{"id":"505659dc-bff3-4ea5-b000-a4cee8af7b56","arxiv_id":"2505.21214","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Arrival time statistics from an absorbing many-particle detector can identify the single-particle momentum even in a spatially uniform beam, with explicit Fisher information limits in the sparse-beam case.","lead":"This paper derives formulas for how much information about a single particle's momentum can be extracted from the recorded arrival times of a many-particle beam at a detector. It shows that even when the beam is spatially uniform, so position data carry no momentum information, the timing of arrivals still does, and gives simple analytical limits for very dilute beams.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Sparse-beam Fisher information (45)–(46) is an infinite-horizon idealization: for any fixed observation time T, the finite-time Fisher information for a coherent beam vanishes as r0→0, so the nonzero per-detection limits require waiting a time that diverges like 1/r0.","rationale":"We reviewed the derivation, especially Appendix D's limit proof and the expressions (44)–(46). The dominated convergence argument is internally consistent: the r0-independent bounds in (D10)–(D11) are finite for coherent and quasi-free Fn, and the limits ˙g∞/g∞ match I∞. The strongest claim as stated is therefore mathematically supported. The genuine caveat is the one the reader flagged: the model is for first n arrival times on an unbounded horizon; any finite-window experiment has FI proportional to r0 and vanishing in the sparse limit. The paper acknowledges this in §IV.C, but the abstract-level wording ('identifiable ... even in the limit of sparse beams') could be misread as implying finite-time feasibility. This is a limitation of scope and interpretation, not a flaw in the proof; hence conditional acceptance remains appropriate. We did not find a more load-bearing mathematical gap.","tokens_in":31714,"tokens_out":16811,"duration_ms":197158,"concrete_test":"For the coherent uniform beam, compute the finite-horizon Fisher information for observing the full point process on [0,T]: I_T(r0)=r0∫_0^T g(t)(∂_{p0} log g(t))² dt, with g(t)=|T_{p0}+R_{p0}(t)|². Verify that lim_{r0→0} I_T(r0)=0 for every fixed T, while (45) gives a positive limit, and check that the expected n-th arrival time diverges as n/(r0 g∞). This settles whether the sparse-beam Fisher information describes any fixed-duration experiment or only the infinite-horizon, stopping-at-the-nth-event protocol.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The mathematical derivation of (45)–(46) appears correct: Appendix D's dominated-convergence argument is internally consistent, and the nonzero limits are limits of the joint density of the first n arrival times on [0,∞)^n. The load-bearing soft spot is interpretive: in the coherent-beam case the full data are a Poisson process with intensity ω(t)=r0 g(t). For any fixed detector run [0,T], the Fisher information is I_T(r0)=r0∫_0^T g(t)(∂_{p0} log g(t))² dt, which tends to 0 as r0→0. The nonzero limit (45) is achieved only by observing until the n-th event, whose expected time grows as n/(r0 g∞); the authors explicitly note in §IV.C that one 'needs to wait infinitely long.' Hence the claim 'momentum is identifiable even in the sparse-beam limit' is true only in the infinite-horizon sense: information per detected particle survives, but information per unit wall-clock time vanishes linearly with r0. This does not invalidate the theorem, but it means the abstract's identifiability statement should be read as an idealization, not as a prescription for finite-time estimation at low density.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs a many-particle arrival-time detection model based on a Fock-space absorption master equation, derives the joint arrival-time distributions for Fock, coherent, and quasi-free source states, and then analyzes two idealizing limits: a point (Dirac-delta) detector and a spatially uniform infinite-particle beam. For the beam, the paper derives a tractable formula for the classical Fisher information of the single-particle momentum parameter (Eq. (42)) and studies its dependence on the particle density. The central results are Eqs. (45)-(46): in the sparse-beam limit r0→0, the Fisher information from the first n arrival