{"id":"10d7633f-156e-46ce-8732-9a0edff9b1f5","arxiv_id":"2411.14582","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Connected two-point functions of quantum trajectories in continuously measured lattice bosons can be recovered by binning trajectories on filtered estimators of the measurement record, with filters derivable from the unconditional dynamics.","lead":"Bosons on a lattice with continuous quadrature measurements are solved analytically, and the paper shows that postselection can be reduced to binning trajectories by one or two filtered numbers from the measurement record. This offers a concrete route to recover hidden two-point correlators in monitored quantum systems without repeating the experiment for every measurement history.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Near-continuum test at ξ=20 likely uses T=10Γ^{-1}, shorter than the memory time τ≈28Γ^{-1}, so Fig. 10(b) does not establish recovery of the steady-state correlator.","rationale":"The reader's weakest assumption correctly identifies steady-state equilibration as the key constraint. My concern sharpens this: the paper's own near-continuum numerical test (Fig. 10(b), ξ=20) does not actually verify that the protocol works at steady state, because the implied observation time T=10Γ^{-1} is shorter than the memory time τ≈28Γ^{-1} and the covariance relaxation time ≈14Γ^{-1}. At T=10Γ^{-1}, the slow modes with h_q≈Γ/200 have not converged to the steady-state covariances used to build the filters in Eq. (30). The figure also lacks a quantitative reference to the analytic steady-state profile, using instead the J0=3Γ curve as a visual contrast. Therefore the demonstrated scope of the protocol is narrower than the abstract implies: it is quantitatively validated only for short-memory-time parameters (e.g., J0=3Γ). This is not a fatal flaw in the analytic solution or the binning idea, but it warrants a conditional acceptance: the authors should either specify the observation time and add the quantitative comparison for the ξ=20 case, or explicitly limit the protocol's demonstrated regime to parameters where equilibration is achieved within the stated time. The central result for J0=3Γ, which the reader's verdict relies on, remains solid, so I do not recommend rejection; the adjustment is to require clarification or a modest revision of the generality claim.","tokens_in":30254,"tokens_out":15823,"duration_ms":154495,"concrete_test":"Repeat the Fig. 10(b) protocol for J0=2.0025Γ, J=Γ with observation times T=10Γ^{-1} and T=100Γ^{-1}, and compare the recovered C^P(r) to the analytic steady-state profile Eq. (32) for these parameters (not the J0=3Γ curve). If the T=10 curve differs from the T=100 curve or from Eq. (32), the near-continuum validation is insufficient; if they agree within error bars, the concern is resolved.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central reduction—estimators Eq. (29) with steady-state kernels Eq. (30) fully determine the conditional state—requires the covariance matrices C^X, C^P, U to have reached their unique steady state before the estimators are used. The paper acknowledges this (Section III C) and the J0>dJ condition. However, the numerical demonstration in the near-continuum regime, Fig. 10(b) with ξ=20 (J0=2.0025Γ, J=Γ), appears to reuse the observation time T=10Γ^{-1} from Section IV A. For ξ=20, the filter memory time is τ≈ξ√(2/(ΓJd))≈28Γ^{-1}, and the slowest covariance relaxation time is about τ/2≈14Γ^{-1} (using v_q≈0.018 from Eq. (31) at q≈0). Thus at T=10Γ^{-1} the system has not equilibrated, so the steady-state kernels are not yet valid. Furthermore, Fig. 10(b) gives no quantitative comparison to the analytic steady-state profile Eq. (32) for J0=2.0025Γ; the gray line is the J0=3Γ correlator from Fig. 8(b). Consequently, the claim that the protocol 'recovers the decaying spatial profile' in the near-continuum regime is not substantiated: the red dots could reflect the time-dependent conditional covariance at T, and the bias from applying steady-state filters out of equilibrium is unquantified. The core result at J0=3Γ (Fig. 8) is well supported, but the generality claim for large ξ is not.