{"id":"7d2432ab-29c8-42af-92c9-78d3023a3ab2","arxiv_id":"2608.05046","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper derives a gauge-invariant excitation-detection weight from left and right Liouvillian eigenoperators and uses its phase-space maps to identify mode-suppressing preparations and projected slow-state routing in driven Kerr resonators.","lead":"A new analysis framework for open quantum systems separates how a slow decay mode is excited from how it deforms the state and how it is seen by measurements, using matched left and right eigenoperators of the Liouvillian. It is applied to driven Kerr resonators to find preparations that suppress the slow switching mode and to map bias-induced routing changes in the slow subspace.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The modal-weight factorization is mathematically sound, but the Fig. 5 routing crossovers rest on a representative-dependent three-coordinate projection that the paper itself flags in App. A3 as unresolved; this is the weakest load-bearing point.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing point: the three-coordinate slow-sector projection and the representative selection govern the routing crossovers and the A_amb partition, with only qualitative stability reported. My independent reading confirms that the formal spectral framework in Sec. II is sound: the modal weight in Eq. (8) is invariant under the mode gauge in Eq. (6), Eq. (9) is the correct diagonalizable expansion, and the response/residue statements in Sec. IIA follow from the same factorization. I found no internal inconsistency in the central definitional claim. The concern is therefore not about the core theory but about whether the Kerr application's headline routing results are robust properties of the full Liouvillian or artifacts of a representative-dependent reduction. The paper is unusually honest about this dependence, but honesty does not remove the need for a quantitative sensitivity check. Because the reader already returned CONDITIONAL on essentially this basis and recommended exactly such validation, my stress-test does not move the verdict; it sharpens the test that would settle the issue. I therefore recommend UNCHANGED, with the concrete test above as the condition that would convert the conditional acceptance into a firm one if the crossings persist, or would downgrade the routing claim if they do not.","tokens_in":21377,"tokens_out":3749,"duration_ms":47752,"concrete_test":"At the biased operating point (Delta/U = 5, G/U = 0.4, kappa/U = 0.1, N = 5), recompute R_+ and R_- from Eq. (72) for three alternative constructions: (i) representatives selected by PCCA+-style fuzzy clustering instead of extremal vertices; (ii) a four-coordinate reduction that includes the next slowest nonstationary mode; and (iii) representatives fixed at F = 0 and continued by the matrix C. If the zero crossings of R_+ and R_- shift by more than the displayed bias step, or disappear under (ii), then Fig. 5 reports a projection artifact rather than a robust routing crossover.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central definitional claim—that a resolved mode is operationally captured by (lambda_k, left eigenoperator, right eigenoperator) and that the excitation-detection product w_k is gauge invariant—does not contain a mathematical gap: it follows directly from the biorthonormal expansion in Eq. (9) and the reciprocal gauge transformation in Eq. (6). The load-bearing risk is instead in the application to projected routing. In Sec. IIID and App. A3, representatives {sigma_c, sigma_+, sigma_-} are chosen as extremal points of the projected slow manifold, and the reduced generator Q = C^{-1} L_slow C is then interpreted as a stochastic three-state model. The paper repeatedly states that the coordinates, the generator, and Fig. 6's A_amb(alpha) are representative dependent (App. A3, Fig. 6 caption), but it does not quantify how the central Fig. 5 diagnostics—the zero crossings of R_+ and R_- in Eq. (72) and the sharp feature in Xi_sw—change under admissible representative choices or under enlargement of the retained slow subspace. Because the biased sweep keeps only the stationary mode plus two tracked nonstationary modes, a third slow mode appearing under bias would invalidate the completeness of the three-coordinate sector and could move or erase the claimed channel-selective crossovers. This is not a disagreement with consensus; it is an internal robustness gap in the argument that the routing crossovers are properties of the slow sector rather than of the chosen projection.