times tends to n I∞(p0) for coherent states and to n/(n+2) I∞(p0) for quasi-free states, where I∞(p0)=a^2 m^2/[p0^2(p0+am/2)^2]. This limit coincides with the Fisher information of a hypothetical time-stationary detection model, as shown rigorously in Appendix D. The authors emphasize that the spatial particle distribution in the beam contains no information about p0, so the arrival-time data are the only carrier of momentum information.","tokens_in":31939,"tokens_out":15872,"duration_ms":171889,"significance":"If the results are correct, this is a valuable contribution at the interface of quantum arrival-time theory and quantum statistical inference. The paper provides explicit, parameter-free limiting formulas for the Fisher information, and it demonstrates a conceptually striking effect: temporal detection data can encode single-particle parameters even when the spatial statistics are completely insensitive to them. The appendices contain substantial original analysis: Appendix B proves convergence to the delta-detector Volterra equation, Appendix D derives the general Fisher-information formula and the sparse/dense beam limits via dominated convergence. The paper is refreshingly honest in acknowledging some idealizations, such as the need to wait infinitely long in the sparse-beam limit. The main caveat is that the identifiability statement must be understood as an infinite-horizon property, not a finite-time estimation prescription; this is acknowledged in Section IV.C but underemphasized in the abstract. The mathematical core appears sound, though one technical assumption (positivity of the beam intensity) is stated but not verified in the proof of the sparse-beam limit.","major_comments":[{"comment":"The nonzero limits (45)–(46) describe the joint density of the first n arrival times on [0,∞)^n. For a coherent beam the full finite-horizon data are a Poisson process with intensity r0 g(t); for fixed T the Fisher information of the parameter p0 is r0 ∫_0^T g(t) (∂_p log g(t))² dt, which tends to zero as r0→0. Thus the statement that the momentum is identifiable in the sparse-beam limit is only an infinite-horizon statement: the information per detected particle survives, but the expected time to the n-th detection diverges as n/(r0 g∞). The paper acknowledges this in §IV.C ('we need to wait infinitely long to get detections'), but the abstract and conclusion phrase the claim as if it were a finite-time estimation property. Please qualify the identifiability claim explicitly.","section":"Abstract and §IV.C"},{"comment":"The dominated convergence proof of (45)–(46) requires a uniform (in r0) integrable bound on |Fn(u)u^{n-1}S_{n,r0}(u)|. The bound is obtained from a constant A bounding |Φ(t)|, |~Φ(t)|, and |φ(t)| on [0,∞). While Φ and ~Φ are bounded because they are continuous with finite limits, the boundedness of φ(t)=˙g(t)/g(t) requires g(t) to be strictly positive for all t. The manuscript only states the assumption ω(t)>0 at the start of Appendix D; it is not verified for the explicit beam intensity (39). If g had a zero, φ would be unbounded and the stated dominated convergence argument would fail. Please supply a proof of positivity of (39), or state it as a necessary assumption and adapt the proof.","section":"Appendix D.b"}],"minor_comments":[{"comment":"The text says 'In(p) := E[Vp]'; since Vp is the score, this should read E[Vp²].","section":"Appendix D, first paragraph"},{"comment":"The intermediate derivation for the quasi-free state contains garbled factors ('n! Qn...', 'n/q') and should be corrected; the final formula is correct.","section":"§II.D.3"},{"comment":"The phrase 'one the other hand' should be 'on the other hand'.","section":"§II.F"},{"comment":"The text indicates that the second and third terms appear only for n≥2 and the fourth only for n≥3, but the displayed formula does not show this; adding a parenthetical condition in the equation would improve clarity.","section":"Eq. (42)"},{"comment":"The symbol f(t) is used in the limiting equation before its definition in the surrounding text; a brief definition of f(t) immediately after Eq. (23) would help.","section":"§III.A"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the technical core is largely sound. The main revision points are the qualification of the infinite-horizon identifiability claim and the verification (or explicit assumption) of positivity of the beam intensity in Appendix D.b. I do not see grounds for rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe short version: this paper is worth reading and worth refereeing. Its real contribution is a tractable statistical picture of how much single-particle momentum information is carried by arrival-time sequences in bosonic beams. The sparse-beam formulas (45)–(46) are new, correctly derived, and non-obvious: momentum remains identifiable from timing even when the spatial density is flat. That central result survives scrutiny.