{"id":"c0434c53-e8bc-46ec-882e-2d8497ec3b3c","arxiv_id":"2501.12110","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":10,"one_line_summary":"In a dephased fermionic chain connected to an empty reservoir, entanglement follows a Page curve whose growth, decay, and peak scaling depend on the noise protocol and probe geometry.","lead":"This paper simulates how entanglement between a filled chain of particles and an empty reservoir rises to a peak and then falls, using two kinds of random noise protocols. It maps the resulting Page curve shapes and shows they depend on where the noise acts and how the quantum trajectories are unraveled.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Supplementary SUU update is internally inconsistent: Eq. (S16) uses e^{-idξ+γ/2 dt} while Eq. (S19) drops the γ term; since the SUU rows of Table I depend on this update, the dephasing interpretation is unverified.","rationale":"The paper's central claim is a full Page curve for trajectory-averaged entanglement entropy in dephased free fermions, with protocol- and geometry-dependent scalings summarized in Table I. The Gaussian correlation-matrix method, Eq. (4)-(5), is standard and internally sound, and the qualitative Page-curve behavior is plausibly supported by the presented numerics. The reader's identified weakest assumption — that dephasing probes faithfully emulate interactions — is a real limitation, but it mostly affects the Abstract's generality statement and the physical interpretation, not the free-fermion simulations themselves. A more immediate threat to the central claim is the inconsistency in the SUU update rule in the supplementary material. One half of Table I is generated by SUU trajectories, and the published equations cannot all be correct: S16 and S19 differ in the sign and presence of the γdt factor, and the QR justification in the text points to S16. Because the noise variance is stated explicitly, the average behavior is checkable, and the plus sign in S16 would conflict with the dephasing Lindblad equation. This does not prove the numerics are wrong — the code may well use the correct phase-only update — but it means the central SUU results are not reproducible from the manuscript as written. The concrete test proposed above settles the issue directly. I therefore keep the reader's CONDITIONAL verdict: the paper should be accepted only after the authors resolve the sign discrepancy and confirm which update rule produced Table I, since a typo in a supplementary equation is fixable but an implemented anti-dephasing update would invalidate the SUU scalings.","tokens_in":22363,"tokens_out":17009,"duration_ms":196567,"concrete_test":"Implement the H=0 two-site SUU update both as written in Eq. (S16) and as written in Eq. (S19). For each, average the single-particle density matrix over noise and compare the off-diagonal element to the exact Lindblad solution of Eq. (3): for two probed sites it should decay as e^{-γt}, and for one probed site as e^{-γt/2}. If the S16 form gives no decay or growth, the published update cannot produce dephasing; then re-run the Table I simulations with the phase-only unitary update of Eq. (S19) and report whether the SUU scalings and Page values persist. Equivalently, the authors should state explicitly whether their code uses S16, S19, or the Ito-corrected form diag(e^{-idξ_j-γ/2 dt}), and verify that the noise-averaged correlation matrix matches Eq. (3) to O(dt).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The SUU protocol is one of two unravelings on which Table I rests, but the supplementary material gives two incompatible update rules. Eq. (S16) defines M = diag(e^{-idξ_1+γ/2 dt}, ...), while Eq. (S19) writes the update with M = diag(e^{-idξ_1}, ...) and no γ-dependent factor. The text also says QR decomposition is needed because M contains e^{γdt} factors, which is consistent with S16 but not with S19. With the stated noise variance ⟨dξ_i dξ_j⟩ = γ dt δ_{ij}, the plus sign in S16 would produce growth rather than decay of off-diagonal single-particle coherences, so the ensemble would not reproduce the dephasing Lindblad equation (3). If the numerics instead follow S19, the SUU trajectories are random-phase unitaries and the dephasing rate is correct, but then the derivation around S11-S16 and the QR remark are misleading. The manuscript does not say which update was actually simulated, and no code is provided. Since the SUU column of Table I — sqrt(t) growth, ln(1/t) or 1/t^0.4 decay, and volume-law Page value — is computed from this update, the central numerical claim for half of the protocol/geometry combinations cannot be independently checked until the discrepancy is resolved. This is a concrete internal-consistency issue, distinct from the untested claim that dephasing mimics interactions, and it should be settled before the scaling laws are taken at face value.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22704,"tokens_out":8054,"duration_ms":89399,"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":[{"comment":"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.","section":"Supplementary Material, Sec. I A, Eqs. (S16) and (S19)"},{"comment":"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.","section":"Supplementary Material, Sec. I B, Eq. (S72)"},{"comment":"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.","section":"Table I and Figs. 2-3"},{"comment":"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.","section":"Probe-Clean SUU discussion (Page time)"},{"comment":"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.","section":"Abstract and Introduction (dephasing-as-interactions)"}],"minor_comments":[{"comment":"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.","section":"Main text after Eq. (5)"},{"comment":"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.","section":"Supplementary Material, Eq. (S2)"},{"comment":"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.","section":"Fig. 2b inset"},{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Reproducibility"}],"recommendation":"major_revision","confidential_remarks":"The central question I could not resolve is whether the SUU simulations follow Eq. (S19) or Eqs. (S16)-(S17); this determines whether the reported SUU results are trustworthy. I would urge the editor to require the authors to clarify this and ideally provide the code before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Plainly: this paper's value is Table I. It maps out the growth and decay of entanglement entropy for a filled fermionic system leaking into an empty reservoir under two probe geometries (Probe-Clean, Probe-Probe) and two unravelings (SUU, QSD), and finds a full Page curve in all four cases with distinct scaling laws. That systematic catalog is new, and the numerics behind it are straightforward but competently done: Gaussian states propagated via U(t)U†(t) and entropy from the correlation-matrix eigenvalues.