We study the entanglement entropy of the quantum trajectories of a free fermion chain under continuous monitoring of local occupation numbers. We propose a simple theory for entanglement entropy evolution from disentangled and highly excited initial states. It is based on generalized hydrodynamics and the quasi-particle pair approach to entanglement in integrable systems. We test several quantitative predictions of the theory against extensive numerics and find good agreement. In particular, the volume law entanglement is destroyed by the presence of arbitrarily weak measurement.We study the entanglement entropy of the quantum trajectories of a free fermion chain under continuous monitoring of local occupation numbers. We propose a simple theory for entanglement entropy evolution from disentangled and highly excited initial states. It is based on generalized hydrodynamics and the quasi-particle pair approach to entanglement in integrable systems. We test several quantitative predictions of the theory against extensive numerics and find good agreement. In particular, the volume law entanglement is destroyed by the presence of arbitrarily weak measurement.

Aug 23, 2019281 citations

Entanglement transition due to particle losses in a monitored fermionic chain

Rafael Diogo Soares, Youenn Le Gal, Marco Schirò

Recently, there has been interest in the dynamics of monitored quantum systems using linear jump operators related to the creation or annihilation of particles. Here, we study the dynamics of the entanglement entropy under quantum jumps that induce local particle losses in a model of free fermions with hopping and ${\mathbb{Z}}_{2}$ pairing. We solve the nonunitary dynamics using the recently developed Faber polynomial method and explore the different steady-state entanglement regimes by tuning the pairing strength, thus interpolating between monitored free fermions coherently driven by a particle-number-conserving Hamiltonian and a parity-conserving one. In the absence of pairing, all quantum trajectories approach the vacuum at long times, with the entanglement entropy showing nonmonotonic behavior over time that we capture with a phenomenological quasiparticle ansatz. In this regime, quantum jumps play a key role, and we highlight this by exactly computing their waiting-time distribution. On the other hand, the interplay between losses and pairing gives rise to quantum trajectories with entangled steady states. We show that by tuning the system parameters, a measurement-induced entanglement transition occurs, where the entanglement entropy scaling changes from logarithmic to area law. We compare this transition with the one derived in the no-click limit and observe qualitative agreement in most of the phase diagram. Furthermore, the statistics of entanglement gain and loss are analyzed to better understand the impact of the linear jump operators.Recently, there has been interest in the dynamics of monitored quantum systems using linear jump operators related to the creation or annihilation of particles. Here, we study the dynamics of the entanglement entropy under quantum jumps that induce local particle losses in a model of free fermions with hopping and ${\mathbb{Z}}_{2}$ pairing. We solve the nonunitary dynamics using the recently developed Faber polynomial method and explore the different steady-state entanglement regimes by tuning the pairing strength, thus interpolating between monitored free fermions coherently driven by a particle-number-conserving Hamiltonian and a parity-conserving one. In the absence of pairing, all quantum trajectories approach the vacuum at long times, with the entanglement entropy showing nonmonotonic behavior over time that we capture with a phenomenological quasiparticle ansatz. In this regime, quantum jumps play a key role, and we highlight this by exactly computing their waiting-time distribution. On the other hand, the interplay between losses and pairing gives rise to quantum trajectories with entangled steady states. We show that by tuning the system parameters, a measurement-induced entanglement transition occurs, where the entanglement entropy scaling changes from logarithmic to area law. We compare this transition with the one derived in the no-click limit and observe qualitative agreement in most of the phase diagram. Furthermore, the statistics of entanglement gain and loss are analyzed to better understand the impact of the linear jump operators.

arXiv:2408.03700Physical review. B./Physical review. BFeb 26, 202516 citations

Entanglement growth in the dark intervals of a locally monitored free-fermion chain