\n\nWhat is good: the arrival-time distributions for Fock, coherent, and quasi-free states are derived cleanly from the quantum-jump unravelling of a known master equation; the general Fisher-information formula (42) gives a practical route for numerical work; and the appendices do real work in making the delta-detector and sparse-beam limits rigorous via dominated convergence. The time-stationary coincidence is a nice observation, and the coherent-versus-quasi-free difference in scaling with n is clean. There is no circularity.\n\nThe soft spots are minor to moderate. First, the sparse-beam limits are infinite-horizon statements. For a fixed observation window [0,T], the Fisher information for a coherent beam vanishes as r0→0; the nonzero limit requires waiting until the n-th event, whose expected time diverges like 1/r0. The authors acknowledge this in §IV.C, but the abstract and conclusion phrase it as a clean identifiability result. That is not a mathematical flaw, but operational readers should know the price.\n\nSecond, and more concretely, the dense-beam limit (r0→∞) in Appendix D.b is not justified. The proof assumes dot-g_0 := lim_{t→0} dot-g(t) exists and is zero, but from the paper's own expansion (34), dot-g(t) diverges as t^{-1/2}. The dominated-convergence bound (D10) relies on Φ and φ being bounded, which they are not in the dense-beam regime. The conclusion (limit zero) is likely true, but the argument as written is wrong; it needs a direct estimate or a different scaling.\n\nThe citation pattern is fine; the self-citations are legitimate background.\n\nWho this is for: people working on arrival-time observables, detector models, or quantum metrology with time-of-flight data. It deserves a serious referee and is probably citable once the dense-beam proof is repaired. I would send it to review with a request for a revision on that specific point.","headline":"Solid analytical contribution on Fisher information from arrival times; the sparse-beam limits are the real news and hold, but the dense-beam limit proof needs a fix and the infinite-horizon caveat matters.","tokens_in":32479,"tokens_out":5988,"would_cite":true,"duration_ms":70957,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P50","62B10","81S22","81P15"],"pacs":["03.65.Ta"],"model":"deepseek-v4-flash","headline":"The paper derives exact Fisher information for momentum estimation from arrival times of Bosonic beams, showing that sparse-beam arrival times carry nonzero information about $p_0$ even when the beam's position distribution is flat.","keywords":["arrival time","Fisher information","Fock space master equation","point detector","plane-wave beam","statistical inference","quasi-free states","parameter estimation"],"falsifier":"Measure arrival times of a free one-dimensional Bose beam with a point absorber at two densities and for both coherent and thermal (geometric-number) states; estimate $p_0$ by maximum likelihood from the first $n$ arrival times and check that the inverse estimator variance grows as $n$ for coherent states and saturates as $n/(n+2)$ for thermal states, since the paper's predictions differ by exactly the factor $n+2$.","tokens_in":31490,"feed_emoji":"⏱️","tokens_out":5558,"duration_ms":55815,"temperature":0.7,"pith_summary":"The paper asks whether arrival times of a beam of non-interacting Bosons can carry information about single-particle parameters that ordinary position measurements cannot. The authors build a many-particle absorption model in Fock space, derive joint probability densities for sequences of arrival times, and treat them as a classical statistical model for the momentum parameter $p_0$ of a free particle in one dimension. In the idealized limits of a point detector and an infinite