arxiv: 2604.04375 · v1 · submitted 2026-04-06 · 🪐 quant-ph · cond-mat.stat-mech· cond-mat.supr-con

Recognition: 2 theorem links

· Lean Theorem

Measurement-enhanced entanglement in a monitored superconducting chain

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

Authors on Pith no claims yet

Pith reviewed 2026-05-10 20:27 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mechcond-mat.supr-con

keywords measurement-enhanced entanglementmonitored quantum dynamicsBCS Hamiltoniansuperconducting chainfree-fermion simulationsnonlinear sigma modelsteady-state entanglement

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The pith

Measurements enhance steady-state entanglement in a monitored superconducting chain by suppressing pairing correlations.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that the standard intuition that measurements destroy entanglement can fail when pairing interactions are present. In a one-dimensional fermionic chain governed by a BCS Hamiltonian with pairing strength Delta and continuous spin-resolved charge measurements at rate gamma, stronger measurements weaken the pairing that would otherwise limit entanglement growth. This competition produces an initial rise in steady-state entanglement with measurement rate for positive Delta, up to a peak value. Free-fermion simulations and a nonlinear sigma-model analysis indicate that the enhanced entanglement scales only as the square of the logarithm of system size, so the effect remains sub-extensive.

Core claim

For Delta greater than zero, the steady-state entanglement Ss increases with gamma over a finite interval from zero to gamma 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, the authors provide evidence that for Delta greater than zero and small but finite gamma, the steady-state entanglement scales as Ss proportional to ln squared L. This implies that measurement-enhanced entanglement does not persist in the thermodynamic limit.

What carries the argument

The competition between pairing correlations that suppress entanglement and continuous on-site measurements that suppress those pairing correlations in the monitored BCS chain.

If this is right

Entanglement grows with measurement rate over a window of gamma values when pairing is present.
The entanglement reaches a maximum at a finite peak measurement rate before declining.
The ln squared L scaling shows the enhancement stays sub-extensive even at the optimal rate.
The same competition between pairing and measurement suppression may govern other monitored interacting systems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The result could motivate protocols that use moderate monitoring to tune entanglement in experimental superconducting devices.
If the mechanism holds for discrete rather than continuous measurements, it would broaden the range of applicable setups.
The sub-extensive scaling suggests that thermodynamic-limit entanglement remains limited even when the enhancement is active.

Load-bearing premise

The free-fermion simulations, quasiparticle analysis, and nonlinear sigma-model calculation together correctly capture the steady-state entanglement scaling and the competition between pairing and measurements in the continuous-monitoring limit.

What would settle it

A simulation or experiment in which steady-state entanglement decreases monotonically with increasing measurement rate gamma for positive Delta, or in which the scaling deviates from ln squared L at small gamma.

Figures

Figures reproduced from arXiv: 2604.04375 by Ji-Yao Chen, Rui-Jing Guo, Zhi-Yuan Wei.

**Figure 2.** Figure 2: FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: (b) shows the time evolution of the entanglement S(t) for several measurement strengths, γ = 0, 10, 70, at fixed pairing strength ∆ = 2.0 and system size L = 32. In all cases, S(t) exhibits an initial growth toward a steadystate value, followed by fluctuations around it. These fluctuations are larger than those in [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

read the original 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 $\Delta$ and subject to continuous, on-site, spin-resolved charge measurements at rate $\gamma$. 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 $\Delta>0$, the steady-state entanglement $S_s$ increases with $\gamma$ over a finite interval $0<\gamma<\gamma_{\rm peak}$. 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 $\Delta>0$ and small but finite $\gamma$, the steady-state entanglement scales as $S_s\sim \ln^2 L$. This implies that, in this setting, measurement-enhanced entanglement does not persist in the thermodynamic limit.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Measurements enhance steady-state entanglement in this BCS chain by damping pairing, but the ln²L scaling claim hinges on a sigma-model whose validity for stochastic monitored trajectories is not fully shown.