Giovanni Di Fresco, Youenn Le Gal, Davide Valenti +2 more

We consider a free fermionic chain with monitoring of the particle density on a single site of the chain and study the entanglement dynamics of quantum jump trajectories. We show that the entanglement entropy grows in time towards a stationary state which display volume law scaling of the entropy, in stark contrast with both the unitary dynamics after a local quench and the no-click limit corresponding to full post-selection. We explain the extensive entanglement growth as a consequence of the peculiar distribution of quantum jumps in time, which display superpoissonian waiting time distribution characterised by a bunching of quantum jumps followed by long dark intervals where no-clicks are detected, akin to the distribution of fluorescence light in a driven atom. We show that the presence of dark intervals is the key feature to explain the effect and that by increasing the number of sites which are monitored the volume law scaling gives away to the Zeno effect and its associated area law.We consider a free fermionic chain with monitoring of the particle density on a single site of the chain and study the entanglement dynamics of quantum jump trajectories. We show that the entanglement entropy grows in time towards a stationary state which display volume law scaling of the entropy, in stark contrast with both the unitary dynamics after a local quench and the no-click limit corresponding to full post-selection. We explain the extensive entanglement growth as a consequence of the peculiar distribution of quantum jumps in time, which display superpoissonian waiting time distribution characterised by a bunching of quantum jumps followed by long dark intervals where no-clicks are detected, akin to the distribution of fluorescence light in a driven atom. We show that the presence of dark intervals is the key feature to explain the effect and that by increasing the number of sites which are monitored the volume law scaling gives away to the Zeno effect and its associated area law.

arXiv:2411.13667QuantumAug 3, 20260 citations

Entanglement transitions and quantum bifurcations under continuous long-range monitoring

Angelo Russomanno, G. Piccitto, Davide Rossini

We study the asymptotic bipartite entanglement entropy of the quantum trajectories of a free-fermionic system, when subject to a continuous nonlocal monitoring. The measurements are described by Gaussian-preserving two-point operators, whose strength decays as a power law with exponent $\ensuremath{\alpha}$. Different behaviors of the entanglement entropy with the system size emerge: for $\ensuremath{\alpha}$ below a given threshold value a volume-law behavior sets in, while for larger $\ensuremath{\alpha}$ we observe a transition from subvolume to area law, whose exact location depends on the measurements rate and on the presence of a Hamiltonian dynamics. We also consider the expectation probability distribution of the measurement operators, and find that this distribution features a transition from a unimodal to a bimodal shape. We discuss the possible connections between this qualitative change of the distribution and the entanglement transition points.We study the asymptotic bipartite entanglement entropy of the quantum trajectories of a free-fermionic system, when subject to a continuous nonlocal monitoring. The measurements are described by Gaussian-preserving two-point operators, whose strength decays as a power law with exponent $\ensuremath{\alpha}$. Different behaviors of the entanglement entropy with the system size emerge: for $\ensuremath{\alpha}$ below a given threshold value a volume-law behavior sets in, while for larger $\ensuremath{\alpha}$ we observe a transition from subvolume to area law, whose exact location depends on the measurements rate and on the presence of a Hamiltonian dynamics. We also consider the expectation probability distribution of the measurement operators, and find that this distribution features a transition from a unimodal to a bimodal shape. We discuss the possible connections between this qualitative change of the distribution and the entanglement transition points.

arXiv:2307.05685Physical review. B./Physical review. BSep 18, 202331 citations

Entanglement phases and phase transitions in monitored free fermion systems of localization