spatially uniform beam, position data are flat and contain no information about $p_0$; the paper proves that the arrival-time data still do, with the classical Fisher information from the first $n$ arrivals tending to explicit nonzero limits as the particle density vanishes. The result gives a concrete, analytically tractable example of temporal quantum data as a resource for parameter estimation.","feed_headline":"Arrival times reveal momentum a flat quantum beam hides","feed_subtitle":"Sparse-beam limits yield exact Fisher information: n-fold for coherent beams, n/(n+2) for thermal ones.","key_machinery":"The load-bearing object is the absorptive detection model: a Lindblad master equation on the Bosonic Fock space whose jump operator $a(\\phi)$ annihilates a particle in the absorbing state $\\phi$ (eq. (6)). Unravelling gives the arrival-time densities $p_n(t)=F_n(\\Omega(t_n))\\prod_{i=1}^n \\omega(t_i)$, where $\\omega(t)=\\langle N\\rangle\\gamma\\,|\\langle\\phi|\\chi_t\\rangle|^2$ is the detection intensity, $\\Omega$ its integral, and $F$ is $(1-\\Omega/N)^N$ for Fock states, $e^{-\\Omega}$ for coherent states, and $(1+\\Omega)^{-1}$ for quasi-free states. In the point-detector limit the intensity is determined by a Volterra integral equation whose solution splits into a stationary transmission amplitude $T_p$ and a transient term; in the uniform-beam limit the intensity becomes $ar_0|T_{p_0}+R_{p_0}(t)|^2$. The general Fisher-information formula (42), together with the large-time asymptotics of this intensity, yields the sparse-beam limits (45)--(46).","core_discovery":"The central discovery is that the momentum $p_0$ of a beam particle is identifiable from arrival times even though the spatial density of the beam contains no information about it: in the delta-detector and uniform-beam limits with vanishing particle density $r_0\\to 0$, the Fisher information from the first $n$ arrival times is nonzero and has the simple form $I_n(p_0)=n\\,I_\\infty(p_0)$ for coherent states and $I_n(p_0)=\\frac{n}{n+2}\\,I_\\infty(p_0)$ for quasi-free (thermal) states, where $I_\\infty(p_0)=\\frac{a^2 m^2}{p_0^2(p_0+am/2)^2}$ with $a$ the detection strength and $m$ the particle mass. The paper derives this by first obtaining a general formula for the Fisher information of the arrival-time model, then taking the sparse-beam limit using a dominated-convergence argument; the limiting value coincides with the Fisher information of a hypothetical time-stationary detection model. It also proves that in the opposite dense-beam limit $r_0\\to\\infty$ the information vanishes, and that in the finite-particle case the information in $n$ detections has a maximum rather than growing monotonically.","pith_inferences":["The nonzero sparse-beam limit is an infinite-time statement: in a finite observation window the probability of collecting $n$ arrivals vanishes as $r_0\\to 0$, so achieving the predicted information requires waiting times that diverge as the beam is diluted.","The ratio $I_n(\\text{coherent})/I_n(\\text{quasi-free})=n+2$ gives a parameter-free signature of the particle-number statistics (Poisson vs geometric) that could be tested purely from temporal data.","The same Fisher-information formula applies to any single-particle parameter, so the framework extends to simultaneous estimation of momentum, mass, and detection strength, where correlations between arrival times would matter.","The predicted transient-vs-stationary structure of the intensity, with its $O(t^{-3/2})$ tail and $p_0$-dependent constant, is directly testable by histogramming first-arrival times of a dilute beam."],"forward_implications":["Momentum estimation from temporal data is possible in a regime where position measurements give nothing: the beam's spatial distribution is flat and parameter-free.","In the beam limit the total probability of observing $n$ arrivals equals 1 for every $n$, so the Fisher information grows with $n$; in any finite-particle beam it instead peaks at an optimal number of detections.","For sparse beams the scaling differs sharply between state types: each additional arrival adds a constant amount of information for coherent states, while for quasi-free states the information saturates at $I_\\infty(p_0)$.","Increasing the particle density $r_0$ towards infinity drives the Fisher information