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arXiv:2604.04375

Measurement-enhanced entanglement in a monitored superconducting chain

Rui-Jing Guo, Ji-Yao Chen, Zhi-Yuan Wei

quant-phcond-mat.stat-mechcond-mat.supr-conSubmitted Apr 6, 2026• Announced Apr 7, 2026• 0 citations• Revised Aug 23, 2026 (v2)
v1 · Apr 6, 2026v2 · Aug 23, 2026
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Abstract

A common view in monitored quantum dynamics is that local measurements suppress entanglement growth. We show that this intuition can fail in a one-dimensional spinful fermionic chain governed by a BCS Hamiltonian with pairing strength $Δ$ and subject to continuous, on-site, spin-resolved charge measurements at rate $γ$. Using free-fermion simulations and quasiparticle analysis, we show that pairing suppresses entanglement growth, while measurements suppress pairing. Their competition yields measurement-enhanced entanglement: for $Δ>0$, the steady-state entanglement $\mathcal{S}_s$ increases with $γ$ over a finite interval $0<γ<γ_{\mathrm{peak}}$. This occurs because stronger measurements suppress pairing correlations, which would otherwise suppress entanglement growth. Using a nonlinear sigma-model calculation and free-fermion simulations, we provide evidence that for $Δ>0$ and small but finite $γ$, the steady-state entanglement scales as $\mathcal{S}_s(L)\sim \ln^2 L$. This implies that, in this setting, measurement-enhanced entanglement does not persist in the thermodynamic limit.

Comments
5 pages of main text (4 figures) and 9 pages of supplemental material (2 figures); companion paper to arXiv:2607.07835. Comments are welcome

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A common view in monitored quantum dynamics is that local measurements suppress entanglement growth. We show that this intuition can fail in a one-dimensional spinful fermionic chain governed by a BCS Hamiltonian with pairing strength ΔΔΔ and subject to continuous, on-site, spin-resolved charge measurements at rate γγγ. Using free-fermion simulations and quasiparticle analysis, we show that pairing suppresses entanglement growth, while measurements suppress pairing. Their competition yields measurement-enhanced entanglement: for Δ>0Δ>0Δ>0, the steady-state entanglement Ss\mathcal{S}_sSs​ increases with γγγ over a finite interval 0<γ<γpeak0<γ<γ_{\mathrm{peak}}0<γ<γpeak​. This occurs because stronger measurements suppress pairing correlations, which would otherwise suppress entanglement growth. Using a nonlinear sigma-model calculation and free-fermion simulations, we provide evidence that for Δ>0Δ>0Δ>0 and small but finite γγγ, the steady-state entanglement scales as Ss(L)∼ln⁡2L\mathcal{S}_s(L)\sim \ln^2 LSs​(L)∼ln2L. This implies that, in this setting, measurement-enhanced entanglement does not persist in the thermodynamic limit.
We study the entanglement dynamics of a one-dimensional spinful sss-wave superconductor subjected to local continuous measurements. In the rare-measurement regime, we derive a replicated Keldysh non-linear sigma model (NLSM) to describe the steady-state entanglement. Within this field-theoretic framework, the interplay between measurement dephasing and superconducting pairing constrains the low-energy, long-wavelength fluctuations to an SO(R)\mathrm{SO(R)}SO(R) target manifold. A one-loop renormalization-group analysis shows that the theory develops a weak-anti-localization flow, stabilizing a critical phase with super-logarithmic scaling of the steady-state entanglement. Our field-theoretic results explain the numerical evidence reported in the companion Letter [arXiv:2604.04375] and demonstrate that such a critical phase can emerge in a one-dimensional topologically trivial superconductor without relying on topological protection.
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.
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=96L=96L=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)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. However, for trajectory- and time-averaged steady-state values, M\mathcal{M}M is nearly a deterministic function of FFF: a pooled curve explains 99.7\% of its variance across all rates, so M\mathcal{M}M carries little independent information. Its main advantage is statistical: its trajectory-to-trajectory coefficient of variation is up to a factor of ∼2\sim2∼2 smaller than that of the entropy or FFF under strong monitoring. Subsystem-scaling comparisons at L=48L=48L=48--969696 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, FFF remains the natural experimentally motivated companion to the entropy, whereas M\mathcal{M}M is best viewed as a variance-reduced numerical summary of essentially the same information.
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.
We utilize the concept of a measurement-induced entanglement transition to analyze the interplay and competition of processes that generate and destroy entanglement in a one-dimensional quantum spin chain evolving under a locally noisy and disordered Hamiltonian. We employ continuous measurements of variable strength to induce a transition from volume to area-law scaling of the steady-state entanglement entropy. While static background disorder systematically reduces the critical measurement strength, this critical value depends nonmonotonically on the strength of nonstatic noise. According to the extracted finite-size scaling exponents, the universality class of the transition is independent of the noise and disorder strength. We interpret the results in terms of the effect of static and nonstatic disorder on the intricate dynamics of the entanglement generation rate due to the Hamiltonian in the absence of measurement, which is fully reflected in the behavior of the critical measurement strength. Our results establish a firm connection between this entanglement growth and the steady-state behavior of the measurement-controlled systems, which therefore can serve as a tool to quantify and investigate features of transient entanglement dynamics in complex many-body systems via a steady-state phase transition.
Recent years have witnessed major progress in the understanding of monitored quantum dynamics, an open quantum setting where the environment is a measurement apparatus that probes the system locally. For the resulting quantum trajectory, unitary evolution and local measurements have competing tendencies in the building of quantum correlations: while the coherent interactions in the former increase the entanglement, the resolving nature of the local measurement collapses the state of the system. Here, the authors study the interplay of monitoring and system dynamics by focusing on the monitored quantum Ising chain as the measurement strength is varied, going from the regime of weak detection, corresponding to the limit of infinite small jumps, to the case of post-selected no-jump case. A sharp transition is found between a logarithmic law phase to an area law phase for the entanglement entropy, which occurs at the same value of the measurement rate in the two limiting situations.