{"id":"6ae494d8-0444-4058-9ae8-3ccd5bc220fc","arxiv_id":"2603.07556","paper_version":2,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Single-mode quantum Fisher information for a coherent-plus-squeezed-vacuum Mach-Zehnder equals the two-mode value near the black fringe, so single-mode readout is optimal and Heisenberg-scaling.","lead":"A squeezing-enhanced interferometer with only one output measured still reaches the same quantum precision limit as measuring both outputs near the dark fringe. This matters because single-mode readout is far simpler in practice for gravitational-wave detectors and other sensors.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified; the black-fringe limiting QFI equals the two-mode bound under the standard Gaussian formula.","rationale":"The strongest claim is the equality Q\theta\theta0+ = F0 near the black fringe, which immediately implies optimality of single-mode readout and Heisenberg scaling. The calculation rests on a transparent Williamson decomposition of a single-mode Gaussian state followed by substitution into a published QFI formula. The only technical subtlety is the pure-state discontinuity when ∂^{2}\theta λ \neq 0; the paper handles it by evaluating both sides of the jump and obtains the expected two-mode value. That procedure is standard and does not introduce circularity or hidden parameters. The abstract's claim about the optimal local measurement (phase-sensitive amplification + photon counting) is not derived in the body, but it is not required for the QFI equality itself. Consequently the reader's ACCEPT / high-confidence verdict is unchanged. The concrete cross-check against an independent Gaussian-QFI formula would further harden the result but is not expected to alter it.","tokens_in":7727,"tokens_out":652,"duration_ms":7686,"concrete_test":"Independently recompute Q\theta\theta from the general single-mode Gaussian formula of Pinel et al. (Phys. Rev. A 88, 040102, 2013) or Gao & Lee (Eur. Phys. J. D 68, 347, 2014) using the same moments d, CN, CA (Eqs. 8–11); confirm that the \theta \to 0 limit recovers |α|^{2} e^{2r} + sinh^{2} r. If it does, the limiting procedure is corroborated and the central claim stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The reader's weakest assumption correctly flags the only soft spot: whether the mixed-state QFIM expressions (Eqs. 15, 17, 18 from Šafránek 2019) remain valid as λ \to 1 with ∂²\theta λ = sinh^{2} r \neq 0, so that the jump value Q\theta\theta0+ = |α|^{2} e^{2r} + sinh^{2} r can be used as the operational bound arbitrarily close to the pure black fringe. The paper itself notes the possible discontinuity and evaluates both the pure-state formula at \theta = 0 and the limiting mixed-state formula, obtaining exactly F0. Because the algebra is elementary once the Williamson parameters (Eqs. 12–14) are substituted, and because the same limiting procedure is standard for Gaussian QFI, this does not undermine the central equality. No other load-bearing gap appears: the two-mode comparison is direct, Heisenberg scaling follows for large |α|, and the abstract's optimal-measurement claim is secondary.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies a Mach–Zehnder interferometer with coherent light and squeezed vacuum at the two inputs and computes the quantum Fisher information (QFI) for the difference phase θ from the reduced single-mode Gaussian state at one output. Using the Williamson decomposition of that state and published formulae for the QFI of single-mode Gaussian states, the authors obtain the limiting value Q_θθ → |α|² e^{2r} + sinh² r as θ approaches the black fringe, which coincides with the known two-mode QFI F_0. They conclude that single-mode readout is therefore optimal for estimating θ in the practically relevant regime and permits Heisenberg scaling of precision for large total photon number. Photon-number (N) precision is also derived and shown to be strictly suboptimal relative to the QFI near the black fringe.","tokens_in":7970,"tokens_out":893,"duration_ms":28305,"significance":"The result is of clear practical interest: two-mode readout is often technically difficult (e.g., in gravitational-wave interferometry), so establishing that the single-mode QFI saturates the two-mode bound near the black fringe is a useful and non-obvious statement. The calculation is parameter-free, relies only on standard Gaussian-state methods and an independent literature value of F_0, and yields a clean equality rather than a numerical bound. The explicit comparison of N-precision versus QFI, including the weak-field regime, further clarifies when an optimal measurement would outperform simple photon counting. These strengths make the manuscript a solid contribution to quantum metrology of squeezing-enhanced interferometers.","major_comments":[{"comment":"The abstract asserts that “the optimal local measurement in the vicinity of the black fringe consists of amplifying the output field in a phase-sensitive way and measuring its photon number.” No derivation, argument, or even discussion of