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Quantum trajectories and Page-curve entanglement dynamics

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Pith's one-line read A filled fermionic chain emptying into an empty reservoir under dephasing probes shows a full Page curve for trajectory-averaged entanglement, with growth, Page time, Page value, and decay all controlled by probe geometry and unraveling…

desk verdict Useful numerical catalog of Page-curve scalings in dephased fermionic chains, but the SUU update rule in the supplementary has an internal sign inconsistency that should be resolved before the numbers are taken at face value. read the letter →

arxiv 2501.12110 v1 pith:SXNOH6BD submitted 2025-01-21 cond-mat.stat-mech cond-mat.mes-hallcond-mat.quant-gashep-thquant-ph

Katha Ganguly , Preethi Gopalakrishnan , Atharva Naik , Bijay Kumar Agarwalla , Manas Kulkarni This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.mes-hallcond-mat.quant-gashep-thquant-ph
keywords Pagecurveentanglemententropyquantumtrajectoriesdephasingfreefermionsstatediffusionstochasticunitaryunravelingparticlecurrent
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper claims that the trajectory-averaged entanglement entropy between a filled finite fermionic system and an empty reservoir shows a complete Page curve — initial rise, peak at a Page time, then decay to zero — when the chain is subjected to dephasing probes. The full time dependence is not universal: the growth law, Page value, and decay law depend on whether only the system is probed (Probe-Clean) or the whole chain is probed (Probe-Probe), and on whether the Lindblad dynamics is unraveled into stochastic unitary evolution (SUU) or quantum state diffusion (QSD). Under SUU the entropy grows diffusively as $\sqrt{t}$ and the Page value is volume-law, while QSD monitoring slows growth to $\ln t$ or $\ln(\ln t)$ and produces sub-volume Page values. The paper also finds that before the Page time the entropy production rate is proportional to the particle current leaving the system under SUU. If correct, this gives a numerically tractable, experimentally relevant way to study entanglement dynamics of effectively interacting systems without simulating interactions directly.

What carries the argument

The central object is the $L \times N$ matrix $U(t)$ whose columns are the occupied single-particle orbitals of the Gaussian state on each trajectory; since $U^\dagger U = I_N$, the correlation matrix factors as $C^\xi_{ij}(t) = [U(t)U^\dagger(t)]_{ji}$, and the entanglement entropy of the $L_S$-site system follows from the eigenvalues $\lambda_k$ of the system block through $S^\xi(t) = -\sum_k [\lambda_k \log_2 \lambda_k + (1-\lambda_k)\log_2(1-\lambda_k)]$. The two unraveling protocols update $U$ differently: SUU multiplies by a diagonal matrix of onsite phase noises $e^{-i d\xi_i}$ (then reorthogonalizes), while QSD multiplies by a diagonal matrix involving $e^{d\xi_i + \frac{\gamma}{2}(2\langle n_i\rangle_t -1)dt}$ with feedback through $\langle n_i\rangle_t$. This machinery carries the argument because it reduces otherwise inaccessible trajectory-averaged entanglement to a stochastic single-particle evolution, and the different diagonal noise factors are what produce the different growth and decay laws in Table I.

What would settle it

Simulate the same domain-wall expansion with a genuinely interacting fermionic chain (for example, a nearest-neighbor interaction term) at comparable parameters and compare trajectory-averaged $S(t)$; if the growth exponents, Page-time scaling, and Page-value scalings differ from Table I, the dephasing-emulates-interactions premise is not the mechanism driving these Page curves.

