subsystem-equilibration time depends on the locality of interactions, scaling linearly and logarithmically in the system size, respectively, for local and nonlocal RUCs. Our work confirms the entanglement asymmetry as a versatile and computable probe of symmetry in many-body physics and yields a phenomenological overview of entanglement-asymmetry evolution in typical nonintegrable dynamics.
We investigate the subsystem entanglement asymmetry in random quantum automaton ensembles, which are generated by permuting the basis states in the Hilbert space and applying global phase shifts. We compute the ensemble average of the U(1)U(1)U(1) subsystem asymmetry in different connectivity geometries, showing that the late-time limit of the ensemble associated to a 2-local circuit geometry coincides with the all-to-all ensemble average. By focusing on different subsystem sizes, we demonstrate that, similarly to Haar-random circuits, the system locally symmetrizes. However, in sharp contrast to the Haar-random setting, the scale at which symmetrization happens depends on the initial state, a phenomenon we associate with the interplay of conservation of the participation entropy and the uniform exploration of charge sectors. Additionally, we connect the growth of the subsystem asymmetry to the subsystem coherence and show that their growth is characterized by the same symmetrization scale.
Entanglement asymmetry provides a quantitative measure of symmetry breaking in many-body quantum states. Focusing on inhomogeneous U(1)U(1)U(1) charges, such as dipole and multipole moments, we show that the typical asymmetry is bounded by a universal fraction of its maximal value. Multipole charges naturally arise in systems with Hilbert-space fragmentation, where the dynamics splits into exponentially many disconnected sectors. Using the commutant algebra formalism, we generalize entanglement asymmetry to account for fragmentation. While the asymmetry grows logarithmically for conventional symmetries, it can scale extensively in fragmented systems and distinguish classical from quantum fragmentation. We derive general upper bounds for the asymmetry and identify states that saturate them. To study the typical behavior of the asymmetry, we consider the ensemble of random matrix product states. By identifying the bond dimension with an effective time parameter, we qualitatively reproduce recent results on asymmetry dynamics in random quantum circuits, suggesting a universal behavior for the asymmetry of U(1)U(1)U(1) charges in local ergodic systems.
The finite parts of a large, locally interacting many-body system prepared out of equilibrium eventually equilibrate. Characterizing the underlying mechanisms of this process and its timescales, however, is particularly hard as it requires one to decouple universal features from observable-specific ones. Recently, additional insight was gained by studying how certain symmetries of the dynamics that are broken by the initial state are restored at the level of the reduced state of a given subsystem. This provides a high-level, observable-independent probe. Until now, this idea has been applied to the restoration of internal symmetries, e.g., U(1) symmetries related to charge conservation. Here we show that the same logic can be applied to the restoration of space-time symmetries, and hence can be used to characterize the relaxation of fully generic systems. We illustrate this idea by considering the paradigmatic example of “generic” many-body dynamics, i.e., a local random unitary circuit, where our method leads to exact results. We show that the restoration of translation symmetry in these systems only happens on timescales that are proportional to the subsystem's volume. In fact, for large enough subsystems, the time of symmetry restoration becomes initial-state independent (as long as the latter breaks the symmetry at time zero) and coincides with the thermalization time. For intermediate subsystems, however, one can observe the “quantum Mpemba effect,” where the state of the system restores a symmetry faster if it is initially more asymmetric. We provide an exact characterization of this effect in a nonintegrable system.
A bstract The entanglement asymmetry measures the extent to which a symmetry is broken within a subsystem of an extended quantum system. Here, we analyse this quantity in Haar random states for arbitrary compact, semi-simple Lie groups, building on and generalising recent results obtained for the U(1) symmetric case. We find that, for any group, the average entanglement asymmetry vanishes in the thermodynamic limit when the subsystem is smaller than its complement. When the subsystem and its complement are of equal size, the entanglement asymmetry jumps to a finite value, indicating a sudden transition of the subsystem from a fully symmetric state to one devoid of any symmetry. For larger subsystem sizes, the entanglement asymmetry displays a logarithmic scaling with a coefficient fixed by the dimension of the group. We also investigate the fluctuations of the entanglement asymmetry, which tend to zero in the thermodynamic limit. We check our findings against exact numerical calculations for the SU(2) and SU(3) groups. We further discuss their implications for the thermalisation of isolated quantum systems and black hole evaporation.
Abstract Entanglement asymmetry (EA) has emerged as a powerful tool for characterizing symmetry breaking in quantum many-body systems. In this Letter, we explore how symmetry is dynamically broken through the lens of EA in two distinct scenarios: a non-symmetric Hamiltonian quench and a non-symmetric random quantum circuit, with a particular focus on U(1) symmetry. In the former case, symmetry remains broken in the subsystem at late times, whereas in the latter case, the symmetry is initially broken and subsequently restored, consistent with the principles of quantum thermalization. Notably, the growth of EA exhibits unexpected overshooting behavior at early times in both contexts, contrasting with the behavior of charge variance. We also consider dynamics of non-symmetric initial states under the symmetry-breaking evolution. Due to the competition of symmetry-breaking in both the initial state and Hamiltonian, the early-time EA can increase and decrease, while quantum Mpemba effects remain evident despite the weak symmetry-breaking in both settings.
The physics of information scrambling in quantum many-body systems is intimately related to thermalisation and emergence of chaos. However, its standard characterisation through bipartite entanglement or operator growth remains inherently coarse-grained, obscuring the spatiotemporal anatomy of how quantum information flows between different regions of the system and across various length scales. In this work, we develop a theory for the dynamics of local information in local random unitary circuits which resolves the fine-grained microscopic structures of local information flow in such systems. Using the framework of information lattice which systematises the information content at each length scale within each subsystem, we identify a distribution of length scales at which information resides and obtain the dynamics of the distribution. For Haar-random circuits with infinite local Hilbert-space dimensions, we map this dynamics onto an exact classical stochastic process which reveals that this distribution is described by a Tracy-Widom form which moves ballistically in time ($ \propto t$) toward larger scales, accompanied by a ∼t1/3\sim t^{1/3}∼t1/3 broadening. We also find the same qualitative behaviour for random Clifford circuits acting on qubits. The similar scaling behaviour in two rather different settings hints strongly towards the universality of our results. In the case of Clifford circuits, we develop a phenomenological Markov process for the dynamics of the stabiliser generators in a specific gauge which confirms the Tracy-Widom distribution and the t1/3t^{1/3}t1/3 scaling of the fluctuations. Ultimately, our results establish the universal properties of the dynamics of local quantum information in a length scale-resolved fashion and provide a possible route towards bridging exact microscopic theories for information dynamics with emergent hydrodynamic descriptions of information flow.