{"id":"acd100ff-bc7a-409d-9ac7-5c4111dec9aa","arxiv_id":"2607.06215","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":4,"one_line_summary":"The optimal sensing time for bath-induced coherence lacks a universal scaling with the Liouvillian gap, but cavity coupling can stabilize this transient metrological advantage into a steady-state resource.","lead":"This paper shows that the optimal time to estimate bath-induced coherence in an open quantum system does not universally scale with the inverse Liouvillian gap, disproving a common heuristic. It also demonstrates that coupling the system to a quantum cavity can turn this transient sensing window into a steady-state resource.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"The exact QFIM decomposition (Eq. 30) and its factorized form (Eq. 38) are never numerically verified against the directly-computed QFI; a direct check would confirm the analytical framework is not just formally correct but numerically stable.","rationale":"The reader correctly identifies the S_ρ invertibility assumption as a potential weak point, but the concern is narrower than the full issue. The more load-bearing problem is that the paper's main analytical contribution—the exact QFIM decomposition in Eq. (30) and its factorized form in Eq. (38)—is never numerically verified against the directly-computed QFI. The mathematical derivation is correct: the Fréchet derivative, spectral decomposition, and Sylvester equation manipulation are all standard and algebraically sound. The factorization F = e^{-2Δt} G(t) is an exact identity. However, formal correctness does not guarantee numerical stability, particularly because S_ρ^{-1} can be ill-conditioned when ρ(t) has small eigenvalues during transient dynamics. The paper uses the analytical framework to interpret the numerical results and to argue for the absence of universal scaling, but never closes the loop by verifying that the decomposition reproduces the QFI curves. That said, the central claim is also supported by the independent numerical evidence (both linear and nonlinear scaling are observed in different regimes) and the two-mode illustrative analysis (Eqs. 49–62). Even without numerical verification of the decomposition, the claim that no single scaling law holds universally is well-motivated. The analytical framework adds generality and interpretive power, but the numerical results alone would support the conclusion. Therefore, the verdict of ACCEPT is appropriate, though correctness_risk should remain 'unknown' or 'moderate' rather than 'low' until the decomposition is numerically cross-checked. The paper would benefit significantly from a single figure overlaying the decomposed QFI (from Eq. 30 or 38) with the directly-computed QFI, which would convert the analytical framework from a formally correct but unverified derivation into a validated tool.","tokens_in":18101,"tokens_out":8464,"duration_ms":449667,"concrete_test":"For the 4-level V-type system with the Liouvillian in Eq. (5), explicitly compute A_{mnkl}(t) from Eq. (29) and G_μν(t) from Eq. (39) at the same parameter values used in Fig. 2 (e.g., p_h = 0.62, p_c = 0.7, T_h = 0.4 and T_h = 3). Then verify that e^{-2Δt} G(t) from Eq. (38) reproduces the directly-computed F_Q(p_h) curves in Fig. 2(a,b) to within numerical precision (e.g., relative error < 10^{-6}). Additionally, check that the stationarity condition Eq. (41), Ġ/G = 2Δ, yields the same optimal times t* as direct numerical maximization of F_Q. If the decomposition deviates from the direct computation, particularly during early transient times where ρ(t) may be near-singular, the analytical framework requires revision or qualification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that no universal scaling exists between t* and 1/Δ—rests on two pillars: (1) numerical results from direct master equation integration (Figs. 2–5) and (2) the general analytical decomposition in Eq. (30), which separates the QFIM into temporal kernels f_{mn}(t) and modal coupling tensors A_{mnkl}(t). The derivation of this decomposition is mathematically sound for diagonalizable L with full-rank ρ. However, the decomposition itself is never numerically evaluated or verified. The numerical QFI curves in Figs. 2–5 are computed directly from ρ(t) and the SLD, not from Eq. (30). Neither the modal coupling tensors A_{mnkl}(t) (Eq. 29) nor the collective function G_μν(t) (Eq. 39) nor the factorization F = e^{-2Δt} G(t) (Eq. 38) is ever explicitly computed and compared against the direct QFI. This matters because the decomposition involves S_ρ^{-1} (Eq. 24), which can become numerically ill-conditioned when eigenvalues of ρ(t) are small during transient dynamics—even if ρ is formally full rank. The modal coupling tensors also involve multiple layers of spectral projection, vectorization, and inversion that could accumulate numerical errors. If the decomposed form fails to reproduce the directly-computed QFI to high precision, the analytical framework used to argue the absence of universal scaling would be unverified in practice, even though it is formally exact. The reader's concern about S_ρ invertibility is related but narrower; the broader issue is that the entire analytical decomposition—the paper's main theoretical contribution—lacks any numerical cross-check.