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an operational spectral theory of Liouvillians, arguing that each resolved mode should be characterized by the triple (eigenvalue, left eigenoperator, right eigenoperator) rather than by the eigenvalue alone. The left eigenoperator determines excitation by an input or source, the right eigenoperator gives the propagated density deformation and readout overlap, and their product defines a gauge-invariant modal weight (Eq. 8). The framework is applied to driven Kerr resonators in three regimes: linear drive yields a switching mode with coherent-state excitation maps whose zero contours suppress the mode; parametric drive separates odd and even symmetry-resolved slow channels; and one-photon bias mixes these channels into a three-coordinate slow sector, from which a projected stochastic generator is reconstructed. The paper reports channel-selective routing crossovers in the biased case and shows that the coherent-preparation partition continues to deform after routing saturates. The conclusions are carefully hedged: the authors explicitly state that the routing crossovers are properties of the projection and that representative choices affect the reduced coordinates.","tokens_in":21653,"tokens_out":3101,"duration_ms":33375,"significance":"If the central claims hold, the paper provides a useful operational vocabulary for open quantum systems, separating spectral persistence from protocol-dependent visibility and giving phase-space tools for mode-selective preparation. The modal-weight factorization in Eqs. (7)-(10) is definitionally sound and gauge invariant under reciprocal eigenoperator rescaling, and the paper is unusually explicit about the conditions under which the reduced slow-sector generator admits a stochastic interpretation. The applications to driven Kerr resonators illustrate the framework concretely and identify falsifiable statements about coherent-state preparations, such as the zero contours of excitation maps. The main limitation, which the authors themselves flag in App. A3, is that the quantitative routing diagnostics in Fig. 5 depend on the representative selection and on the retained three-dimensional slow subspace; the paper does not currently quantify how robust those diagnostics are. This is a correctness-risk concern for the application section rather than a flaw in the core spectral theory, which appears mathematically sound.","major_comments":[{"comment":"The routing crossovers in Fig. 5, defined by the zero crossings of R_+ and R_- in Eq. (72), are computed from the projected generator Q = C^{-1} L_slow C, where the representatives {sigma_c, sigma_+, sigma_-} are selected as extremal points of the sampled slow manifold (App. A3). The paper repeatedly states that the coordinates, generator, and Fig. 6 partition are representative dependent, but it does not quantify how the crossover locations or the sharp feature in Xi_sw change under admissible variations of the representative set or under enlargement of the retained slow subspace. This is load-bearing for the central claim of bias-induced channel-selective routing, because a different representative choice or an additional slow mode could shift or erase the claimed crossovers. The authors should provide a stability analysis: perturb the representative construction, include a third nonstationary mode in the slow sector, and report the resulting changes in R_+, R_-, and Xi_sw.","section":"Sec. IIID, Fig. 5, App. A3"},{"comment":"The numerical results are presented without quantitative convergence data. The paper states that the reported stationary states, retained spectral data, and phase-space maps are 'stable under enlargement of the numerical Hilbert space' (Sec. IIIA), but no truncation dimension, solver tolerance, or residual values are reported. The residual definitions in Eqs. (A14)-(A15) and the admissibility checks in Eqs. (A33)-(A34) are described, yet no actual numbers or convergence plots are provided. Without these, the reader cannot assess whether the zero contours in Fig. 4 and the routing rates in Fig. 5 are converged with respect to Hilbert-space truncation. Please report the Fock-space dimension used, the numerical tolerances, and a representative convergence scan (e.g., eigenvalue and excitation-map values versus truncation size).","section":"Sec. IIIC, App. A2, App. A4"},{"comment":"The reconstruction protocol for excitation maps is described as a direct consequence of the modal-weight factorization, but it is not demonstrated on data. The protocol in Eqs. (27)-(29) requires joint fitting of transients to common poles and then separating excitation maps from detection factors; this involves nontrivial numerical fitting and error propagation. Since the paper does not claim experimental implementation, this