Continuous measurement of quantum systems provides a standard route to quantum trajectories through the successive acquisition of information which further results in measurement back-action. In this work, we harness this back-action as a resource for global $U(1)$ symmetry restoration where continuous measurement is combined with a $U(1)$-preserving unitary evolution. Starting from a $U(1)$ symmetry-broken initial state, we simulate quantum trajectories generated by continuous measurements of both global and local observables. We show that under global monitoring, states containing superpositions of distant charge sectors restore symmetry faster than those involving nearby sectors. We establish the universality of this behavior across different measurement protocols. Finally, we demonstrate that local monitoring can further accelerate symmetry restoration for certain states that relax slowly under global monitoring.
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13 pages including end matter and supplementary material, 8 figures
Continuous measurement of quantum systems provides a standard route to quantum trajectories through the successive acquisition of information which further results in measurement back-action. In this work, we harness this back-action as a resource for global U(1) symmetry restoration where continuous measurement is combined with a U(1)-preserving unitary evolution. Starting from a U(1) symmetry-broken initial state, we simulate quantum trajectories generated by continuous measurements of both global and local observables. We show that under global monitoring, states containing superpositions of distant charge sectors restore symmetry faster than those involving nearby sectors. We establish the universality of this behavior across different measurement protocols. Finally, we demonstrate that local monitoring can further accelerate symmetry restoration for certain states that relax slowly under global monitoring.
Monitoring a quantum system can profoundly alter its dynamical properties, leading to non-trivial emergent phenomena. In this work, we demonstrate that dynamical measurements strongly influence the evolution of symmetry in many-body quantum systems. Specifically, we demonstrate that monitored systems governed by non-Hermitian dynamics exhibit a quantum Mpemba effect, where systems with stronger initial asymmetry relax faster to a symmetric state. Crucially, this phenomenon is purely measurement-induced: in the absence of measurements, we find states where the corresponding unitary evolution does not display any Mpemba effect. Furthermore, we uncover a novel measurement-induced symmetry restoration mechanism: below a critical measurement rate, the symmetry remains broken, but beyond a threshold, it is fully restored in the thermodynamic limit-along with the emergence of the quantum Mpemba effect.
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.
This thesis considers quantum systems which interact with their surrounding environment. Specifically we consider the underlying stochastic dynamics which arise from the back-action of measurements performed in the environment. These quantum trajectories can be tracked by continuously monitoring the environment, as is now possible in modern experiments. We consider phenomena understood at the average level of the system state density matrix, and explore these at this more information rich level. This is especially relevant for experimental setups and implementation of quantum technologies. Firstly, we consider how systems exhibiting metastability behave at the level of stochastic quantum trajectories. In doing so we make comparisons to classical theories of metastability, where trajectories exhibit fast relaxation into distinct phases and slow transitions between them. We also illustrate when this phenomenology does not hold, with instead the emergence of a slow manifold in the form of a decoherence free subspace. We also highlight the useful role that quantum reset processes can play in the analysis of stochastic quantum trajectories. The dynamics of the stochastic quantum trajectories depends on the choice of environmental measurements. There are many choices which leave the average dynamics unchanged, but give different statistical ensembles of quantum trajectories. This gauge freedom of the average dynamics is well known. We derive the analogous result at the level of quantum trajectory ensembles; giving the measurement protocols which leave the generator of the stochastic trajectory dynamics unchanged. We also extend this to include joint trajectories of conditional system states and their associated measurement records, either full records or coarse-grained. Using these results on gauge freedoms of quantum trajectories, we consider the conditions under which the trajectory dynamics exhibit a weak unitary symmetry. We then consider under what conditions the joint state of the system and environment exhibit symmetric dynamics, including under various levels of dephasing. We find that the joint dynamics always features a separable symmetry directly related to that of the quantum master equation, but for quantum trajectories the corresponding symmetry is present only if the counting measurement satisfies certain conditions.
A quantum trajectory is the natural response of a quantum system subject to external perturbations due to continuous indirect measurement. We completely characterize the asymptotic behavior of continuously monitored quantum systems in finite dimensions and show that generically, spontaneous irreversible localization transitions on the level of individual realizations occur, where the evolution becomes effectively constrained to one of the irreducible components of the total Hilbert space. More generally, localization can be either complete, where the strongest possible confinement is achieved, or incomplete, where localization terminates prematurely. On the trajectory level, symmetries and conserved quantities are no longer respected and localization transitions occur concurrently with violations of ergodicity. As a result, a generalized update rule emerges that effectively projects the system onto one of several possible time evolutions. The update comes equipped with a generalized Born rule that assigns probabilities to these irreversible events. Spontaneous transitions thus occur probabilistically and can deviate considerably from the behavior of the ensemble. In particular, time and ensemble average no longer commute which gives rise to global violations of ergodicity, while on a local level, ergodicity is restored. We explicitly illustrate the asymptotic behavior of continuously monitored systems in a series of examples demonstrating stabilization of many-body scar states and generation of Bell states from local measurement. Finally, we present two algorithms, one based on simultaneous block diagonalization and one based on quantum trajectories to identify all the minimal orthogonal subspaces of the Lindblad equation and all the extremal stationary states which are supported on them.
We study the critical properties of random quantum circuits with a 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)-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)-symmetric Haar and stabilizer random circuits.
We study the critical properties of random quantum circuits with a 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)-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)-symmetric Haar and stabilizer random circuits.