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"The manuscript investigates the estimation of bath-induced coherence (BIC) parameters in a four-level V-type open quantum system coupled to thermal reservoirs. The central claim is that the optimal interrogation time for coherence estimation does not obey any universal scaling with the inverse Liouvillian gap. The authors support this claim through numerical simulations of the quantum Fisher information matrix (QFIM) for a specific model system and through a general analytical decomposition of the QFIM into temporal kernels (spectral contribution) and modal coupling tensors (statistical contribution) derived from standard Markovian master equation theory. The paper further demonstrates that coupling the system to a quantum cavity can shift the transient metrological advantage to a sustained steady-state resource under appropriate thermal bias.","tokens_in":18431,"tokens_out":1029,"duration_ms":238501,"significance":"The paper addresses a timely question in open-system quantum metrology: whether the Liouvillian gap universally determines the optimal sensing time. The analytical QFIM decomposition (Eq. 30) into spectral and statistical components is a parameter-free derivation that provides a clear framework for understanding why inverse-gap scaling can fail. The numerical results illustrating that linear (or nonlinear) scaling is not a reliable signature of unimodal (or multimodal) dynamics are instructive. The demonstration of cavity-induced steady-state metrological enhancement adds practical value. The self-citations are appropriate for the model system employed.","major_comments":[{"comment":"§V, Eqs. (27)-(30): The exact QFIM decomposition and its factorized form (Eq. 38) are never numerically verified against the directly computed QFI from the SLD. The numerical QFI curves in Figs. 2-5 are computed directly from ρ(t), not from the decomposed form. A direct comparison—evaluating the right-hand side of Eq. (30) or Eq. (38) and checking agreement with the directly computed QFI—would confirm that the analytical framework is not only formally exact but numerically stable. This is particularly important because the decomposition involves S_ρ^{-1} (Eq. 24), which can become ill-conditioned when eigenvalues of ρ(t) are small during transient dynamics. Without this verification, the analytical framework used to argue the absence of universal scaling remains unvalidated in practice.","section":null},{"comment":"§V, Eq. (38): The factorization F = e^{-2Δt} G(t) extracts a universal exponential envelope set by the Liouvillian gap. While mathematically exact, the claim that the optimal time is determined by Ġ/G = 2Δ (Eq. 41) is then used to argue that no universal scaling exists. However, this factorization itself shows that the gap always sets one component of the dynamics. The manuscript should clarify more precisely what 'no universal scaling' means in light of this exact factorization: is the claim that G(t) has no universal form, or that the solution to Eq. (41) has no universal dependence on Δ? The current phrasing risks understating the role of Δ that Eq. (38) itself establishes.","section":null}],"minor_comments":[{"comment":"§II, Eq. (5): The Liouvillian matrix is written in the reduced picture ρ = {ρ_11, ρ_22, ρ_aa, ρ_bb, ℜ(ρ_12)}. It would help the reader to explicitly state the basis ordering and clarify whether the imaginary part of ρ_12 is decoupled or zero.","section":null},{"comment":"§III: The observation that one mode always satisfies c_m = 1/√2 is intriguing but its origin is not explained. A brief comment on whether this is a structural feature of the model or a numerical coincidence would strengthen the discussion.","section":null},{"comment":"§IV, Fig. 8: The caption for panel (b) references 'Optimal estimation time t*(p_c)' but the text discusses saturation time t_∞ in panel (c). The panel labeling and caption should be checked for consistency.","section":null},{"comment":"§V, Eq. (33): The two-mode truncation uses f_i(t) = t e^{-γ_i t}, which corresponds to the degenerate kernel (λ_m = λ_n case in Eq. 18). The manuscript should state whether this degenerate form is assumed for simplicity or whether the two dominant modes are genuinely degenerate.","section":null},{"comment":"Throughout: Several typographical issues (e.g., 'coeherences', 'photosytnthetic', 'sill' for 'still', 'magnetude') should be corrected.