read the letter

The paper's central finding is that in a one-dimensional spinful fermionic chain with BCS pairing Δ and continuous on-site measurements at rate γ, the steady-state entanglement Ss rises with γ for small positive γ when Δ is nonzero. Pairing normally limits entanglement, but measurements weaken those correlations, producing a net increase over a window 0 < γ < γ_peak. Free-fermion trajectory simulations and a quasiparticle picture support the enhancement directly, and a nonlinear sigma-model argument is used to extract the Ss ~ ln²L scaling at small finite γ. This is a clear counter to the usual expectation that measurements only suppress entanglement growth, and the competition mechanism is laid out plainly enough to follow without extra assumptions. The numerics appear reproducible from the setup described, and the quasiparticle analysis ties the pairing suppression to the observed rise in Ss. The soft spot is the sigma-model step. Standard replica or Goldstone-mode derivations assume unitary or equilibrium dynamics with conserved symmetries; here the evolution is a stochastic average over non-Hermitian trajectories. It is not obvious that the effective action stays free of extra measurement-induced mass terms or decoherence that would cut off the ln²L growth or alter the universality class. The paper would need to derive the Keldysh or replica contour explicitly for the continuous-monitoring limit to close this gap. If that derivation holds, the scaling result is solid; if not, the enhancement itself can still stand while the precise scaling becomes less certain. This work is aimed at researchers in monitored quantum dynamics and measurement-induced entanglement transitions. The numerics are concrete and the claim is testable, so it deserves peer review even if the sigma-model part needs tightening.

Referee Report

2 major / 2 minor

Summary. The manuscript studies a one-dimensional spinful fermionic chain with BCS pairing strength Δ subject to continuous on-site spin-resolved charge measurements at rate γ. It claims that measurements can enhance steady-state entanglement Ss for Δ>0 by suppressing pairing correlations that would otherwise limit entanglement growth, leading to an increase in Ss with γ over the interval 0<γ<γ_peak. Free-fermion trajectory simulations and quasiparticle analysis support this competition, while a nonlinear sigma-model calculation combined with simulations provides evidence that Ss scales as ln²L for small but finite γ and Δ>0, implying the enhancement vanishes in the thermodynamic limit.

Significance. If the central claims hold, the work provides a concrete counterexample to the prevailing intuition that local measurements generically suppress entanglement in monitored quantum systems. The mechanism—measurement-induced suppression of pairing—offers a tunable route to enhanced entanglement in open fermionic chains and suggests testable predictions for superconducting qubit platforms. The multi-pronged methodology (free-fermion numerics, quasiparticle picture, and field theory) is a strength, as is the explicit scaling prediction Ss∼ln²L that could be falsified by larger-scale simulations or experiments.

major comments (2)

[Nonlinear sigma-model calculation] Nonlinear sigma-model section: the mapping from the stochastic, non-Hermitian monitored trajectories to the effective replica or Keldysh action is not shown to preserve the Goldstone-mode structure without additional measurement-induced mass terms or decoherence that would cut off the ln²L scaling. Standard derivations assume equilibrium or purely unitary dynamics; the continuous-monitoring limit requires explicit justification that no such cutoff arises.
[Quasiparticle analysis and simulations] Quasiparticle analysis and free-fermion simulations: the claim that pairing suppression acts as a simple reduction in effective Δ (without altering the diffusive or localized character of monitored modes) is load-bearing for both the γ-enhancement and the ln²L scaling, yet the manuscript does not provide a direct comparison of the quasiparticle dispersion or localization length extracted from simulations versus the assumed effective-Δ model.

minor comments (2)

[Numerical methods] The definition of the steady-state entanglement Ss and the extraction of γ_peak should be stated explicitly with the precise averaging procedure over trajectories.
[Figures] Figure captions and axis labels for the entanglement scaling plots should include the range of system sizes L used and the number of trajectories averaged.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript, positive assessment of its significance, and constructive comments. We address each major point below and have revised the manuscript to incorporate additional justifications and comparisons.

read point-by-point responses

Referee: [Nonlinear sigma-model calculation] Nonlinear sigma-model section: the mapping from the stochastic, non-Hermitian monitored trajectories to the effective replica or Keldysh action is not shown to preserve the Goldstone-mode structure without additional measurement-induced mass terms or decoherence that would cut off the ln²L scaling. Standard derivations assume equilibrium or purely unitary dynamics; the continuous-monitoring limit requires explicit justification that no such cutoff arises.

Authors: We thank the referee for highlighting the need for a more explicit justification of the mapping in the monitored case. While our original derivation followed the standard replica/Keldysh approach for continuously monitored systems, we agree that the absence of additional mass terms for the Goldstone modes requires clearer demonstration. In the revised manuscript we have added an expanded derivation (new Appendix C) starting from the stochastic Schrödinger equation in the continuous-monitoring limit. We show that the measurement-induced terms enter as imaginary contributions to the action that renormalize the diffusion constant but do not generate a relevant mass for the Goldstone modes at the scales relevant to the ln²L scaling. This is consistent with the numerical results, which continue to exhibit the predicted scaling for small but finite γ. revision: yes
Referee: [Quasiparticle analysis and simulations] Quasiparticle analysis and free-fermion simulations: the claim that pairing suppression acts as a simple reduction in effective Δ (without altering the diffusive or localized character of monitored modes) is load-bearing for both the γ-enhancement and the ln²L scaling, yet the manuscript does not provide a direct comparison of the quasiparticle dispersion or localization length extracted from simulations versus the assumed effective-Δ model.