this measurement appears in the body (the N-precision section treats only direct photon counting, which is shown to be suboptimal). Either the derivation must be added or the claim must be removed from the abstract; as written the manuscript overstates what is demonstrated.","section":null},{"comment":"Introduction and Conclusions state that “the single-mode QFI on the difference phase is also given by F_0,” while the explicit evaluation (Eqs. 19) and the abstract restrict the equality to the vicinity of the black fringe. Because single-mode QFI cannot exceed the two-mode value, equality only at the operating point is sufficient for the optimality claim, but the wording should be made consistent throughout so that the reader is not left unsure whether Q_θθ_θ = F_0 for all θ.","section":null}],"minor_comments":[{"comment":"Notation “20rlge” (Figs. 2–3 and surrounding text) is unintelligible; replace by the conventional expression for squeezing in dB (e.g., −10 log10(e^{−2r}) or 8.686 r dB).","section":null},{"comment":"The Heisenberg-scaling remark (“Q ∼ ⟨N+⟩²”) is correct only when the total photon number is increased while the coherent/squeezed allocation is optimized; a brief clause clarifying this would prevent misreading for fixed r.","section":null},{"comment":"Fig. 3 plots QFI for all θ yet the analytic expression used for intermediate θ is never written down; a short formula or reference to the substituted Eq. (15) would aid reproducibility.","section":null},{"comment":"Typographical inconsistencies: “modesaandb”, “phase shiftsθ a andθ b”, missing spaces after commas in several equations, and “20rlge=12.5 dB”.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The central algebraic result is sound and the soft spot flagged by the stress test (validity of the mixed-state QFI formulae as λ → 1) is handled carefully by the authors themselves. The only real obstacle to acceptance is the unsupported optimal-measurement sentence in the abstract; once that is fixed the paper is ready. Scope is appropriate for a quant-ph / quantum-optics journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is simple: for the standard Caves setup (coherent + squeezed vacuum into an MZI), the quantum Fisher information of the reduced single-mode state on the difference phase jumps, in the vicinity of the black fringe, to exactly the two-mode value F0 = |α|² e^{2r} + sinh² r. Single-mode readout is therefore optimal and still permits Heisenberg scaling. That equality was not previously written down.\n\nWhat they do well is the algebra. They give the output field, extract the mean and the complex covariance, perform the Williamson decomposition (λ, r_out, S), and plug straight into Šafránek’s mixed-Gaussian QFI formulae. The off-diagonal Q_Φθ vanishes, the pure-state value at θ=0 is |α|² e^{2r}, and the limiting mixed-state value is F0. The comparison with ordinary photon-number precision is plotted cleanly and shows the expected gap, especially at modest |α|. Citations to the two-mode literature and to the Gaussian-QFI toolkit are accurate and non-circular.\n\nSoft spots are minor and do not break the result. The abstract asserts that the optimal local measurement is phase-sensitive amplification followed by photon counting; the body only shows that QFI exceeds N-precision and never derives that particular measurement. The handling of the discontinuity at λ→1 (where ∂²λ ≠ 0) relies on the limiting procedure already present in Šafránek 2019; they evaluate both sides and obtain the expected jump, so the operational claim near the fringe is standard rather than novel. Neither issue affects the central equality.\n\nThis is for people who design or analyse squeezing-enhanced interferometers (LIGO-style, nonlinear SU(1,1), mid-IR spectroscopy). The calculation is short, reproducible by hand, and settles a practical question that experimentalists actually care about. I would send it to a serious referee without hesitation; the math is solid enough that any remaining discussion will be about presentation and the un-derived measurement claim, not about correctness.","headline":"Clean, transparent calculation that single-mode QFI near the black fringe equals the known two-mode F0, so discarding one port costs nothing asymptotically.","tokens_in":8593,"tokens_out":525,"would_cite":true,"duration_ms":58227,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Single-mode readout of a squeezed-light interferometer reaches the same quantum precision limit for the difference phase as measuring both outputs, near the black fringe.","keywords":["quantum Fisher information","squeezing-enhanced interferometry","Mach-Zehnder interferometer","single-mode readout","Heisenberg limit","Gaussian states","phase estimation","black fringe"],"falsifier":"Measure or compute the classical Fisher information of phase-sensitive amplification plus photon counting on the single kept mode for a sequence of small difference phases approaching zero, and check whether it approaches |α|² e^{2r} + sinh² r rather than the pure-state black-fringe value or ordinary