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Entropy

Entanglement Crossovers Under Time-Dependent U(1) Symmetry Breaking in Monitored Quantum Circuits

Terrence Lingzhu Zhao, Zhou Shangnan

Published Oct 13, 2026• 0 citations
Publisher

Abstract

Conservation laws reshape quantum information dynamics by restricting how scrambling can occur. In monitored circuits, this raises a natural dynamical question: when a symmetry constraint is gradually relaxed, how quickly do the entanglement dynamics forget the symmetric regime? We address this question in one-dimensional monitored random circuits by driving the two-qubit gate ensemble from exact U(1) conservation toward unconstrained Haar dynamics. The response is not instantaneous: the entanglement retains a finite-depth memory of the conserving dynamics and crosses over toward the unconstrained behavior over a rate-dependent time window. Operational 25%, 50%, and 75% crossover times all move to later circuit depths for slower drives. At p=0.4, the same rate ordering is also observed through L=32 in a finite-time matrix-product-state consistency check with a limited trajectory ensemble. By contrast, the weak fixed perturbations studied here, ϵ(t)=ϵ0≤0.1, remain close to the conserving reference over the same finite observation window. Within this finite-time window, the driven crossover is therefore not reproduced merely by waiting under the fixed perturbations tested. Purification dynamics from a maximally mixed state exhibit the same qualitative response. We therefore interpret the results as a finite-time dynamical crossover from symmetry-constrained dynamics toward less constrained scrambling, rather than as evidence for a universal crossover exponent or a microscopic noise law. The study provides a controlled setting for asking how long symmetry-dependent information-processing behavior persists as a conservation law is deliberately weakened.

