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arXiv:2306.16595 · Quantum

Free fermions under adaptive quantum dynamics

Vikram Ravindranath, Zhi-Cheng Yang, Xiao Chen

Published Apr 3, 2025• 5 citations
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Abstract

We study free fermion systems under adaptive quantum dynamics consisting of unitary gates and projective measurements followed by corrective unitary operations. We further introduce a classical flag for each site, allowing for an active or inactive status which determines whether or not the unitary gates are allowed to apply. In this dynamics, the individual quantum trajectories exhibit a measurement-induced entanglement transition from critical to area-law scaling above a critical measurement rate, similar to previously studied models of free fermions under continuous monitoring. Furthermore, we find that the corrective unitary operations can steer the system into a state characterized by charge-density-wave order. Consequently, an additional phase transition occurs, which can be observed at both the level of the quantum trajectory and the quantum channel. We establish that the entanglement transition and the steering transition are fundamentally distinct. The latter transition belongs to the parity-conserving (PC) universality class, arising from the interplay between the inherent fermionic parity and classical labelling. We demonstrate both the entanglement and the steering transitions via efficient numerical simulations of free fermion systems, which confirm the PC universality class of the latter.

DOI
10.22331/q-2025-04-03-1685

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We study free fermion systems under adaptive quantum dynamics consisting of unitary gates and projective measurements followed by corrective unitary operations. We further introduce a classical flag for each site, allowing for an active or inactive status which determines whether or not the unitary gates are allowed to apply. In this dynamics, the individual quantum trajectories exhibit a measurement-induced entanglement transition from critical to area-law scaling above a critical measurement rate, similar to previously studied models of free fermions under continuous monitoring. Furthermore, we find that the corrective unitary operations can steer the system into a state characterized by charge-density-wave order. Consequently, an additional phase transition occurs, which can be observed at both the level of the quantum trajectory and the quantum channel. We establish that the entanglement transition and the steering transition are fundamentally distinct. The latter transition belongs to the parity-conserving (PC) universality class, arising from the interplay between the inherent fermionic parity and classical labelling. We demonstrate both the entanglement and the steering transitions via efficient numerical simulations of free fermion systems, which confirm the PC universality class of the latter.
The intensely studied measurement-induced entanglement phase transition has become a hallmark of nonunitary quantum many-body dynamics. Usually, such a transition only appears at the level of each individual quantum trajectory, and is absent for the density matrix averaged over measurement outcomes. In this work, we introduce a class of adaptive random circuit models with feedback that exhibit transitions in both settings. After each measurement, a unitary operation is either applied or not depending on the measurement outcome, which steers the averaged density matrix towards a unique state above a certain measurement threshold. Interestingly, the transition for the density matrix and the entanglement transition in the individual quantum trajectory in general happen at different critical measurement rates. We demonstrate that the former transition belongs to the parity-conserving universality class by explicitly mapping to a classical branching-annihilating random-walk process.
We analyze the quantum trajectory dynamics of free fermions subject to continuous monitoring. For weak monitoring, we identify a novel dynamical regime of subextensive entanglement growth, reminiscent of a critical phase with an emergent conformal invariance. For strong monitoring, however, the dynamics favors a transition into a quantum Zeno-like area-law regime. Close to the critical point, we observe logarithmic finite size corrections, indicating a Berezinskii-Kosterlitz-Thouless mechanism underlying the transition. This uncovers an unconventional entanglement transition in an elementary, physically realistic model for weak continuous measurements. In addition, we demonstrate that the measurement aspect in the dynamics is crucial for whether or not a phase transition takes place.
© 2019 authors. Published by the American Physical Society. We define dynamical universality classes for many-body systems whose unitary evolution is punctuated by projective measurements. In cases where such measurements occur randomly at a finite rate p for each degree of freedom, we show that the system has two dynamical phases: "entangling" and "disentangling." The former occurs for p smaller than a critical rate pc and is characterized by volume-law entanglement in the steady state and "ballistic" entanglement growth after a quench. By contrast, for p>pc the system can sustain only area-law entanglement. At p=pc the steady state is scale invariant, and in 1+1D, the entanglement grows logarithmically after a quench. To obtain a simple heuristic picture for the entangling-disentangling transition, we first construct a toy model that describes the zeroth Rényi entropy in discrete time. We solve this model exactly by mapping it to an optimization problem in classical percolation. The generic entangling-disentangling transition can be diagnosed using the von Neumann entropy and higher Rényi entropies, and it shares many qualitative features with the toy problem. We study the generic transition numerically in quantum spin chains and show that the phenomenology of the two phases is similar to that of the toy model but with distinct "quantum" critical exponents, which we calculate numerically in 1+1D. We examine two different cases for the unitary dynamics: Floquet dynamics for a nonintegrable Ising model, and random circuit dynamics. We obtain compatible universal properties in each case, indicating that the entangling-disentangling phase transition is generic for projectively measured many-body systems. We discuss the significance of this transition for numerical calculations of quantum observables in many-body systems.
We study a free fermion model where two sets of non-commuting non-projective measurements stabilize area-law entanglement scaling phases of distinct topological order. We show the presence of a topological phase transition that is of a different universality class than that observed in stroboscopic projective circuits. In the presence of unitary dynamics, the two topologically distinct phases are separated by a region with sub-volume scaling of the entanglement entropy. We find that this entanglement transition is well identified by a combination of the bipartite entanglement entropy and the topological entanglement entropy. We further show that the phase diagram is qualitatively captured by an analytically tractable non-Hermitian model obtained via post-selecting the measurement outcome. Finally we introduce a partial-post-selection continuous mapping, that uniquely associates topological indices of the non-Hermitian Hamiltonian to the distinct phases of the stochastic measurement-induced dynamics.
Measurement-induced phase transitions are nonequilibrium transitions between phases characterized by distinct entanglement scaling behaviors, driven by the competition between unitary dynamics and measurements. Despite recent numerical efforts, how quenched disorder affects these transitions remains unclear. In this work, we study a ddd-dimensional noninteracting fermionic system subject to both quenched disorder and continuous monitoring of the local particle density, and derive an effective field theory describing its long-time universal behaviors. We find that the system is governed by the same nonlinear sigma model as in the case of clean monitored fermions, with disorder entering only through a modification of model parameters. This result suggests that the presence or absence of a measurement-induced phase transition is unaffected by the introduction of disorder. In spatial dimension d=1d=1d=1, the system exhibits only an area law phase and no transition, whereas in d>1d>1d>1, a transition persists between an area×\times×log law phase and an area law phase, with the critical measurement rate suppressed by disorder. Numerical simulation confirms the absence of MIPT in d=1d=1d=1, and the linear shift of the critical measurement rate by disorder scattering rate in higher dimensions predicted by analytical theory.
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 Z2{\mathbb{Z}}_{2}Z2​ 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.