is not a fatal gap, but the manuscript should state more clearly whether the protocol is a proposal or a demonstrated reconstruction. If it is a central contribution, a numerical demonstration on synthetic data with known modes would strengthen the claim; otherwise it should be moved to the outlook section as a future step.","section":"Sec. IIB"}],"minor_comments":[{"comment":"The parity covariance in Eq. (44) is written as Z_2(ρ) = Π ρ Π^†, which is clear, but the subsequent text uses 'Z2' both as a superoperator and as a label for the symmetry group. Please use a consistent notation, e.g., calligraphic script for the superoperator.","section":"Eq. (44)"},{"comment":"There is a typographical error in the full text: 'anoperational spectral theory' should read 'an operational spectral theory'. Please correct this and scan for similar spacing issues.","section":"Sec. II, paragraph after Eq. (6)"},{"comment":"The caption contains the placeholder text '□10□1 □10□6' in the axis labels, which appears to be a rendering artifact. The intended values ('10^{-1}' and '10^{-6}' on the logarithmic decay-rate axis) should be inserted.","section":"Fig. 1 caption"},{"comment":"The definition of chi_e in Eq. (67) uses absolute values that are then dropped in the final equality assuming nonnegative rates. The text should state explicitly that the absolute values are redundant under the nonnegativity assumption, or define chi_e without absolute values and then state the nonnegativity condition separately.","section":"Eq. (67) and surrounding text"},{"comment":"The quantity A_amb(alpha) is called an 'ambiguity' measure, but its interpretation as a 'top-two margin complement' is only clear after reading the definition. Please add a sentence explaining that it is large when the two leading coordinates are nearly equal, and that it is not a physical probability or a distance.","section":"Sec. IIID, Eq. (81)"}],"recommendation":"major_revision","confidential_remarks":"The core spectral theory in Sec. II is sound and the paper is well written, but the application section's main quantitative claim—the bias-induced routing crossovers—rests on a representative-dependent projection whose robustness is not demonstrated. This is fixable with additional numerical analysis and does not require changing the theoretical framework. I would also encourage the author to consider whether the reconstruction protocol should be presented as a demonstrated tool or explicitly labeled as a proposal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: the paper's core claim is definitionally sound. The modal weight w_k = Tr(O r_k) Tr(l_k† X) is gauge-invariant, and the separation into excitation and detection factors follows directly from the biorthonormal expansion. That framing is genuinely useful, even though much of it is implicit in existing response theory. The new content is the coherent-state excitation maps E_k(X,P) from left eigenoperators, their zero contours as mode-suppression manifolds, and the projected three-state routing analysis for the biased Kerr resonator.\n\nThe paper is careful and honest. It flags mode-gauge freedom, cluster-level Riesz projectors, Petermann conditioning, and the representative-dependence of the reduced coordinates. The phase-space maps give a concrete operational picture that should help experimentalists on Kerr-cat platforms. The distinction between spectral persistence, protocol weight, and metastable organization is clearly drawn. The derivations are standard but correct.\n\nThe soft spots are in the application, not the framework. Fig. 5's routing crossovers—the zeros of R_+ and R_- and the sharp feature in Xi_sw—are properties of a three-coordinate projection whose representatives come from extremal points of the sampled slow manifold. The paper admits the construction is representative-dependent but never quantifies how the crossovers shift or vanish under admissible representative choices or a larger slow subspace. That is a real robustness gap for a headline result. The numerics also lack convergence data and error bars, and the reconstruction protocol in Sec. IIB is described but not demonstrated. These issues do not break the central factorization, which holds as stated, but they make the Kerr conclusions suggestive rather than definitive.\n\nThis paper deserves a serious referee. A competent one will ask for code and data, convergence thresholds, and a sensitivity analysis of the routing diagnostics to representative selection. The author's explicit caveats make that evaluation straightforward.