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arXiv:2607.16278

Number Fluctuations and Entanglement-Spectrum Participation in Monitored Free Fermions

Enso O. Torres Alegre

quant-phSubmitted Jul 8, 2026• Announced Jul 10, 2026• 0 citations
v1 · Jul 8, 2026v2 · Jul 21, 2026
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Abstract

Monitored free-fermion chains show a crossover in entanglement scaling as the measurement rate is increased, from a subextensive weak-monitoring regime to an area law at strong monitoring. I test, using trajectory-resolved correlation-matrix simulations of chains up to $L=96$, whether two diagnostics built from the single-particle entanglement spectrum add practical value: the bipartite particle-number fluctuation $F_A=\sum_kν_k(1-ν_k)$ and a participation-style effective number of entangling modes $\mathcal{M}_A=\exp[-\sum_k w_k\ln w_k]$, with $w_k\proptoν_k(1-ν_k)$. Both quantities track the crossover, decreasing monotonically with measurement probability and losing system-size dependence in the strongly monitored regime. However, at the level of trajectory- and time-averaged steady-state values, $\mathcal{M}$ is almost a deterministic function of $F$: a pooled curve captures 99.7\% of its variance across all measurement rates, so $\mathcal{M}$ carries little independent information in this model. Its main advantage is statistical: the trajectory-to-trajectory coefficient of variation of $\mathcal{M}$ is up to a factor of about 2 smaller than that of the entropy or $F$ at strong monitoring. A model comparison of subsystem scaling at $L=48$--$96$ further shows that, at weak monitoring, a pure logarithmic law is not statistically preferred at these sizes; a small residual quasi-extensive component and a drifting effective logarithmic coefficient remain. I conclude that $F$ is the natural experimentally motivated companion to the entropy, while $\mathcal{M}$ is best viewed as a variance-reduced numerical summary of largely the same information, not an independent diagnostic.