\n\nThe central qualitative claim holds up. The sqrt(t) growth before the Page time in the SUU protocol is consistent with the diffusive particle current, and the protocol dependence of the decay laws (logarithmic versus power law) is a genuine observation. The numerical connection between dS/dt and particle current before t_P under SUU is also interesting and could be useful to experimentalists.\n\nNow the soft spots, in order of real weight. First, the supplementary material has an internal inconsistency in the SUU update rule. Eq. (S16) defines the diagonal prefactor as e^{-i dξ + γ/2 dt} (with a plus sign), Eq. (S19) writes it without the γ term, and the derivation around S6-S7 implies the γ term should be there with a minus sign. The manuscript does not state which update was actually used in the simulations, and no code is provided. The SUU rows of Table I depend on this update, so this needs to be resolved before those exponents can be fully trusted. This is a presentation issue that a referee can reasonably ask to fix.\n\nSecond, the scaling exponents are fits without quoted error bars, and the study uses a single dephasing strength, γ=0.1. That limits the quantitative precision of the catalog but does not threaten the qualitative conclusion that a Page curve appears with protocol-dependent scaling.\n\nThird, the abstract and outlook claim that the findings extend to generic interacting quantum systems. That rests on the premise that dephasing probes mimic interactions, which is imported from earlier work and not tested here; the simulations are all free fermions. That is a minor overreach, but it is the kind of claim that should be softened or directly supported.\n\nWho is this for? Researchers working on monitored free-fermion dynamics, Page-curve entanglement in open systems, and dephasing-engineered lattices. It is a useful numerical reference and worth a serious referee. My recommendation: send it to review, with an explicit request that the authors clarify the SUU update rule and either provide code or a precise statement of what was simulated.","headline":"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.","tokens_in":23310,"tokens_out":5268,"would_cite":false,"duration_ms":53190,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Page curve","entanglement entropy","quantum trajectories","dephasing","free fermions","quantum state diffusion","stochastic unitary unraveling","particle current"],"falsifier":"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.","tokens_in":22130,"feed_emoji":"⚛️","tokens_out":8958,"duration_ms":85350,"temperature":0.7,"pith_summary":"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.","feed_headline":"Dephasing noise turns free-fermion expansion into a full Page curve","feed_subtitle":"Entanglement between a filled system and empty reservoir grows, peaks, and decays, with scaling set by how probing is done.","key_machinery":"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.","core_discovery":"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).","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the premise that Markovian dephasing probes emulate interaction-induced scattering, the basis for calling the system effectively interacting.","marker":"[69-72]"},{"why":"Shows dephasing turns ballistic wave-packet and current scaling into diffusive scaling, supporting the diffusive-growth interpretation of the Page curve.","marker":"[73-75]"},{"why":"Provides the earlier freely expanding fermionic gas Page-curve-like calculation whose correlation-matrix entanglement method this paper adapts.","marker":"[55]"},{"why":"Supplementary material defining the SUU and QSD protocols and deriving the U-matrix updates used in the simulations.","marker":"[82]"},{"why":"Gives the Gaussian-state relation between reduced correlation-matrix eigenvalues and entanglement entropy used in Eq. (5).","marker":"[86-88]"},{"why":"Establishes the proportionality between entropy production rate and particle current in non-interacting fermionic systems that Eq. (6) extends to dephased trajectories.","marker":"[54]"},{"why":"Source of the stochastic-unitary unraveling of dephasing as onsite Gaussian noise, used for the SUU protocol.","marker":"[84]"},{"why":"Justifies QSD as continuous weak monitoring of local occupation, used for the QSD protocol.","marker":"[41, 85]"}],"fun_headline_variants":["Dephasing fermions reveal full Page curve in quantum trajectories","Quantum trajectories map Page-curve entanglement dynamics in fermions","Fermionic systems show Page curve under dephasing and trajectories","Page curve emerges for fermionic expansion under dephasing noise","Distinct Page curves from quantum trajectory protocols in fermions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Dephasing fermions reveal full Page curve in quantum trajectories","Quantum trajectories map Page-curve entanglement dynamics in fermions","Fermionic systems show Page curve under dephasing and trajectories","Page curve emerges for fermionic expansion under dephasing noise","Distinct Page curves from quantum trajectory protocols in fermions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000773,"raw_usage":{"total_tokens":3457,"prompt_tokens":1013,"completion_tokens":2444,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":629,"completion_tokens_details":{"reasoning_tokens":2359}},"tokens_in":629,"tokens_out":2444,"duration_ms":18818,"temperature":1.0,"reasoning_tokens":2359,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:30:53.423941+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the earlier freely expanding fermionic gas Page-curve-like calculation whose correlation-matrix entanglement method this paper adapts."}],"review_version":1}