\n\nBottom line: worth reading and citing if you work on metastable open quantum systems or Kerr-cat qubits. The framework is a useful organizational tool. The routing crossovers need more quantitative support before I would build on them.","headline":"Carefully framed operational spectral toolkit for Liouvillian modes; the central factorization is sound, but the routing crossovers in the Kerr application rest on a representative-dependent projection that is never quantified.","tokens_in":22145,"tokens_out":2830,"would_cite":true,"duration_ms":26769,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81S22","81Q12"],"pacs":["03.65.Yz"],"model":"deepseek-v4-flash","headline":"Every resolved Liouvillian mode should be described by a triple: eigenvalue for timing, left eigenoperator for excitation, right eigenoperator for detection and deformation; their product is a gauge-invariant modal weight.","keywords":["Liouvillian spectral theory","modal weight","left and right eigenoperators","driven Kerr resonator","metastability","phase-space excitation maps","slow-sector routing","anomalous relaxation acceleration"],"falsifier":"A direct test in a driven Kerr resonator: prepare a coherent state on the predicted zero contour of the switching-mode excitation map, measure the transient of an observable, and check whether the slow exponential tail at the switching rate is absent. If a tail at that rate persists, the excitation-detection factorization for that mode is wrong; equivalently, independently reconstruct left and right eigenoperators from transient fits and compare the predicted gauge-invariant modal weight with the measured amplitude.","tokens_in":21169,"feed_emoji":"⚛️","tokens_out":10055,"duration_ms":98504,"temperature":0.7,"pith_summary":"The paper's central claim is that a Liouvillian eigenvalue says how long a mode persists, but not whether a chosen preparation excites it or a chosen observable detects it. For each spectrally resolved mode, the eigenvalue fixes the temporal or frequency dependence, the left eigenoperator determines excitation by an input or source, and the right eigenoperator determines the propagated density deformation and readout overlap; the product of the two overlaps is a gauge-invariant modal weight $w_k$. This separates persistence from protocol visibility: a slow mode may be invisible to a given protocol, and a faster mode can dominate a transient when its weight is larger. In driven Kerr resonators the framework predicts coherent preparations whose excitation vanishes, giving mode-selective suppression and, when the slowest mode is the only gap mode, anomalously fast relaxation; it also separates symmetry-resolved relaxation channels and, under one-photon bias, yields a three-coordinate slow-sector stochastic model with channel-selective routing crossovers. The upshot is that preparation geometry and slow-sector propagation are complementary sources of operational information beyond the eigenvalue spectrum.","feed_headline":"Eigenvalues alone don't tell which mode a protocol sees","feed_subtitle":"Left eigenoperators set excitation, right eigenoperators set detection; their gauge-invariant product predicts which modes dominate.","key_machinery":"The machinery is the biorthonormal left-right spectral decomposition of the Liouvillian: for each mode the eigenvalue $\\lambda_k$ fixes the time or frequency dependence, the left eigenoperator $\\ell_k$ fixes excitation by an input or source, the right eigenoperator $r_k$ gives the propagated density deformation and readout overlap, and $w_k = \\operatorname{Tr}(O r_k)\\,\\operatorname{Tr}(\\ell_k^\\dagger X)$ is the gauge-invariant modal weight. For coherent states, the excitation factor is the phase-space symbol of $\\ell_k$ and its zero contour is the mode-suppression manifold, while the right-mode symbol of $r_k$ displays the deformation. For resolved clusters, the Riesz projector and restricted propagator become the stable objects. Slow-sector coordinates are constructed from the extremal geometry of the projected trace-one manifold, and the restricted Liouvillian in the representative basis, $Q = C^{-1} \\Lambda C$, becomes a stochastic routing generator when the representatives are positive and the off-diagonal entries are nonnegative.","core_discovery":"On the paper's own terms, the discovery is that the operational content of a resolved Liouvillian mode is the matched triple $\\lambda_k$, $\\ell_k$, $r_k$ rather than the eigenvalue alone. After biorthonormalization, the excitation factor $E_k(X)=\\operatorname{Tr}(\\ell_k^\\dagger X)$ and the detection factor $D_k(O)=\\operatorname{Tr}(O r_k)$ separately carry the mode gauge, while $w_k = D_k E_k$ is gauge invariant and controls transients, correlation residues, and response spectra. For coherent preparations the excitation factor becomes a phase-space map whose