photon-counting precision.","tokens_in":8627,"feed_emoji":"🔭","tokens_out":861,"duration_ms":17827,"temperature":0.7,"pith_summary":"This paper asks whether you really need to measure both outputs of a squeezing-enhanced interferometer to reach the best possible precision on the phase difference. The authors compute the quantum Fisher information carried by the mixed Gaussian state that appears in just one output mode when the inputs are coherent light and squeezed vacuum. Near the black fringe that information jumps to exactly the known two-mode value, so discarding the other mode costs nothing asymptotically. The same bound scales with the square of the total photon number, i.e., Heisenberg scaling is still available. An optimal practical measurement is phase-sensitive amplification of the kept mode followed by photon counting. The result matters because two-mode readout is often technically hard, especially in gravitational-wave detectors and other strong-field instruments.","feed_headline":"One output mode matches two-mode quantum phase limit","feed_subtitle":"Near the black fringe, discarding one arm costs nothing in Fisher information for squeezed interferometers.","key_machinery":"The compact quantum-Fisher-information formula for a single-mode Gaussian state given its Williamson decomposition (symplectic eigenvalue λ and symplectic matrix S). Applied to the output covariance matrix of the discarded-mode interferometer, it produces a Q_θθ that equals the two-mode bound just off the pure-state black fringe.","core_discovery":"In the vicinity of the black fringe the single-mode quantum Fisher information element for the difference phase equals the two-mode value F_0 = |α|² e^{2r} + sinh² r. Therefore single-mode readout is optimal for phase estimation in squeezing-enhanced interferometry and still permits Heisenberg scaling of precision.","pith_inferences":["The same single-mode optimality may carry over to high-gain nonlinear interferometers used for mid-infrared sensing with undetected photons, where one mode is routinely discarded.","A direct laboratory test would compare maximum-likelihood estimates from amplified single-mode counts against simultaneous two-mode photon-number data at identical power and squeezing.","If the formal jump in quantum Fisher information at the pure-state points is mainly a mathematical discontinuity, continuous adaptive locking slightly off the black fringe could still harvest the full F_0 bound in practice."],"forward_implications":["Optimal estimators can be built that act only on one output mode without asymptotic loss relative to full two-mode readout.","Near the black fringe the single-mode precision bound scales as the square of the total mean photon number (Heisenberg limit).","Plain photon-number detection is suboptimal; phase-sensitive amplification before counting saturates the single-mode quantum bound near the black fringe.","With balanced beam splitters the sum and difference phases remain independently estimable from the single kept mode."],"fun_headline_variants":["Single-mode readout matches two-mode quantum phase limit near black fringe","Squeezed interferometer single output reaches full Heisenberg scaling","Near black fringe one mode equals two-mode Fisher information for phase","Single-mode readout optimal for squeezing-enhanced phase estimation","Discarding one arm costs nothing in phase precision near black fringe"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The Gaussian-state quantum Fisher information formula stays valid and operationally meaningful when the state purity approaches one and the second derivative of the symplectic eigenvalue is nonzero, so the limiting value just off the black fringe can be used as the attainable precision bound.","fun_headline_variants_meta":{"raw":{"variants":["Single-mode readout matches two-mode quantum phase limit near black fringe","Squeezed interferometer single output reaches full Heisenberg scaling","Near black fringe one mode equals two-mode Fisher information for phase","Single-mode readout optimal for squeezing-enhanced phase estimation","Discarding one arm costs nothing in phase precision near black fringe"]},"model":"grok-4.5","effort":"low","cost_usd":0.004808,"raw_usage":{"total_tokens":1292,"prompt_tokens":688,"num_sources_used":0,"completion_tokens":90,"cost_in_usd_ticks":48080000,"prompt_tokens_details":{"text_tokens":688,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":514,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":688,"tokens_out":90,"duration_ms":4781,"temperature":1.0,"reasoning_tokens":514,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-15T13:11:05.961256+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Measure or compute the classical Fisher information of phase-sensitive amplification plus photon counting on the single kept mode for a sequence of small difference phases approaching zero, and check whether it approaches |α|² e^{2r} + sinh² r rather than the pure-state black-fringe value or ordinary photon-counting precision.","supporting_citations":[],"review_version":1}