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Monitored free-fermion chains show a crossover in entanglement scaling as the measurement rate is increased, from a subextensive weak-monitoring regime to an area law at strong monitoring. I test, using trajectory-resolved correlation-matrix simulations of chains up to L=96L=96L=96, whether two diagnostics built from the single-particle entanglement spectrum add practical value: the bipartite particle-number fluctuation FA=∑kνk(1−νk)F_A=\sum_kν_k(1-ν_k)FA​=∑k​νk​(1−νk​) and a participation-style effective number of entangling modes MA=exp⁡[−∑kwkln⁡wk]\mathcal{M}_A=\exp[-\sum_k w_k\ln w_k]MA​=exp[−∑k​wk​lnwk​], with wk∝νk(1−νk)w_k\proptoν_k(1-ν_k)wk​∝νk​(1−νk​). Both quantities track the crossover, decreasing monotonically with measurement probability and losing system-size dependence in the strongly monitored regime. However, at the level of trajectory- and time-averaged steady-state values, M\mathcal{M}M is almost a deterministic function of FFF: a pooled curve captures 99.7% of its variance across all measurement rates, so M\mathcal{M}M carries little independent information in this model. Its main advantage is statistical: the trajectory-to-trajectory coefficient of variation of M\mathcal{M}M is up to a factor of about 2 smaller than that of the entropy or FFF at strong monitoring. A model comparison of subsystem scaling at L=48L=48L=48--969696 further shows that, at weak monitoring, a pure logarithmic law is not statistically preferred at these sizes; a small residual quasi-extensive component and a drifting effective logarithmic coefficient remain. I conclude that FFF is the natural experimentally motivated companion to the entropy, while M\mathcal{M}M is best viewed as a variance-reduced numerical summary of largely the same information, not an independent diagnostic.
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 ααα. For α≲1α\lesssim 1α≲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 LLL is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small α≲1/2α\lesssim 1/2α≲1/2. For α>3/2α> 3/2α>3/2, the EE is in the area-law phase, namely, no scaling with LLL, for any monitoring strength. For 1<α≲3/21 < α\lesssim 3/21<α≲3/2, we identify an ααα-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.
Continuous monitoring of one-dimensional free fermionic systems can generate phenomena reminiscent of quantum criticality, such as logarithmic entanglement growth, algebraic correlations, and emergent conformal invariance, but in a nonequilibrium setting. However, whether these signatures reflect a genuine phase of nonequilibrium quantum matter or persist only over finite length scales is an active area of research. We address this question in a free fermionic chain subject to continuous monitoring of lattice-site occupations. An unraveling phase φ interpolates between measurement schemes, corresponding to different stochastic unravelings of the same Lindblad master equation; for φ = 0 , measurements disentangle lattice sites, while for φ = π / 2 they act as unitary random noise, yielding volume-law steady-state entanglement. Using replica Keldysh field theory, we obtain a nonlinear sigma model describing the long-wavelength physics. This analysis shows that for 0 ≤ φ < π / 2 , entanglement ultimately obeys an area law, but only beyond the exponentially large scale ln ( l φ , * ) ∼ J / [ γ cos ( φ ) ] , where J is the hopping amplitude and γ the measurement rate. Resolving l φ , * in numerical simulations is difficult for γ / J → 0 or φ → π / 2 . However, the theory also predicts that critical-like behavior appears below a crossover scale that grows only algebraically in J / γ , making it numerically accessible. Our simulations confirm these predictions, establishing the absence of measurement- or unraveling-induced entanglement transitions in this model.
Entanglement fluctuations associated with the Schrödinger evolution of wave functions offer a unique perspective on various fundamental issues ranging from quantum thermalization to state preparation in quantum devices. Very recently, a subset of the present authors showed that, in a class of free-fermion lattice models and interacting spin chains, entanglement dynamics enters into a new regime at long time, with entanglement probes displaying persistent temporal fluctuations, whose statistics falls into the seemingly disparate paradigm of mesoscopic fluctuations in condensed-matter physics. This motivates us to revisit here entanglement dynamics of a canonical bosonic model in many-body physics, i.e., a coupled harmonic oscillator chain. We find that when the system is driven out of equilibrium, the long-time entanglement dynamics exhibits strictly the same statistical behaviors as those of free-fermion models. Specifically, irrespective of entanglement probes and microscopic parameters, the statistical distribution of entanglement fluctuations is flanked by asymmetric tails: sub-Gaussian for upward fluctuations and sub-Gamma for downward; moreover, the variance exhibits a crossover from the scaling ∼ 1 / L to ∼ L A 3 / L 2 as the subsystem size L A increases ( L the total system size). This insensitivity to the particle statistics, dubbed boson-fermion universality, is contrary to the common wisdom that statistical phenomena of a many-body nature depend strongly on particle statistics. Together with our previous work, the present study indicates rich fluctuation phenomena in long-time entanglement dynamics that await in-depth exploration.
Entanglement phase transitions driven by quantum measurements have emerged as a central paradigm in open quantum many-body physics. Such phase transitions are well established for systems with finite local Hilbert-space dimensions, such as qubits and fermions, while their realization in bosonic systems with unbounded local occupation numbers remains poorly understood. Even in the absence of interactions, number states of bosons are intrinsically non-Gaussian, preventing the use of standard correlation-matrix approaches. To address this problem, we develop a replica-free Keldysh field-theoretic framework that expresses the Renyi entropy of bosonic systems initialized in on-site Fock states in terms of permanents of matrices constructed from single-particle Green's functions. Applying this framework to a continuously monitored one-dimensional cross-stitch lattice conditioned on the no-click trajectory, we uncover a transition from volume-law to logarithmic entanglement scaling. We show that the transition is controlled by a restructuring of the non-Hermitian spectrum that changes the number of long-lived modes from extensive to finite. In the strongly monitored regime, bosons dynamically condense into a microscopic number of slowest-decaying modes, producing logarithmic entanglement scaling, whereas an extensive manifold of long-lived modes at weak monitoring gives rise to volume-law entanglement. Our results establish a distinct mechanism for measurement-induced entanglement transitions in free bosonic systems and provide a computationally efficient diagnostic of the measurement-induced bosonic condensation.
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.
For one-dimensional non-interacting complex fermions, we compute numerically the probability distribution of the change in the entanglement entropy (EE) after saturation, resulting from a single measurement of the occupation number by using different measurements protocols. In the thermodynamic limit, the system is in the area-law phase for any monitoring strength, however we can also study the distribution in the critical phase characterized by a logarithmic scaling of the EE with system size by considering a sufficiently weak monitoring strength. For a quantum state diffusion protocol, where the change in EE is defined between two consecutive time steps, the distribution, Gaussian for weak monitoring, gradually develops symmetric exponential tails. For strong monitoring, the core turns from Gaussian to strongly peaked at zero suggesting the dominance of quantum Zeno effect. For the quantum jump and the projective measurement protocols, we observe clear deviations from Gaussianity characterized by broader and asymmetric tails, exponential for positive values of the change, and a peak at zero, likely a precursor of Zeno effect, that increases with the system size and the monitoring strength. Intriguingly, the distribution is spatially inhomogeneous. For sites around the boundary separating the subsystems defining the EE, the distribution is close to Gaussian with a broad support while for the rest of sites has asymmetric exponential tails and a much narrower support. As the monitoring strength increases, the full distribution is controlled by the boundary sites.