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a one-dimensional lattice of bosons with local continuous quadrature measurements, a Gaussian model that is exactly solvable. It shows that, after the covariance matrices reach their steady state, the conditional means at each site are determined by linear filtered versions of the measurement record (Eq. (29) with kernels Eq. (30)). The authors exploit this to reduce postselection: instead of conditioning on the full measurement record, one can bin trajectories by one or two scalar estimators per site and still recover connected two-point correlators C^X and C^P that are inaccessible from the unconditional state. The paper derives the filters analytically, shows how the same filters can be estimated from the unconditional dynamics or from record correlations, verifies the protocol numerically in a way that mimics experimental access, discusses the continuum or large-correlation-length regime, and proposes cavity-QED and circuit-QED implementations. Several limitations are acknowledged in the text, including the unproved spatial-averaging assumption in Section IV C and the heuristic cancellation of divergent terms in Appendix C.","tokens_in":30528,"tokens_out":6653,"duration_ms":67321,"significance":"If the claims hold, this is a valuable exactly solvable example in which the postselection barrier for local observables is explicitly broken: the conditional state depends on the record only through a few estimators, and those estimators can be constructed from experimentally accessible data. The analytic derivation is careful, the numerical protocol in Figure 8 matches the analytic steady-state profile at J0=3Γ, and the paper is unusually explicit about what is assumed and what is demonstrated. The authors correctly distinguish between observables that are linear in the conditional state and genuine nonlinear correlators, and they provide concrete experimental implementations. The central result at J0=3Γ is convincing. However, the near-continuum numerical demonstration and two admitted gaps—the divergent-term cancellation in Appendix C and the spatial-averaging assumption in Section IV C—prevent the paper from being fully self-contained in its broader claims.","major_comments":[{"comment":"The claim that the postselection protocol \"can still recover the decaying spatial profile\" for ξ=20 is not substantiated by the data shown. At J0=2.0025Γ and J=Γ, the observation time T=10Γ^{-1} is shorter than the memory/equilibration time: from Section III C the slowest covariance relaxation is τ/2, and with τ≈ξ√(2/(ΓJd)) one obtains τ≈28Γ^{-1} at ξ=20, so equilibration requires about 14Γ^{-1}. The steady-state kernels in Eq. (30) therefore need not be valid at T=10Γ^{-1}. Moreover, the gray line in Fig. 10(b) is the J0=3Γ profile from Fig. 8(b), not the analytic steady-state profile Eq. (32) for the simulated parameters, so the red points are not compared to the correct curve. Please rerun the near-continuum test with T substantially larger than τ, or explicitly benchmark the time-dependent conditional covariance at the observation time, and compare against Eq. (32) at the same J0, J, and Γ.","section":"§IV B, final paragraph; Fig. 10(b)"},{"comment":"The derivation of the filter from the record-record correlator assumes, without proof, that the time-non-invariant and ∝T terms in the record-record correlator cancel against analogous contributions on the right-hand side of Eq. (15); the text says \"hoping that the diverging terms will cancel against a similar contribution on the right-hand side of Eq. (15)\". Because this is one of the routes used to establish that the filter can be obtained from the record correlation alone, this step should be made rigorous or explicitly regularized. Please provide a well-defined T→∞ limit or prove that the decaying solution of Eq. (C11) with Eq. (C8) is the unique filter following from the minimization problem in Section III B.","section":"Appendix C, Eqs. (C4)-(C11)"},{"comment":"The single-sample protocol rests on replacing measurement-realization averages by spatial averages over regions separated by more than a few correlation lengths. The manuscript states in step 2 that \"Although we do not present a formal proof, we justify this assumption by arguing that spatial regions separated by more than few correlations lengths are uncorrelated in practical terms.