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The stress-test concern about numerical verification of the QFIM decomposition is well-placed and is the primary reason for the minor revision recommendation. The analytical derivation itself appears correct for diagonalizable Liouvillians with full-rank ρ, but the practical stability of S_ρ^{-1} during transient dynamics is a legitimate concern that a single verification plot would address. The paper is otherwise a solid contribution to the open-system metrology literature."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful reading and constructive feedback. Both major comments are well-taken and will be addressed in the revised manuscript.","responses":[{"response":"The referee is correct that the analytical decomposition in Eq. (30) and its factorized form in Eq. (38) are not numerically verified against the directly computed QFI in the present manuscript. This is a legitimate gap, and we will address it in the revision. Specifically, we will add a new figure (or panel) in which we evaluate the right-hand side of Eq. (30) using the Liouvillian spectral projectors, modal response vectors, and the Sylvester superoperator S_ρ, and overlay the result on the directly SLD-computed QFI curves from Figs. 2–5. This will confirm numerical agreement (or reveal any discrepancies arising from truncation or conditioning). Regarding the ill-conditioning concern: the referee rightly notes that S_ρ^{-1} can become problematic when eigenvalues of ρ(t) are small during transient dynamics. In our model system, the density matrix remains full-rank throughout the evolution (the four-level system does not develop zero eigenvalues at any finite time for the parameter regimes considered), so S_ρ is invertible. However, we agree that this should be stated explicitly and that the condition number of S_ρ should be monitored during the comparison. We will include a brief discussion of the conditioning and note that for systems where ρ(t) becomes rank-deficient, the Moore–Penrose pseudoinverse (as mentioned in the manuscript) should be used. We will also verify that the pseudoinverse formulation gives consistent results in any near-singular regimes encountered.","revision_made":"yes","referee_comment":"§V, Eqs. (27)-(30): The exact QFIM decomposition and its factorized form (Eq. 38) are never numerically verified against the directly computed QFI from the SLD. The numerical QFI curves in Figs. 2-5 are computed directly from ρ(t), not from the decomposed form. A direct comparison—evaluating the right-hand side of Eq. (30) or Eq. (38) and checking agreement with the directly computed QFI—would confirm that the analytical framework is not only formally exact but numerically stable. This is particularly important because the decomposition involves S_ρ^{-1} (Eq. 24), which can become ill-conditioned when eigenvalues of ρ(t) are small during transient dynamics. Without this verification, the analytical framework used to argue the absence of universal scaling remains unvalidated in practice."},{"response":"We agree that the current phrasing could be misread as understating the role of Δ. The referee's observation is accurate: Eq. (38) shows that Δ always sets the universal exponential envelope, and Eq. (41) shows that Δ appears explicitly in the stationarity condition. Our claim is not that Δ is irrelevant—clearly it is not. Rather, the claim is twofold: (1) G(t) has no universal functional form, as it depends on the full Liouvillian spectrum (through the reduced kernels f̃_mn), the modal coupling tensors A^{μν}_{mnkl}(t), and the instantaneous quantum statistical metric, all of which are system- and parameter-dependent; and (2) the solution t* to Eq. (41) therefore has no universal dependence on Δ alone, because the ratio Ġ/G depends on quantities that are not determined by Δ. In other words, while Δ always enters the stationarity condition, it does not uniquely determine t*—the latter depends on the interplay between Δ and the multimode/statistical content of G(t). This is why both linear and nonlinear t*(1/Δ) behavior can arise depending on the system parameters, as our numerical results demonstrate. We will revise the manuscript to state this more precisely, replacing the somewhat loose phrase 'no universal scaling' with the more specific statement that 'the optimal interrogation time is not uniquely determined by the Liouvillian gap, because the stationarity condition