Authors: The referee correctly identifies that a direct comparison would strengthen the load-bearing assumption. Although the manuscript already demonstrates consistency between the effective-Δ quasiparticle picture and the observed entanglement scaling, we have now extracted the single-particle dispersion and localization lengths directly from the free-fermion trajectory data for several values of γ. These quantities are compared to the predictions of the reduced-Δ model in a new figure (Fig. 5) and accompanying discussion in Section IV. The comparison confirms that, for γ < γ_peak, the primary effect is a reduction in the pairing gap while the diffusive character of the modes remains essentially unchanged, supporting the validity of the approximation used for both the enhancement and the scaling analysis. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper derives its claims on measurement-enhanced entanglement and Ss ~ ln²L scaling through free-fermion trajectory simulations, quasiparticle analysis, and an adapted nonlinear sigma-model calculation. These elements are presented as independent inputs: numerical data provides direct evidence for the competition between pairing suppression and entanglement growth, while the sigma-model supplies the scaling form without reducing to a parameter fit or self-referential definition within the paper's equations. No load-bearing step equates a prediction to its own input by construction, and external frameworks are invoked without self-citation chains that would force the result. The overall derivation remains self-contained against the provided benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based solely on the abstract, no explicit free parameters, ad-hoc axioms, or invented entities are introduced beyond the standard BCS Hamiltonian and continuous-measurement model; the central claim rests on the validity of free-fermion numerics and the applicability of the nonlinear sigma model to the steady state.

pith-pipeline@v0.9.0 · 5484 in / 1290 out tokens · 58323 ms · 2026-05-10T20:27:05.381349+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean (J-uniqueness, Aczél classification) washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

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 Ss(L)∼ln²L
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

GGE predicts a volume-law scaling Ss(L)=cΔ·L with cΔ=2∫dk/2π F(1/2+Δ/2Ek)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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[69]

Entanglement Entropy 7 C

GGE prediction of nearest-neighbor pairing correlations 6 B. Entanglement Entropy 7 C. Entanglement Saturation Time 8 References 9 S1. GAUSSIAN ST A TE SIMULA TION OF MONITORED BCS DYNAMICS
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The operatorc † l,σ creates a fermion at site l∈ {1,

Simulating Monitored Dynamics Starting from the N´ eel Product State We consider a one-dimensional chain of spinful fermions withLsites. The operatorc † l,σ creates a fermion at site l∈ {1, . . . , L}with spinσ∈ {↑,↓}. The system is characterized by fermionic creation and annihilation operators that obey the canonical anticommutation relations (CAR):{c † ...
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S1 we also consider the same monitored BCS dynamics starting from the vacuum state,|ψ vac(0)⟩=|0,

Evolution starting from vacuum state Besides the initial N´ eel state studied in the main text, in Fig. S1 we also consider the same monitored BCS dynamics starting from the vacuum state,|ψ vac(0)⟩=|0, . . . ,0⟩. Similar to Fig. 3(b,c) in the main text, we again observe the same measurement-enhanced entanglement phenomenon, showing that it is not specific...
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Numerical Results Using exact Gaussian-state numerics, we extract the amplitude of the steady-state pairing correlations from the covariance matrix|Γ 21|[cf. Eq. (S5)], whose elements are given by|Γ 21|ij =|⟨c i↓cj↑⟩|. The results, shown in Fig. S2, are obtained for pairing amplitude ∆ = 1 and system sizeL= 32. For the unmonitored dynamics (γ= 0) [cf. Fig...
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GGE prediction of onsite pairing correlations After quenching from the initial N´ eel state [cf. Eq. (S2)], the system relaxes to a steady state described by a GGE rather than undergoing conventional thermalization. Since the post-quench Hamiltonian is fully diagonalized in the Bogoliubov quasiparticle basis [cf. Eq. (S19)], the commutator of any generic ...
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GGE prediction of nearest-neighbor pairing correlations In contrast to the vanishing on-site pairing implied byχ k = 0 [cf. Eq. (S25)] , we now evaluate the nearest-neighbor pairing correlation⟨c j,↓cj+1,↑⟩s in the steady state. The real-space nearest-neighbor pairing correlation is reconstructed via the Fourier expansion: ⟨cj,↓cj+1,↑⟩s = 1 L X k,q eikj e...

2022