Watch

Extended reading notes

Core claim

Starting from a domain-wall initial state — system fully filled, reservoir empty — the authors show that each quantum trajectory remains a Gaussian fermionic state, so the entanglement entropy of the system-reservoir split can be obtained from the eigenvalues of the system block of the single-particle correlation matrix $C^\xi = U U^\dagger$, where $U$ is the $L \times N$ isometry of occupied orbitals evolved stochastically. Averaging over trajectories yields a Page curve in every combination studied: Probe-Clean with SUU gives $\sqrt{t}$ growth and $\ln(1/t)$ decay, with $S_P \propto L_S$ and $t_P \sim L_S^2$; Probe-Clean with QSD gives $\ln t$ growth and the same logarithmic decay, with a sub-volume $S_P \propto \ln L_S$ that crosses over to an area law; Probe-Probe with SUU gives $\sqrt{t}$ growth and a power-law $t^{-0.4}$ decay; Probe-Probe with QSD gives $\ln(\ln t)$ growth and $1/\sqrt{t}$ decay. In the SUU protocol, $dS/dt$ is proportional to the reservoir particle current up to the Page time. These results are summarized in Table I and are interpreted as the entanglement dynamics of an effectively interacting fermionic gas expanding into vacuum (Probe-Clean) or of a monitored interacting system (Probe-Probe).

Load-bearing premise

Dephasing noise on a non-interacting fermionic chain behaves like genuine particle interactions for the purpose of entanglement growth, although the paper does not simulate an interacting model or vary the dephasing strength.

Editorial extensions

If this is right

  • Under SUU, the Page value $S_P$ grows linearly with $L_S$ and the Page time scales as $t_P \sim L_S^2$, so the full rising side of the Page curve is fixed by diffusive particle transport.
  • Under QSD, continuous weak monitoring slows entanglement growth and reduces the Page value: $S_P$ is sub-volume ($\propto \ln L_S$), and in the Probe-Clean case it crosses over to an area law as $L_S$ grows.
  • The decay side of the Page curve is faster when only the system is probed ($\ln(1/t)$) than when the whole chain is probed (power laws $t^{-0.4}$ or $1/\sqrt{t}$), because probes throughout the reservoir delay the emptying of the system.
  • Up to the Page time, the SUU entropy production rate is proportional to the particle current into the reservoir, giving a master-equation-accessible proxy for a purely trajectory-dependent quantity.
  • In the Probe-Probe with QSD case, the entanglement grows as $\ln(\ln t)$, an ultra-slow growth tied to monitoring every site.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the dephasing-emulates-interactions premise holds, the $t_P \sim L_S^2$ scaling and the $\sqrt{t}$ growth of SUU make a sharp prediction for cold-atom experiments: entanglement in an effectively interacting expansion should lag ballistic expansion and peak at a diffusive time set by $L_S^2/\gamma$.
  • The area-law crossover seen in the Probe-Clean QSD Page value at $L_S \approx 160$ suggests an effective entanglement phase transition as a function of system size and monitoring rate; a scaling collapse in $\gamma L_S$ would test whether it is universal.
  • The Supplementary bipartite result — saturation value follows a different, sub-volume scaling from the Page value — warns that steady-state bipartite entanglement measurements cannot be used to infer the Page value or Page time in this setup.
  • Since only $\gamma = 0.1$ is simulated, the Table I exponents should be checked as $\gamma$ is varied; if they change, the relevant control parameter may be a dimensionless ratio such as $\gamma L_S^2 / g$ rather than $\gamma$ alone.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. This manuscript studies the trajectory-resolved entanglement entropy S(t) of a filled fermionic system coupled to an empty reservoir, for two probe geometries (Probe-Clean and Probe-Probe) and two unraveling schemes (stochastic unitary unraveling and quantum state diffusion). The central results are the full Page-curve-like time dependence of S(t) with distinct growth and decay regimes, the system-size scaling of the Page value, and the proportionality between entropy-production rate and particle current before the Page time, all summarized in Table I and Figs. 2-4.