\" Since this assumption is load-bearing for the \"application to single samples\" discussion and for the claimed phase-of-matter framing, please either provide numerical evidence of convergence at the system sizes used, or explicitly demote this to a conjecture with the required separation length identified as an open condition.","section":"§IV C, step 2"}],"minor_comments":[{"comment":"The condition \"J0 > J2\" should be written as \"J0 > dJ\", and the later phrase \"J0 → J + 2\" should be replaced by the correct d-dimensional expression (or \"J0→J+2\" only after specifying d=1).","section":"§IV, first paragraph"},{"comment":"The notation \"fixed p ΓJ d/2 T* = 10\" is unclear; please write √(ΓJd/2) T* = 10 or equivalent, matching the exponent in Eq. (35).","section":"Fig. 9(b) caption"},{"comment":"The statement that the correlators decay with \"a correlation length of the order of the lattice size\" would be clearer if the value of ξ from Eq. (34) for J0=3Γ, J=Γ were quoted explicitly, since the text later defines ξ and discusses the continuum limit.","section":"Fig. 7(c) and surrounding text"},{"comment":"The sentence \"This is enough to determine f(t)\" relies on an implicit boundary-condition argument for the fourth-order differential equation; please spell out why the exponentially growing solutions are discarded and why the two normalization conditions in Eq. (20) give a unique filter.","section":"§III B, Eq. (19)"}],"recommendation":"major_revision","confidential_remarks":"The central analytic and numerical results for J0=3Γ are solid and suitable for publication after revision. The main issue is the near-continuum numerical claim in Fig. 10(b), which is currently underdetermined: the observation time is too short for steady-state kernels to apply, and the comparison curve is not the correct analytic profile. The two admitted gaps (Appendix C and Section IV C) should be either fixed or clearly labeled as assumptions rather than parts of the main claim. I do not see a circularity problem or a citation concern; the self-citations to earlier Gaussian-state filtering results are appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read it. The central result is real: for a Gaussian boson lattice with continuous quadrature measurement, they derive explicit space-time filter kernels and show that conditional correlators can be recovered by binning trajectories on one or two estimators per site instead of the full record. The numerical experiment in Fig. 8, with 3e4 runs, reproduces the analytic profile at J0=3Γ. That is a genuine step toward mitigating postselection in a solvable, experimentally relevant class.\n\nThe construction of the filters from unconditional dynamics or record correlations alone (Section III B) is the most valuable part; it separates the protocol from the exactly-solvable model. The experimental sections are thoughtful, with plausible cavity-QED and circuit-QED routes.\n\nNow the soft spots. The near-continuum test in Fig. 10(b) is not as clean as the text claims. The paper states J0=2.0025Γ, J=Γ, and ξ=20, but Eq. (34) gives ξ≈0.7 for those parameters. One of those numbers is wrong. If ξ=20 is intended, then the memory time is τ≈28/Γ, and the numerical run at T=10/Γ has not equilibrated; the steady-state filter kernels are not yet valid, and the gray comparison curve is from the J0=3Γ run, not the analytic profile at the new parameters. So the recovery of the decaying profile in that regime is not established. This is fixable by extending T and/or providing the direct comparison, and it does not affect the core J0=3Γ result.\n\nThe unproved spatial-averaging assumption in Section IV C and the heuristics in Appendix C are flagged by the authors; they are real but secondary. The paper's own limitations are mostly stated honestly.\n\nWho is this for: people working on measurement-induced dynamics, quantum trajectories, and postselection in bosonic/continuous-measurement systems. It deserves serious refereeing. I would suggest the referee focus on the parameter inconsistency and the equilibration condition, and ask for a direct analytic comparison in the continuum regime.","headline":"Strong core result on postselection in monitored bosons; the near-continuum test has a parameter inconsistency and likely too short integration time.","tokens_in":31055,"tokens_out":4158,"would_cite":true,"duration_ms":35761,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Continuously measured lattice bosons can be postselected using one or two per-site estimators instead of the full measurement record.","keywords":["postselection","continuous measurement","quantum trajectories","lattice bosons","Gaussian states","estimators","two-point correlators","cavity QED"],"falsifier":"Take a one-dimensional lattice at $J_0 = 2.0025\\Gamma$, $J=\\Gamma$ so that $\\xi=20$, evolve from vacuum for a fixed time $T=10\\Gamma^{-1}$, and apply the binning protocol with $N_{\\rm trial}=3\\times 10^4$; compare the recovered $C^P(r)$ against the exact conditional covariance computed by directly integrating Eq. (26). If the recovered profile deviates from the exact profile by more than the statistical error bars, the steady-state-filter assumption has broken down. A sharper test increases $\\xi$ (by moving $J_0$ closer to $2J$) at fixed $T$: the protocol should fail once the equilibration time $\\sim \\tau$ exceeds $T$.","tokens_in":1907,"feed_emoji":"⚛️","tokens_out":3336,"duration_ms":60120,"temperature":0.7,"pith_summary":"This paper claims that the postselection problem in a solvable model of lattice bosons under continuous quadrature measurement can be dramatically reduced. Instead of requiring identical full measurement records to build an ensemble of identical quantum states, the late-time conditional state depends on just one or two numbers per lattice site, the estimators. The authors show that connected two-point correlators, which are nonlinear in the conditional density matrix and invisible in the unconditional state, can be recovered by binning trajectories according to these estimators. They further show that the estimators can be constructed from information available in the unconditional dynamics or in the measurement record alone, and they verify the protocol in numerical experiments mimicking experimental conditions. A sympathetic reader would take away that efficient postselection is possible in this Gaussian bosonic model and that the structural features identified may guide mitigation in more general monitored systems.","feed_headline":"Two numbers per site beat the postselection barrier","feed_subtitle":"For continuously measured bosons on a lattice, connected correlators hidden behind postselection are recovered by record-derived estimators.","key_machinery":"The central objects are the per-site estimators defined as spatiotemporal convolutions of the measurement record with filter kernels, $(\\hat{x}_i)_{\\rm est}(t)=2\\sqrt{\\Gamma}\\sum_j\\int_{-\\infty}^{t} K_x(i-j,t-s)\\,dI_j(s)$ and similarly for $\\hat{p}_i$, with $K_x$ and $K_p$ given in Eq. (30) in terms of the momentum-space steady-state covariances $v_q$, $u_q$, $w_q$. These filters encode a memory time $\\tau=(2\\Gamma v^\\infty_x)^{-1}$ and, on the lattice, a correlation length $\\xi=\\sqrt{J/[2(J_0-dJ)]}$, so that only the measurement record in a spacetime correlation volume around a site matters. The machinery is Gaussian Kalman-Bucy filtering: the conditional state is described by means and covariances, the means are linear functionals of the record at late times, and the estimators convert postselection on the full record into postselection on one or two numbers per site.","core_discovery":"The paper establishes that in a lattice of bosons with local continuous measurements of the $\\hat{x}_i$ quadrature, the conditional quantum state at late times is fully determined by per-site linear estimators of the measurement record: $(\\hat{x}_i)_{\\rm est}$ and $(\\hat{p}_i)_{\\rm est}$ given by Eq. (29) with time-translation-invariant filter kernels $K_x(i-j,t-s)$ and $K_p(i-j,t-s)$ from Eq. (30). Because these estimators capture the full dependence of the state on the record, connected two-point functions $C^X_{ij}$ and $C^P_{ij}$, which are nonlinear in the conditional density matrix, can be obtained by binning trajectories on the estimators rather than on the entire measurement history. The filter kernels can be derived analytically from the steady-state covariances $v_q$, $u_q$, $w_q$, and the paper demonstrates numerically that the same kernels can be inferred from the record-record and system-record correlations of the unconditional dynamics. The protocol recovers the exponentially decaying spatial profile of $C^P(r)$ that is absent in the unconditional correlators, using only experimentally accessible data and a few tens of thousands of repetitions.","pith_inferences":["The estimator framework suggests that postselection can be reformulated as a classical regression problem: if good filters can be learned from unconditional response functions, the same binning idea may apply to non-Gaussian monitored systems where the state-to-estimator map is not linear but still low-dimensional.","Near the critical point $J_0 = dJ$, the diverging correlation length and memory time imply that the steady-state filter assumption fails at fixed observation time; a testable prediction is that the recovered correlators will be biased when $\\xi$ exceeds the system size or the observation window.","The cavity-QED and circuit-QED implementations proposed in the paper could be used to probe the robustness of the estimator protocol against finite detection efficiency and nonlinearities, with the expectation that imperfect filters produce systematically biased estimates of $C^P$.","If this method extends to area-law phases of more general monitored circuits, it would convert a fundamental obstruction into a practical signal-processing