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arXiv:2506.10873

Asymptotic Fate of Continuously Monitored Quantum Systems

Finn Schmolke

Published Jun 12, 2025• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2506.10873

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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.

Abstract The dynamics of a quantum system, undergoing unitary evolution and continuous monitoring, can be described in term of quantum trajectories. Although the averaged state fully characterizes expectation values, the entire ensemble of stochastic trajectories goes beyond simple linear observables, keeping a more attentive description of the entire dynamics. Here we go beyond the Lindblad dynamics and study the probability distribution of the expectation value of a given observable over the possible quantum trajectories. The measurements are applied to the entire system, having the effect of projecting the system into a product state. We develop an analytical tool to evaluate this probability distribution at any time t. We illustrate our approach by analyzing two paradigmatic examples: a single qubit subjected to magnetization measurements, and a free hopping particle subjected to position measurements.

Quantum evolution is generally expected to drive a quantum many-body system toward equilibrium. This expectation is often justified by the Hilbert-space ergodicity of generic quantum dynamics, namely, the idea that pure-state evolution explores Hilbert space uniformly up to physical constraints. Such a statement can be made rigorous by requiring the associated state ensemble to form the Haar-random ensemble, or its more structured generalization, the Scrooge ensemble. In this Letter, we report the emergence of exact Hilbert-space ergodicity in a continuously monitored quantum many-body system. For any target density matrix

σ

, we construct a continuously monitored system for which we rigorously prove that the Scrooge ensemble of

σ

is the unique late-time equilibrium distribution of quantum trajectories. Remarkably, this requires only that the jump operators in the monitoring form a deformed unitary 1-design, a seemingly much weaker condition than full ergodicity. We numerically demonstrate our predictions by simulating continuously monitored systems whose equilibrium states are thermal states. Our results establish a rigorous mechanism for the emergence of Hilbert-space ergodicity and provide a practical route for its investigation on quantum devices.

Monitored quantum system have sparked great interest in recent years due to the possibility of observing measurement-induced phase transitions (MIPTs) in the full-counting statistics of quantum trajectories. Here, we investigate the dynamics of the Lipkin-Meshkov-Glick model, composed of

N

all-to-all interacting spins

1/2

, under a weak external monitoring. In the thermodynamic limit, we derive a set of semiclassical stochastic equations describing the evolution of the expectation values of global spin observables. Our results show that the limit $N\ensuremath{\rightarrow}\ensuremath{\infty}$ does not commute with the long-time limit: while for any finite

N

the average over trajectories is expected to converge towards a trivial steady state, in the thermodynamic limit a MIPT appears. The transition is not affected by postselection issues, as it is already visible at the level of ensemble averages. We derive a quantitative theoretical picture explaining the nature of the transition within our semiclassical approach.

The thermodynamics of quantum systems driven out of equilibrium has attracted increasing attention in the last decade, in connection with quantum information and statistical physics, and with a focus on non-classical signatures. While a first approach can deal with average thermodynamics quantities over ensembles, in order to establish the impact of quantum and environmental fluctuations during the evolution, a continuous quantum measurement of the open system is required. Here, we provide an introduction to the general theoretical framework to establish and interpret the thermodynamics for quantum systems whose nonequilibrium evolution is continuously monitored. We review the formalism of quantum trajectories and its consistent application to the thermodynamic scenario, where primary quantities such as work, heat, and entropy production can be defined at the stochastic level. The connection to irreversibility and fluctuation theorems is also discussed together with some recent developments, and we provide some simple examples to illustrate the general theoretical framework.

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.

Measurements are able to fundamentally affect quantum dynamics. We here show that a continuously measured quantum many-body system can undergo a spontaneous transition from asynchronous stochastic dynamics to noise-free stable synchronization at the level of single trajectories. We formulate general criteria for this quantum phenomenon to occur and demonstrate that the number of synchronized realizations can be controlled from none to all. We additionally find that ergodicity is typically broken, since time and ensemble averages may exhibit radically different synchronization behavior. We further introduce a quantum type of multiplexing that involves individual trajectories with distinct synchronization frequencies. Measurement-induced synchronization appears as a genuine nonclassical form of synchrony that exploits quantum superpositions.