Significance. The paper addresses a genuinely nonlinear-in-state quantity (entanglement entropy) using Gaussian trajectory numerics, so the reported curves are outputs of a well-defined protocol rather than inputs. If the scaling laws and their protocol/geometry dependence are correct, this is a useful free-fermion benchmark for Page-curve dynamics in open systems, and the current-entropy connection is a testable statement. The presentation, however, currently leaves key implementation details and fitting procedures undocumented, no code is provided, and the extension to generic interacting systems is asserted rather than demonstrated.

major comments (5)
  1. [Supplementary Material, Sec. I A, Eqs. (S16) and (S19)] There is an internal inconsistency in the definition of the SUU update. Eq. (S16) defines M with diagonal entries exp(-i dξ + γ/2 dt), while Eq. (S19) implements the update using only exp(-i dξ) with no γ-dependent factor, and the text states that QR decomposition is needed because M contains e^{γdt} factors, which is true for S16 but false for S19. Since the SUU columns of Table I are generated by this update, please state explicitly which update was simulated, correct the sign of the γ/2 dt term in S16 (the Itô expansion of e^{-i dξ} under variance γ dt produces a -γ/2 dt term, not +), and indicate whether QR was actually applied; if S19 is the implemented rule, no QR is required and the surrounding derivation is misleading.
  2. [Supplementary Material, Sec. I B, Eq. (S72)] The QSD update written for the U matrix is not manifestly norm-preserving: the M in Eq. (S72) contains state-dependent diagonal factors e^{dξ+γ/2(2⟨n⟩-1)dt} that are not unitary, yet no normalization or QR re-orthonormalization step is mentioned. Since the correlation-matrix formula and Eq. (5) assume U†U=I_N, please specify how the isometry constraint is enforced at each time step; an unnormalized U would directly affect all QSD rows of Table I.
  3. [Table I and Figs. 2-3] The central temporal scaling exponents (√t, ln t, ln ln t, ln(1/t), t^{-0.4}, 1/√t) are stated from inspection of log-log or semi-log plots, but no fitting protocol is given: no fit ranges, no functional forms with adjustable parameters, no residuals, and no uncertainty estimates. Because the separation between e.g. ln(1/t) and a weak power law, or between t^{-0.4} and t^{-1/2}, is exactly what Table I claims, please provide a reproducible fitting/collapse analysis for each regime.
  4. [Probe-Clean SUU discussion (Page time)] The statement that the Page time scales as t_P ∼ L_S^2 and is 'confirmed in our numerics' is not supported by any displayed data: no plot of t_P versus L_S or fit is shown, and the relation is derived from combining the fitted S∼√t growth with the fitted S_P∼a L_S. Please present the direct t_P measurements and their fit, with error bars, for the systems considered.
  5. [Abstract and Introduction (dephasing-as-interactions)] The interpretation of dephasing probes as emulating interactions, and the closing claim that the findings carry over to generic interacting quantum systems, rest on an untested transfer of Refs. [69-75] from transport and wave-packet spreading to entanglement entropy. All simulations are free fermions at γ=0.1 and no direct comparison with an interacting model is made. Please either add such a comparison or explicitly restrict the claims to dephased free fermions and soften the generalization statement.
minor comments (5)
  1. [Main text after Eq. (5)] The notation S(t) = Sξ(t) should be S(t) = ⟨Sξ(t)⟩ over noise realizations; as written it could be misread as a single-trajectory quantity.
  2. [Supplementary Material, Eq. (S2)] The Hamiltonian in Eq. (S2) is written with the sum going to L-1, whereas the main-text Hamiltonian in Eq. (1) sums to L; please make the two conventions consistent.
  3. [Fig. 2b inset] The log fit g1(LS)=a ln(LS)+b with a=6.2, b=-2.3 gives a negative Page value at LS=1, which illustrates the need for error bars and a clearly stated fit range.
  4. [Throughout] There are small typographical errors, including 'dephashing' in the Setup section and 'seizes to exist' for 'ceases to exist' in the discussion after Eq. (6); please proofread the manuscript.
  5. [Reproducibility] No code or data-availability statement is provided; given the ambiguity in the SUU update, releasing the simulation code would materially help readers verify the reported scaling laws.