Eq. (41) depends on the full spectral and statistical structure of G(t), which has no universal form.' We will also explicitly acknowledge that Δ sets the exponential decay envelope and appears in the stationarity condition, so it always plays a role—just not a sole or universally predictive one.","revision_made":"yes","referee_comment":"§V, Eq. (38): The factorization F = e^{-2Δt} G(t) extracts a universal exponential envelope set by the Liouvillian gap. While mathematically exact, the claim that the optimal time is determined by Ġ/G = 2Δ (Eq. 41) is then used to argue that no universal scaling exists. However, this factorization itself shows that the gap always sets one component of the dynamics. The manuscript should clarify more precisely what 'no universal scaling' means in light of this exact factorization: is the claim that G(t) has no universal form, or that the solution to Eq. (41) has no universal dependence on Δ? The current phrasing risks understating the role of Δ that Eq. (38) itself establishes."}],"tokens_in":17825,"tokens_out":1342,"duration_ms":101085,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"The main thing to know: this paper proves analytically that the optimal interrogation time for estimating bath-induced coherence in a Markovian open system does not universally scale with the inverse Liouvillian gap, and it provides a spectral decomposition of the QFIM (Eq. 30) that separates the gap's exponential envelope from a multimode function G(t). That decomposition is the real contribution here — it is parameter-free, derived from standard master equation theory, and gives a transparent reason why linear or nonlinear scaling between t* and 1/Δ can occur under either unimodal or multimodal dynamics. The two-mode truncation analysis (Eqs. 49–58) is particularly clear: it shows exactly when inverse-gap scaling emerges and when it breaks down, depending on the relative statistical weights and relaxation rates. The numerical work in Section III (Figs. 2–5) supports the analytical claims by computing QFI directly from the master equation and examining modal expansion coefficients. The observation that a mode with c_m = 1/√2 always exists is a nice structural detail. The cavity-coupling extension showing transient-to-steady-state conversion under thermal bias is a useful addition, though it is more illustrative than rigorous. The stress-test concern lands: the analytical decomposition in Eq. (30) and its factorized form in Eq. (38) are never numerically evaluated against the directly computed QFI. The numerical QFI curves come from ρ(t) and the SLD, not from the spectral decomposition. This matters because the decomposition involves S_ρ^{-1}, which can become ill-conditioned when eigenvalues of ρ(t) are small during transient dynamics. The derivation is formally exact for diagonalizable L with full-rank ρ, and the pseudoinverse fallback is mentioned, but without a numerical cross-check we cannot confirm the decomposition is stable in practice. This is the main gap between formal correctness and verified utility. The reader's concern about S_ρ invertibility is real but standard for this literature — it does not undermine the central result, which is the structural decomposition itself. The self-citations are appropriate; they establish the model system and prior numerical work that this paper generalizes. This paper is for researchers in dissipative quantum metrology and open quantum systems who work with Liouvillian spectral methods. It deserves a serious referee who can verify the algebra in Section V and push for the missing numerical cross-check. I recommend accepting for peer review.","headline":"QFIM spectral decomposition is a clean new analytical result; the numerical cross-check gap is the main weakness","tokens_in":18937,"tokens_out":566,"would_cite":false,"duration_ms":147282,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["03.65.Yz","03.67.-a","05.70.Ln"],"model":"glm-5.2","headline":"Optimal sensing time decouples from Liouvillian gap","keywords":["quantum metrology","Liouvillian gap","quantum Fisher information","bath-induced coherence","open quantum systems","spectral decomposition","optimal interrogation time"],"falsifier":"A concrete demonstration that the QFIM decomposition in Eq. (27) fails or gives qualitatively wrong predictions when the density matrix becomes rank-deficient during transient dynamics, showing that the pseudoinverse replacement does not preserve the analytical conclusions.","tokens_in":18265,"feed_emoji":"🔬","tokens_out":875,"duration_ms":134480,"temperature":0.7,"pith_summary":"The paper proves that the optimal interrogation time for estimating bath-induced coherence in a finite open quantum system does not obey any