We explore, both analytically and numerically, the quantum dynamics of a many-body free-fermion system subjected to local density measurements. We begin by extending the mapping to the nonlinear sigma-model (NLSM) field theory for the case of finite evolution time T and different classes of initial states, which lead to different NLSM boundary conditions. The analytical formalism is then used to study how quantum correlations gradually develop, with increasing T , from those determined by the initial state towards their steady-state form. The analytical results are confirmed by numerical simulations for several types of initial states. We further consider the long-time limit, when the system in d + 1 space-time dimensions becomes quasi-one-dimensional, and analyze the scaling of the “localization” time (which is simultaneously the purification time and the charge-sharpening time for this class of problems). The analytical predictions for scaling properties are fully confirmed by numerical simulations in a d = 2 model around the measurement-induced phase transition. We use this dynamical approach to determine numerically the measurement-induced transition point and the associated correlation-length critical exponent.
Quantum science and engineering increasingly target the exploitation and control of large-scale dynamical systems. As system size grows, novel phenomena and quantum advantages emerge, but the resulting many-body dynamics become correspondingly harder to monitor and characterize. In particular, the underlying Hamiltonian often lacks symmetries, and the induced many-body dynamics quickly become hard to track. Focusing on one of the most important quantum features---entanglement---we demonstrate that, for a broad class of common Hamiltonian models, one can design effective witnesses that detect both the presence and structure of entanglement over relatively long evolution times. Inspired by stabilizer-based simulation, we track the evolution of generalized stabilizers of the underlying state and use them to construct linear entanglement witnesses. Through numerical experiments on Heisenberg-type Hamiltonians with up to twelve qubits, we show that the required number of measurement settings can be kept at a modest scale as the system size increases, and entanglement detection remains effective over evolution times that significantly exceed those accessible to naive approaches. The corresponding calculations can be carried out largely symbolically. Furthermore, unlike accurate state simulation, the stabilizer-based design of entanglement witnesses can be performed in a much more coarse-grained, digitized manner. We hope these results will stimulate further exploration of the quantum control of large systems.
We present a theoretical and numerical study of measurement-induced phase transitions and quantum Zeno dynamics in continuously monitored open many-body systems. Moving beyond closed unitary systems, we investigate the competition between chaotic unitary evolution and continuous weak measurements by employing the stochastic Schrödinger equation (quantum jump Monte Carlo) formalism and non-hermitian effective Hamiltonians. Our analysis examines the transition from area-law to volume-law entanglement scaling, mutual information properties, and the critical scaling hypothesis near the measurement-rate threshold. Furthermore, we derive original analytical estimates for the quantum correlation damping decrement in the regime of strong Zeno suppression and validate the theoretical findings using stochastic trajectory numerical simulation protocols with finite-size scaling.
We consider the statistical properties of eigenstates of the time-evolution operator in chaotic many-body quantum systems. Our focus is on correlations between eigenstates that are specific to spatially extended systems and that characterize entanglement dynamics and operator spreading. In order to isolate these aspects of dynamics from those arising as a result of local conservation laws, we consider Floquet systems in which there are no conserved densities. The correlations associated with scrambling of quantum information lie outside the standard framework established by the eigenstate thermalization hypothesis (ETH). In particular, ETH provides a statistical description of matrix elements of local operators between pairs of eigenstates, whereas the aspects of dynamics we are concerned with arise from correlations among sets of four or more eigenstates. We establish the simplest correlation function that captures these correlations and discuss features of its behavior that are expected to be universal at long distances and low energies. We also propose a maximum-entropy ansatz for the joint distribution of a small number n of eigenstates. In the case n = 2 , this ansatz reproduces ETH. For n = 4 it captures both the growth with time of entanglement between subsystems, as characterized by the purity of the time-evolution operator, and also operator spreading, as characterized by the behavior of the out-of-time-order correlator. We test these ideas by comparing results from Monte Carlo sampling of our ansatz with exact diagonalization studies of Floquet quantum circuits. Published by the American Physical Society 2024
Quantum circuits under sufficiently weak monitoring purify on a timescale
TP exponentially long in the system size. This slowness underlies a universal purification dynamics, whose quantitative description has so far required the replica trick, with a delicate analytic continuation. We show that monitored Clifford circuits on
L qudits of prime dimension
q bypass this construction entirely: in the scaling limit at fixed
x=t/TP(L), purification reduces to the Markovian decay of the density-matrix rank, an exactly solvable death process descending from infinity. We compute the full scaling functions in compact form: all Rényi entropies collapse onto a universal curve
⟨S(x)⟩. Exact stabilizer simulations at
q=2,3,5 confirm the predictions, with no fitting parameter for the global model and
TP as the only fitted scale for local brick-wall circuits. Also, the replica problem amounts to a tilted version of the same Markov process, in agreement with exact computations from the Clifford commutant. Finally, the quantization of the rank leaves two hallmarks that distinguish Clifford dynamics from generic monitored circuits: the entropy fluctuations saturate at short scaled times
x→0 to an
O(1) variance, instead of vanishing, and observables develop a temporal modulation periodic in
logqx, which cannot be captured by the replica approach.
We study an exactly solvable model of monitored dynamics in a system of
N spin-
1/2 particles with pairwise all-to-all noisy interactions, where each spin is continuously weakly measured along a random direction. Using the replica trick to incorporate the Born-rule weighting of measurement outcomes, we obtain an exact large-
N description of purification and of the statistics of local observables. We find that the nature of the phase transition strongly depends on the number
n of replicas: non-perturbative logarithmic corrections appear in the physically relevant
n→1 limit and destroy the purifying phase present at finite integer
n. As a consequence, the purification time of an initially mixed state is always exponentially long in the system size, even at arbitrarily large measurement rate.