Circularity Check

1 steps flagged · score 2.0 of 10

Table I scalings are genuine trajectory outputs; only t_P ~ L_S^2 is a stated consequence of combining separately fitted growth and Page-value scalings, giving one minor non-central reduction.

  1. fitted input called prediction [Main text, PC-SUU discussion after Fig. 2 (first Page curve section)]
    "We also find that the Page time tP ∼ L2 S. This is a combined consequence of diffusive √t growth of EE and volume law scaling for the Page value. In other words, S(t) ∼ D√t and SP ∼ a LS along with the fact that S(tP ) = SP yields tP ∼ L2 S. This is confirmed in our numerics, where we find excellent quadratic behaviour."

    The Page-time scaling is not an independent measurement or prediction: it is obtained by substituting the separately fitted growth law S(t) ~ D sqrt(t) and the separately fitted Page-value law S_P ~ a L_S into the definitional identity S(t_P)=S_P, giving t_P ~ (a L_S / D)^2. Calling this 'confirmed in our numerics' is therefore a consistency check on the same fits, not a new result. The reduction is explicit in the paper's own sentence ('This is a combined consequence...'), so the presentation is partly circular, but it concerns only t_P, not the growth/decay exponents or the Page-value scalings that constitute the central claim.

full rationale

The central derivation chain is self-contained: the SUU/QSD stochastic updates produce a Gaussian U matrix, Eq. (4) gives C = U U^dagger, and Eq. (5) computes trajectory entanglement entropy from the correlation spectrum. The reported growth laws, decay laws, and Page-value scalings in Table I are outputs of trajectory averaging, not fitted inputs renamed as predictions; they are therefore not circular. The premise that dephasing probes emulate interactions is imported from Refs. [69-75] and is an untested physical assumption (and a correctness risk), but it is external support rather than a self-referential reduction, so it does not count as circularity under the rules. The only identified reduction is the t_P ~ L_S^2 statement, which the manuscript itself labels a 'combined consequence' of the fitted sqrt(t) growth and the fitted volume-law Page value; that is a derived consistency identity, not an independent check. The SUU update inconsistency between Eq. (S16) and Eq. (S19) in the supplementary material is a reproducibility/correctness issue for half of Table I, but it is not a circularity and does not raise the circularity score; it should be resolved separately. Ref. [55] from the same author group is used only alongside standard references [86-88] for the Gaussian correlation-matrix entropy formula, so it is not load-bearing self-citation. Overall score 2: one minor non-central reduction, while the central Page-curve results have independent numerical content.

Assumptions & free parameters 10 free parameters · 5 assumptions · 0 invented entities

The ledger shows that the paper's results rest on standard Gaussian-trajectory machinery and on the physical mapping from dephasing probes to interactions. The fitted numerical coefficients and exponents in Table I and the insets are empirical characterizations, not derived quantities. No new particles, forces, or exotic entities are introduced. The main epistemic weight falls on the dephasing-to-interaction analogy and on the unverified numerical convergence assumptions.