task, with the measurement record itself acting as a classical shadow of the conditional state."],"forward_implications":["Connected two-point functions of monitored quantum trajectories become experimentally accessible with roughly $10^4$ repetitions, instead of the exponentially many repetitions required to reproduce a full measurement record.","The filters needed for postselection can be designed from the unconditional dynamics, specifically from record-record and system-record correlation functions, without solving the conditional evolution.","The same set of measured trajectories can be reused to recover all two-point correlators $C^P_{ij}$ by rebinning according to the estimators at different site pairs.","Locality in time and space emerges dynamically: only the measurement record within a memory time and correlation length of a site is relevant, which reduces the postselection overhead in extended systems.","In the continuum limit the filter kernels exhibit ballistic light-cone-like structure with power-law tails, providing a concrete classical postprocessing rule for near-critical parameters."],"supporting_citations":[{"why":"Supplies the continuous-measurement stochastic Schrödinger equation formalism and the Kalman-Bucy filter framework used to solve the single-site and lattice Gaussian dynamics.","marker":"[58]"},{"why":"Provides the route to design the optimal filter from the unconditional dynamics alone, which the paper adapts to construct estimators without solving conditional evolution.","marker":"[60]"},{"why":"Establishes continuous Gaussian measurements of free boson CFTs as an exactly solvable model of measurement-induced dynamics, the setting this paper builds on.","marker":"[30]"},{"why":"Gives the quantum theory of field-quadrature measurements, including the record-record correlation formulas used to derive the filter equation.","marker":"[69]"},{"why":"Provides the standard quantum trajectories formalism and measurement-record equations that define the conditional dynamics and the output data model.","marker":"[15]"}],"fun_headline_variants":["Two per site numbers beat postselection","Record compression to two numbers unlocks hidden correlations","Two estimators per site tame the postselection cost","Beat postselection with two numbers per site","Two record summaries expose unseen correlators"],"cache_read_input_tokens":33152,"weakest_assumption_plain":"The protocol assumes that the covariance matrices reach their unique Gaussian steady state within the observation time, so the time-translation-invariant filter kernels derived from steady-state covariances are valid over the entire postselected evolution; this requires $h_q = J_0 - J\\sum_\\mu \\cos(q_\\mu) > 0$ for all $q$, i.e. $J_0 > dJ$.","fun_headline_variants_meta":{"raw":{"variants":["Two per site numbers beat postselection","Record compression to two numbers unlocks hidden correlations","Two estimators per site tame the postselection cost","Beat postselection with two numbers per site","Two record summaries expose unseen correlators"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001142,"raw_usage":{"total_tokens":4781,"prompt_tokens":1026,"completion_tokens":3755,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":642,"completion_tokens_details":{"reasoning_tokens":3700}},"tokens_in":642,"tokens_out":3755,"duration_ms":28725,"temperature":1.0,"reasoning_tokens":3700,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:07:48.647585+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a one-dimensional lattice at $J_0 = 2.0025\\Gamma$, $J=\\Gamma$ so that $\\xi=20$, evolve from vacuum for a fixed time $T=10\\Gamma^{-1}$, and apply the binning protocol with $N_{\\rm trial}=3\\times 10^4$; compare the recovered $C^P(r)$ against the exact conditional covariance computed by directly integrating Eq. (26). If the recovered profile deviates from the exact profile by more than the statistical error bars, the steady-state-filter assumption has broken down. A sharper test increases $\\xi$ (by moving $J_0$ closer to $2J$) at fixed $T$: the protocol should fail once the equilibration time $\\sim \\tau$ exceeds $T$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the continuous-measurement stochastic Schrödinger equation formalism and the Kalman-Bucy filter framework used to solve the single-site and lattice Gaussian dynamics."},{"cited_title":"Kokail, R","cited_arxiv_id":null,"evidence_quote":"Provides the route to design the optimal filter from the unconditional dynamics alone, which the paper adapts to construct estimators without solving conditional evolution."}],"review_version":1}