Yu-Jun Zhao, Xuyang Huang, Yong Zhang +2 more

In recent years, the presence of local potentials has significantly enriched and diversified the entanglement patterns in monitored free fermion systems. In our approach, we employ the stochastic Schrödinger equation to simulate a one-dimensional spinless fermion system under continuous measurement and local potentials. By averaging the steady-state entanglement entropy over many quantum trajectories, we investigate its dependence on measurement and localization parameters. We used a phenomenological model to interpret the numerical results, and the results show that the introduction of local potentials does not destroy the universality class of the entanglement phase transition, and that the phase boundary is jointly characterized by the measurement process and the localization mechanism. This work offers a new perspective on the characterization of the entanglement phase boundary arising from the combined effects of measurement and localization, and provides criteria for detecting this novel phase transition in cold atom systems, trapped ions, and quantum dot arrays.In recent years, the presence of local potentials has significantly enriched and diversified the entanglement patterns in monitored free fermion systems. In our approach, we employ the stochastic Schrödinger equation to simulate a one-dimensional spinless fermion system under continuous measurement and local potentials. By averaging the steady-state entanglement entropy over many quantum trajectories, we investigate its dependence on measurement and localization parameters. We used a phenomenological model to interpret the numerical results, and the results show that the introduction of local potentials does not destroy the universality class of the entanglement phase transition, and that the phase boundary is jointly characterized by the measurement process and the localization mechanism. This work offers a new perspective on the characterization of the entanglement phase boundary arising from the combined effects of measurement and localization, and provides criteria for detecting this novel phase transition in cold atom systems, trapped ions, and quantum dot arrays.

arXiv:2509.09538Physical review. B./Physical review. BJan 16, 20262 citations

Entangled multiplets and spreading of quantum correlations in a continuously monitored tight-binding chain

Federico Carollo, Vincenzo Alba

We analyze the dynamics of entanglement in a paradigmatic noninteracting system subject to continuous monitoring of the local excitation densities. Recently, it was conjectured that the evolution of quantum correlations in such system is described by a semiclassical theory, based on entangled pairs of ballistically propagating quasiparticles and inspired by the hydrodynamic approach to unitary (integrable) quantum systems. Here, however, we show that this conjecture does not fully capture the complex behavior of quantum correlations emerging from the interplay between coherent dynamics and continuous monitoring. We unveil the existence of multipartite quantum correlations which are inconsistent with an entangled-pair structure and which, within a quasiparticle picture, would require the presence of larger multiplets. We also observe that quantum information is highly delocalized, as it is shared in a collective nonredundant way among adjacent regions of the many-body system. Our results shed light onto the behavior of correlations in quantum stochastic dynamics and further show that these may be enhanced by a (weak) continuous monitoring process.We analyze the dynamics of entanglement in a paradigmatic noninteracting system subject to continuous monitoring of the local excitation densities. Recently, it was conjectured that the evolution of quantum correlations in such system is described by a semiclassical theory, based on entangled pairs of ballistically propagating quasiparticles and inspired by the hydrodynamic approach to unitary (integrable) quantum systems. Here, however, we show that this conjecture does not fully capture the complex behavior of quantum correlations emerging from the interplay between coherent dynamics and continuous monitoring. We unveil the existence of multipartite quantum correlations which are inconsistent with an entangled-pair structure and which, within a quasiparticle picture, would require the presence of larger multiplets. We also observe that quantum information is highly delocalized, as it is shared in a collective nonredundant way among adjacent regions of the many-body system. Our results shed light onto the behavior of correlations in quantum stochastic dynamics and further show that these may be enhanced by a (weak) continuous monitoring process.

arXiv:2206.07806Physical review. B./Physical review. BDec 9, 202248 citations

{\mathbb{Z}}_{2}

pairing. We solve the nonunitary dynamics using the recently developed Faber polynomial method and explore the different steady-state entanglement regimes by tuning the pairing strength, thus interpolating between monitored free fermions coherently driven by a particle-number-conserving Hamiltonian and a parity-conserving one. In the absence of pairing, all quantum trajectories approach the vacuum at long times, with the entanglement entropy showing nonmonotonic behavior over time that we capture with a phenomenological quasiparticle ansatz. In this regime, quantum jumps play a key role, and we highlight this by exactly computing their waiting-time distribution. On the other hand, the interplay between losses and pairing gives rise to quantum trajectories with entangled steady states. We show that by tuning the system parameters, a measurement-induced entanglement transition occurs, where the entanglement entropy scaling changes from logarithmic to area law. We compare this transition with the one derived in the no-click limit and observe qualitative agreement in most of the phase diagram. Furthermore, the statistics of entanglement gain and loss are analyzed to better understand the impact of the linear jump operators.