free parameters (10)
  • a_PC_SUU = 0.5
    Slope of the linear fit S_P = a L_S + b for the Probe-Clean SUU Page value, Fig. 2a inset.
  • b_PC_SUU = 1.1
    Intercept of the same linear fit for the Probe-Clean SUU Page value, Fig. 2a inset.
  • a_PP_SUU = 0.6
    Slope of the linear fit S_P = a L_S + b for the Probe-Probe SUU Page value, Fig. 3a inset.
  • b_PP_SUU = 1.0
    Intercept of the same linear fit for the Probe-Probe SUU Page value, Fig. 3a inset.
  • a_PC_QSD = 6.2
    Coefficient of the fit S_P = a ln(L_S) + b for the Probe-Clean QSD Page value, Fig. 2b inset.
  • b_PC_QSD = -2.3
    Intercept of the same logarithmic fit for the Probe-Clean QSD Page value, Fig. 2b inset.
  • a_PP_QSD = 3.2
    Coefficient of the fit S_P = a ln(L_S) + b for the Probe-Probe QSD Page value, Fig. 3b inset.
  • b_PP_QSD = -3.4
    Intercept of the same logarithmic fit for the Probe-Probe QSD Page value, Fig. 3b inset.
  • decay_exponent_PP_SUU = 0.4
    Empirical exponent in the t^-0.4 decay after the Page time for the Probe-Probe SUU case, Fig. 3a and Table I; reported without an uncertainty estimate.
  • decay_exponent_PP_QSD = 0.5
    Empirical 1/sqrt(t) decay after the Page time for the Probe-Probe QSD case, Fig. 3b and Table I; reported without an uncertainty estimate.
assumptions (5)
  • domain assumption Each quantum trajectory remains Gaussian under both SUU and QSD, so entanglement entropy can be computed from the correlation matrix [U U†] via Eq. (5).
    Used in the main text between Eqs. (4) and (5) and in the supplementary material after Eqs. (S17) and (S64). This is standard for quadratic Hamiltonians and Gaussian initial states, but the QSD evolution includes nonlinear expectation values and the preservation of Gaussianity is asserted rather than proven.
  • domain assumption Markovian dephasing probes faithfully mimic the scattering effects of many-body interactions, so the dephased free-fermion model represents an effectively interacting system.
    This mapping is the bridge to the abstract's claim of applicability to generic interacting systems. It is cited to Refs. [69-75] and is not directly tested in this paper.
  • standard math The noise averaging of the stochastic trajectories reproduces the Lindblad dynamics in Eq. (3).
    This is the standard unravelling property relied on for both SUU and QSD. However, the sign of the gamma term in Eq. (S16) as typeset appears inconsistent with Eq. (S7), so the average-equivalence is not cleanly demonstrated in the text.
  • domain assumption The reservoir is large enough, with LR up to 6000, that boundary reflections do not affect the Page curve within the simulated time window.
    Stated in the Setup section; no explicit convergence test in LR is shown.
  • domain assumption Averaging over 100 noise trajectories gives converged entanglement entropy.
    Stated in the Setup section; no convergence plot or trajectory-number scan is provided.

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Cite this review

Pith. "Pith review of Quantum trajectories and Page-curve entanglement dynamics." pith.science (2026). https://pith.science/paper/SXNOH6BD

@misc{pith2026250112110,
  author       = {Pith},
  title        = {Pith review of: Quantum trajectories and Page-curve entanglement dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SXNOH6BD}},
  note         = {Machine review of arXiv:2501.12110}
}
read the original abstract

We consider time dynamics of entanglement entropy between a filled fermionic system and an empty reservoir. We consider scenarios (i) where the system is subjected to a dephasing mechanism and the reservoir is clean, thereby emulating expansion of effectively interacting fermions in vacuum, and (ii) where both the system and the reservoir are subjected to dephasing and thereby enabling us to address how the entanglement between the part of the effectively interacting system and its complement evolves in time. We consider two different kinds of quantum trajectory approaches, namely stochastic unitary unraveling and quantum state diffusion. For both protocols, we observe and characterize the full Page curve-like dynamics for the entanglement entropy. Depending on the protocol and the setup, we observe very distinct characteristics of the Page curve and the associated Page time and Page value. We also compute the number of fermions leaking to the reservoir and the associated current and shed light on their plausible connections with entanglement entropy. Our findings are expected to hold for a wide variety of generic interacting quantum systems.

Figures

Figures reproduced from arXiv: 2501.12110 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic showing a finite-size filled fermionic system [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Plot of the entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Plot of the entanglement entropy [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Plot of [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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Forward citations

Cited by 2 Pith papers

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  2. Sharp Page transitions in generic Hamiltonian dynamics

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