universal scaling with the inverse Liouvillian gap. The author decomposes the quantum Fisher information matrix into temporal kernels (spectral) and modal coupling tensors (statistical), showing that the optimal time depends on the full Liouvillian spectrum and the evolving quantum statistical metric, not just the gap.","feed_headline":"Sensing time breaks free from Liouvillian gap","feed_subtitle":"Optimal quantum coherence estimation depends on the full spectral structure, not just the slowest decay rate.","key_machinery":"Liouvillian spectral decomposition, Sylvester superoperator, temporal kernels f_{mn}(t), modal coupling tensors A_{mnkl}(t), quantum Fisher information matrix (QFIM), bath-induced coherence parameters (p_h, p_c)","core_discovery":"The paper's central analytical result is the exact decomposition of the QFIM into temporal kernels f_{mn}(t) determined by the Liouvillian spectrum and modal coupling tensors A_{mnkl}(t) encoding the instantaneous quantum statistical metric. This decomposition shows that the QFIM factorizes as F(t) = e^{-2Δt} G(t), where Δ is the Liouvillian gap and G(t) is a genuinely multimode quantity. The optimal interrogation time satisfies Ġ/G = 2Δ, meaning it is set by the balance between the universal decay rate and the internal redistribution of information among all Liouvillian modes, not by the gap alone.","pith_inferences":["The factorization F(t) = e^{-2Δt} G(t) suggests that systems with small gaps but rich multimode structure could exhibit optimal sensing times much shorter or longer than 1/Δ, depending on whether G(t) evolves faster or slower than the gap decay.","The observation that one mode always carries a fixed weight c_m = 1/√2 regardless of parameters hints at a structural invariant in the Liouvillian eigenspace of V-type systems, which could constrain the achievable metrological precision.","If the Sylvester superoperator S_ρ becomes singular during transient dynamics (rank-deficient density matrix), the pseudoinverse replacement may not faithfully capture the true QFIM behavior, potentially limiting the analytical proof to full-rank regimes."],"forward_implications":["If the result generalizes, inverse-gap scaling of optimal sensing times observed near dissipative phase transitions may be coincidental rather than fundamental, and the actual scaling could depend on the full spectral structure.","The finding that linear scaling can arise from multimode dynamics while nonlinear scaling can arise from unimodal dynamics means that observing either scaling behavior cannot be used to infer the underlying mode structure.","The cavity-coupling result suggests that engineering a thermal bias combined with coherent control could convert transient metrological advantages into sustained steady-state resources in other open quantum systems.","The decomposition F(t) = e^{-2Δt} G(t) provides a practical tool for identifying which Liouvillian modes actually contribute to sensing, by examining G(t) independently of the gap envelope."],"fun_headline_variants":["Quantum sensing time set by multimode decay, not Liouvillian gap","Optimal coherence sensing depends on full Liouvillian spectrum","Coherence sensing time uncoupled from Liouvillian gap","Multimode dynamics dictate optimal quantum coherence sensing","Liouvillian gap does not fix quantum coherence sensing time"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The analytical proof assumes the Liouvillian is diagonalizable and the density matrix is full rank throughout the transient dynamics, so that the Sylvester superoperator S_ρ is invertible. If the density matrix becomes rank-deficient, the Moore-Penrose pseudoinverse may not capture the true metrological behavior.","fun_headline_variants_meta":{"raw":{"variants":["Quantum sensing time set by multimode decay, not Liouvillian gap","Optimal coherence sensing depends on full Liouvillian spectrum","Coherence sensing time uncoupled from Liouvillian gap","Multimode dynamics dictate optimal quantum coherence sensing","Liouvillian gap does not fix quantum coherence sensing time"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":884,"prompt_tokens":474,"completion_tokens":410,"prompt_tokens_details":null},"tokens_in":474,"tokens_out":410,"duration_ms":22160,"temperature":1.0,"reasoning_tokens":340,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-08T12:58:46.191376+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"A concrete demonstration that the QFIM decomposition in Eq. (27) fails or gives qualitatively wrong predictions when the density matrix becomes rank-deficient during transient dynamics, showing that the pseudoinverse replacement does not preserve the analytical conclusions.","supporting_citations":[],"review_version":1}