DOI
10.3390/e28101120

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Conservation laws reshape quantum information dynamics by restricting how scrambling can occur. In monitored circuits, this raises a natural dynamical question: when a symmetry constraint is gradually relaxed, how quickly do the entanglement dynamics forget the symmetric regime? We address this question in one-dimensional monitored random circuits by driving the two-qubit gate ensemble from exact U(1) conservation toward unconstrained Haar dynamics. The response is not instantaneous: the entanglement retains a finite-depth memory of the conserving dynamics and crosses over toward the unconstrained behavior over a rate-dependent time window. Operational 25%, 50%, and 75% crossover times all move to later circuit depths for slower drives. At p=0.4, the same rate ordering is also observed through L=32 in a finite-time matrix-product-state consistency check with a limited trajectory ensemble. By contrast, the weak fixed perturbations studied here, ϵ(t)=ϵ0≤0.1, remain close to the conserving reference over the same finite observation window. Within this finite-time window, the driven crossover is therefore not reproduced merely by waiting under the fixed perturbations tested. Purification dynamics from a maximally mixed state exhibit the same qualitative response. We therefore interpret the results as a finite-time dynamical crossover from symmetry-constrained dynamics toward less constrained scrambling, rather than as evidence for a universal crossover exponent or a microscopic noise law. The study provides a controlled setting for asking how long symmetry-dependent information-processing behavior persists as a conservation law is deliberately weakened.
We study the critical properties of random quantum circuits with a U(1)U(1)U(1) symmetry subject to local projective measurements that explicitly break this symmetry. We find that, at the measurement-induced phase transition, symmetry-breaking measurements act as a relevant perturbation at large scales, leading to the same universal critical properties as the corresponding monitored random circuit with non-symmetric unitary dynamics. In particular, we consider monitored U(1)U(1)U(1)-symmetric Haar-random circuits in the limit of large local Hilbert-space dimension, where the trajectory-averaged entanglement entropy can be exactly obtained in terms of a classical statistical mechanics model. In this model, the charge associated with the conservation law follows a symmetric simple exclusion process, in which symmetry-breaking measurements correspond to disordered defects that create and destroy charges. We prove that the charge correlation length remains finite for any measurement rate, ruling out a charge-sharpening transition, in contrast to the case of symmetry-preserving measurements. We further support our predictions at finite local Hilbert-space dimension through numerical finite-size scaling analyses of the entanglement transition in monitored U(1)U(1)U(1)-symmetric Haar and stabilizer random circuits.
We study the critical properties of random quantum circuits with a U(1)U(1)U(1) symmetry subject to local projective measurements that explicitly break this symmetry. We find that, at the measurement-induced phase transition, symmetry-breaking measurements act as a relevant perturbation at large scales, leading to the same universal critical properties as the corresponding monitored random circuit with non-symmetric unitary dynamics. In particular, we consider monitored U(1)U(1)U(1)-symmetric Haar-random circuits in the limit of large local Hilbert-space dimension, where the trajectory-averaged entanglement entropy can be exactly obtained in terms of a classical statistical mechanics model. In this model, the charge associated with the conservation law follows a symmetric simple exclusion process, in which symmetry-breaking measurements correspond to disordered defects that create and destroy charges. We prove that the charge correlation length remains finite for any measurement rate, ruling out a charge-sharpening transition, in contrast to the case of symmetry-preserving measurements. We further support our predictions at finite local Hilbert-space dimension through numerical finite-size scaling analyses of the entanglement transition in monitored U(1)U(1)U(1)-symmetric Haar and stabilizer random circuits.
Monitored quantum circuits can exhibit an entanglement transition as a function of the rate of measurements, stemming from the competition between scrambling unitary dynamics and disentangling projective measurements.We study how entanglement dynamics in nonunitary quantum circuits can be enriched in the presence of charge conservation, using a combination of exact numerics and a mapping onto a statistical mechanics model of constrained hard-core random walkers.We uncover a charge-sharpening transition that separates different scrambling phases with volume-law scaling of entanglement, distinguished by whether measurements can efficiently reveal the total charge of the system.We find that while Rényi entropies grow sub-ballistically as ffiffi t p in the absence of measurement, for even an infinitesimal rate of measurements, all average Rényi entropies grow ballistically with time ∼t.We study numerically the critical behavior of the charge-sharpening and entanglement transitions in U(1) circuits, and show that they exhibit emergent Lorentz invariance and can also be diagnosed using scalable local ancilla probes.Our statistical mechanical mapping technique readily generalizes to arbitrary Abelian groups, and offers a general framework for studying dissipatively stabilized symmetry-breaking and topological orders.
Entanglement equilibration need not mark the end of quantum-information dynamics in quantum thermalization. We show that the complete Schmidt spectra of a fixed tripartition can become stationary while its multipartite correlation structure, distinguished by local-unitary invariants, continues to evolve. This separation occurs exactly within a finite window in self-dual kicked-Ising dynamics and asymptotically in Haar-random circuits. We further demonstrate an independently tunable timescale for multipartite dynamics using quantum-error-correcting codes. When coupled to a locally scrambling bath, ordinary entanglement approaches equilibrium through rapid local scrambling, while weak logical--bath coupling drives a slower reorganization of multipartite correlations. These results reveal a deeper distinction than the ordering of equilibration times: bipartite entanglement and multipartite correlation structure can probe distinct physical channels of quantum-information dynamics, rather than merely different stages of the same relaxation process.
We study how conservation laws shape the spreading of quantum coherence in many-body dynamics. Focusing on U(1)U(1)U(1)-symmetric random circuits, charge-and-dipole conserving circuits, as well as ergodic Hamiltonian dynamics, we probe coherences both globally, via the participation entropy, and locally, via the relative entropy of coherence. Combining exact vector evolution, matrix product state simulations, and replica tensor networks methods, we find that conservation laws replace the logarithmic saturation of unconstrained circuits with slow hydrodynamic relaxation of the global coherence measures. Locally, symmetry-constrained circuits show a clean rise-peak-fall structure whose peak time grows algebraically with subsystem size. In contrast, ergodic Hamiltonians broaden the peak into an extended plateau at larger subsystems, highlighting a qualitatively distinct mechanism. Coherence thus emerges as a sensitive probe of symmetry-constrained thermalization, linking quantum resource dynamics to many-body transport.
Abstract Monitored quantum circuits offer great perspectives for exploring the interplay of quantum information and complex quantum dynamics. These systems exhibit entanglement and purification phase transitions, along with various symmetry-enforced and ordered non-equilibrium phases. The central question is whether these phases can persist in real-world noisy devices. We study the fate of the symmetry-enforced absorbing state and charge-sharpening transitions in the presence of noise, and establish that noise results in coherent and incoherent symmetry-breaking effects. The coherent effects blur the distinction between phases, turning sharp transitions into crossovers, but states far from phase boundaries largely retain their essential character. We find, corrective feedback and postselected measurements can mitigate noise, stabilizing the absorbing and charge-sharp phases. Hence, if challenges like postselection are addressed, errors do not prevent the observation of symmetry-enforced phases in noisy intermediate-scale quantum (NISQ) devices. Additionally, we propose a symmetry-based method to characterize noise and gate fidelity.