We explore statistical fluctuations over the ensemble of quantum trajectories in a model of two-dimensional free fermions subject to projective monitoring of local charge across the measurement-induced phase transition. Our observables are the particle-number covariance between spatially separated regions, G A B , and the two-point density correlation function, C ( r ) . Our results exhibit a remarkable analogy to Anderson localization, with G A B corresponding to two-terminal conductance and C ( r ) to two-point conductance, albeit with different replica limits and unconventional symmetry class, geometry, and boundary conditions. In the delocalized phase, G A B exhibits “universal,” nearly Gaussian, fluctuations with variance of order unity. In the localized phase, we find a broad distribution of G A B with − ln G A B ¯ ∼ L (where L is the system size) and the variance var ( ln G A B ) ∼ L μ , and similarly for C ( r ) , with μ ≈ 0.5 . At the transition point, the distribution function of G A B becomes scale invariant and C ( r ) exhibits multifractal statistics, C q ( r ) ¯ ∼ r − q ( d + 1 ) − Δ q . We characterize the spectrum of multifractal dimensions Δ q . Our findings lay the groundwork for mesoscopic theory of monitored systems, paving the way for various extensions.
We explore statistical fluctuations over the ensemble of quantum trajectories in a model of two-dimensional free fermions subject to projective monitoring of local charge across the measurement-induced phase transition. Our observables are the particle-number covariance between spatially separated regions, G A B , and the two-point density correlation function, C ( r ) . Our results exhibit a remarkable analogy to Anderson localization, with G A B corresponding to two-terminal conductance and C ( r ) to two-point conductance, albeit with different replica limits and unconventional symmetry class, geometry, and boundary conditions. In the delocalized phase, G A B exhibits “universal,” nearly Gaussian, fluctuations with variance of order unity. In the localized phase, we find a broad distribution of G A B with − ln G A B ¯ ∼ L (where L is the system size) and the variance var ( ln G A B ) ∼ L μ , and similarly for C ( r ) , with μ ≈ 0.5 . At the transition point, the distribution function of G A B becomes scale invariant and C ( r ) exhibits multifractal statistics, C q ( r ) ¯ ∼ r − q ( d + 1 ) − Δ q . We characterize the spectrum of multifractal dimensions Δ q . Our findings lay the groundwork for mesoscopic theory of monitored systems, paving the way for various extensions.
We investigate the entanglement structure and wave function characteristics of continuously monitored free fermions with U(1) symmetry in two spatial dimensions. By deriving the exact fermion replica-quantum master equation, we line out two approaches: (i) a nonlinear sigma model analogous to disordered free fermions, resulting in an SU(R)-symmetric field theory of symmetry class AIII in (2+1) space-time dimensions, or (ii) for bipartite lattices, third quantization leading to a non-Hermitian SU(2R)-symmetric Hubbard model. Using exact numerical simulations, we explore the phenomenology of the entanglement transition in two-dimensional (2D) monitored fermions, examining entanglement entropy and wave function inverse participation ratio. At weak monitoring, we observe characteristic LlogL entanglement growth and multifractal dimension Dq=2, resembling a metallic Fermi liquid. Under strong monitoring, wave functions localize and the entanglement saturates towards an area law. Between these regimes, we identify a high-symmetry point exhibiting both entanglement growth indicative of emergent conformal invariance and maximal multifractal behavior. While this multifractal behavior aligns with the nonlinear sigma model of the Anderson transition, the emergent conformal invariance is an unexpected feature not typically associated with Anderson localization. These discoveries add a new dimension to the study of 2D monitored fermions and underscore the need to further explore the connection between nonunitary quantum dynamics in D dimensions and quantum statistical mechanics in D+1 dimensions.
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.
Repeated local measurements of quantum many-body systems can induce a phase transition in their entanglement structure. These measurement-induced phase transitions (MIPTs) have been studied for various types of dynamics, yet most cases yield quantitatively similar critical exponents, making it unclear how many distinct universality classes are present. Here, we probe the properties of the conformal field theories governing these MIPTs using a numerical transfer-matrix method, which allows us to extract the effective central charge, as well as the first few low-lying scaling dimensions of operators at these critical points for (1+1)-dimensional systems. Our results provide convincing evidence that the generic and Clifford MIPTs for qubits lie in different universality classes and that both are distinct from the percolation transition for qudits in the limit of large on-site Hilbert space dimension. For the generic case, we find strong evidence of multifractal scaling of correlation functions at the critical point, reflected in a continuous spectrum of scaling dimensions.
We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent α. For α≲1, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size L is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small α≲1/2. For α>3/2, the EE is in the area-law phase, namely, no scaling with L, for any monitoring strength. For 1<α≲3/2, we identify an α-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.
We study the statistical properties of a single two-level system (qubit) subject to repetitive ancilla-based measurements. This setup is a fundamental minimal model for exploring the intricate interplay between the unitary dynamics of the system and the nonunitary stochasticity introduced by quantum measurements, which is central to the phenomenon of measurement-induced phase transitions. We demonstrate that this “toy model” harbors remarkably rich dynamics, manifesting in the distribution function of the qubit's quantum states in the long-time limit. We uncover a compelling analogy with the phenomenon of Anderson localization, albeit governed by distinct underlying mechanisms. Specifically, the state distribution function of the monitored qubit, parameterized by a single angle on the Bloch sphere, exhibits diverse types of behavior familiar from the theory of Anderson transitions, spanning from complete localization to almost uniform delocalization, with fractality occurring between the two limits. By combining analytical solutions for various special cases with two complementary numerical approaches, we achieve a comprehensive understanding of the structure delineating the “phase diagram” of the model. We categorize and quantify the emergent regimes and identify two distinct phases of the monitored qubit: ergodic and nonergodic. Furthermore, we identify a genuinely localized phase within the nonergodic phase, where the state distribution functions consist of delta peaks, as opposed to the delocalized phase characterized by extended distributions. Identification of these phases and demonstration of transitions between them in a monitored qubit are our main findings. Published by the American Physical Society 2024
On finite system sizes, monitored one-dimensional free-fermion chains display a broad crossover in entanglement scaling as the measurement rate increases, from sub-extensive behavior at weak monitoring toward an area law at strong monitoring. Analytical field theory and recent large-scale simulations indicate that this apparent change is not a finite-rate transition in the thermodynamic limit. I test, using trajectory-resolved correlation-matrix simulations (chains up to L=96 and up to 128 Born-rule trajectories per parameter point), whether two quantities built from the single-particle entanglement spectrum are useful finite-size diagnostics: the bipartite particle-number fluctuation FA=∑kνk(1−νk) and a participation-style effective number of entangling modes, MA=exp[−∑kwklnwk], with wk∝νk(1−νk). Both quantities track the crossover. However, for trajectory- and time-averaged steady-state values, M is nearly a deterministic function of F: a pooled curve explains 99.7\% of its variance across all rates, so M carries little independent information. Its main advantage is statistical: its trajectory-to-trajectory coefficient of variation is up to a factor of ∼2 smaller than that of the entropy or F under strong monitoring. Subsystem-scaling comparisons at L=48--96 also show that a pure logarithmic law is not statistically preferred under weak monitoring; a small residual quasi-extensive component and a drifting logarithmic coefficient are instead consistent with a crossover rather than a critical phase. Thus, F remains the natural experimentally motivated companion to the entropy, whereas M is best viewed as a variance-reduced numerical summary of essentially the same information.