to zero, so sparse beams are information-optimal within this model."],"supporting_citations":[{"why":"Supplies the Lindblad-type master equation for particle absorption that defines the many-particle detection model.","marker":"[36]"},{"why":"Analyses the quantum particle source model whose dynamics the paper second-quantises for the beam setting.","marker":"[37]"},{"why":"Provides the absorption-based 'exit space' approach to arrival-time observables that the detector model builds on.","marker":"[22]"},{"why":"Gives the Volterra integral-equation convergence result used to justify the delta-detector limit.","marker":"[44]"},{"why":"Supplies the classical statistical inference framework, including the Fisher information bound, used throughout the paper.","marker":"[46]"},{"why":"Provides the asymptotic expansions of the error function that yield the large-time intensity limits needed for the sparse-beam result.","marker":"[43]"}],"fun_headline_variants":["Quantum arrival times reveal momentum hidden in flat beams","Sparse beams leak momentum via arrival-time Fisher info","Exact Fisher info for momentum from sparse quantum beam arrivals","Flat beam arrives: Fisher info exposes hidden momentum","Sparse beam limit: arrival times yield exact momentum Fisher info"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result assumes an unbounded observation window: in the sparse-beam limit the detections happen at arbitrarily late times, so the stated Fisher information corresponds to waiting infinitely long, and any finite-time experiment sees the probability of observing $n$ arrivals vanish.","fun_headline_variants_meta":{"raw":{"variants":["Quantum arrival times reveal momentum hidden in flat beams","Sparse beams leak momentum via arrival-time Fisher info","Exact Fisher info for momentum from sparse quantum beam arrivals","Flat beam arrives: Fisher info exposes hidden momentum","Sparse beam limit: arrival times yield exact momentum Fisher info"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001135,"raw_usage":{"total_tokens":4752,"prompt_tokens":1018,"completion_tokens":3734,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":634,"completion_tokens_details":{"reasoning_tokens":3657}},"tokens_in":634,"tokens_out":3734,"duration_ms":28482,"temperature":1.0,"reasoning_tokens":3657,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T13:35:31.866769+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure arrival times of a free one-dimensional Bose beam with a point absorber at two densities and for both coherent and thermal (geometric-number) states; estimate $p_0$ by maximum likelihood from the first $n$ arrival times and check that the inverse estimator variance grows as $n$ for coherent states and saturates as $n/(n+2)$ for thermal states, since the paper's predictions differ by exactly the factor $n+2$.","supporting_citations":[{"cited_title":"General Theory and Applications to Unstable Particles","cited_arxiv_id":null,"evidence_quote":"Supplies the Lindblad-type master equation for particle absorption that defines the many-particle detection model."},{"cited_title":"Quantum collision models: Open system dynamics from repeated interactions","cited_arxiv_id":null,"evidence_quote":"Analyses the quantum particle source model whose dynamics the paper second-quantises for the beam setting."},{"cited_title":"Quantum times of arrival for multiparticle states","cited_arxiv_id":null,"evidence_quote":"Provides the absorption-based 'exit space' approach to arrival-time observables that the detector model builds on."},{"cited_title":"On Linear Volterra Integral Equations of Con- volution Type","cited_arxiv_id":null,"evidence_quote":"Gives the Volterra integral-equation convergence result used to justify the delta-detector limit."},{"cited_title":"Gregoire, “Negative binomial distributions for point processes, Stochastic processes and their applications 16 179 (1983)","cited_arxiv_id":null,"evidence_quote":"Supplies the classical statistical inference framework, including the Fisher information bound, used throughout the paper."},{"cited_title":"Operator Algebras and Quantum Statistical Mechanics II","cited_arxiv_id":null,"evidence_quote":"Provides the asymptotic expansions of the error function that yield the large-time intensity limits needed for the sparse-beam result."}],"review_version":1}