zero set marks preparations that do not excite the mode, and the matched right eigenoperator gives the corresponding signed phase-space deformation. When a slow sector is spectrally resolved but a mode is not isolated, the Riesz projector and restricted propagator replace the single eigenpair; on a slow manifold, reconstructed coordinates and, under positivity and Markov admissibility, a projected stochastic generator describe propagation. In the driven Kerr resonator this structure identifies switching-mode suppression, separates odd lobe-imbalance and even bright-central channels that reorganize differently across the bright-central crossover, and reveals bias-induced crossovers from center-first to opposite-lobe-first routing while the coherent-preparation partition continues to deform.","pith_inferences":["The zero-contour suppression idea carries over to finite-time control: a temporary one-photon bias during a ramp could preferentially address one outer phase-space region before symmetry is restored, a control problem the paper introduces but does not solve.","Because modal weights separate excitation from detection, a diagnostic based purely on the Liouvillian gap can be corrected by weighting each sector mode by its excitation-detection product; this is a testable way to reconcile gap-versus-relaxation discrepancies in other dissipative systems.","The representative dependence of the projected coordinates suggests that a fully protocol-independent slow-sector description would need to average over representative sets or fix them by independent first-passage data, of which the paper reports only qualitative stability.","The same matched left-right structure should transfer to other bosonic or fermionic dissipative platforms, with coherent-state maps replaced by the appropriate phase-space symbols; exact suppression on zero contours would then depend on the preparation family surviving the symbol transform."],"forward_implications":["Coherent preparations on the zero contour of a resolved mode suppress that mode in the selected protocol; when that mode is the unique slowest mode, the state relaxes anomalously fast rather than at the Liouvillian gap rate.","A faster mode can dominate a transient or a frequency window over a slower one whenever its modal weight or residue is larger, so gap ordering alone does not predict relaxation times.","Joint fits of coherent-state transients to common poles can reconstruct the excitation map up to one mode-dependent detection scale, and repeating with different readouts fixes relative detection factors without full generator reconstruction.","In the biased Kerr-cat regime, the projected three-state generator yields transition rates whose routing log-odds locate channel-selective crossovers between center-first and opposite-lobe-first competition, while the coherent-preparation partition keeps deforming after the branchings have nearly saturated.","Near internal degeneracies, the resolved Riesz subspace and its restricted propagator, not individual eigenpairs, are the protocol-relevant objects, so operational predictions remain stable where individual eigenpairs are ill conditioned."],"supporting_citations":[{"why":"supplies the damping-basis method for writing the Liouvillian in matched right and left eigenmodes","marker":"[14]"},{"why":"gives the spectral resolution of the Lindblad master equation used for the mode expansion","marker":"[16]"},{"why":"fixes the biorthonormal structure and symmetry handling for Lindblad operators","marker":"[23]"},{"why":"establishes that source-and-readout residues, not only poles, control response through residues $R_{AB}^k$","marker":"[8]"},{"why":"shows residue reorganization can change chirality-resolved response while the Liouvillian gap stays finite","marker":"[9]"},{"why":"provides the anomalously fast relaxation result that suppressing the slowest mode accelerates approach to stationarity","marker":"[13]"},{"why":"supplies the metastable-reduction method and representative-coordinate construction used for slow sectors","marker":"[5]"},{"why":"supports the three-coordinate slow-sector description of the two-photon driven Kerr oscillator","marker":"[40]"}],"fun_headline_variants":["Eigenvalues alone miss which modes a protocol excites and detects","Left and right eigenoperators determine which Liouvillian modes dominate","Gauge-invariant modal weights expose switching suppression and routing crossovers","Operational spectral theory: matched eigenoperators decide mode visibility"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the retained three-coordinate slow sector, spanned by the stationary state and two continuation-tracked modes with representatives chosen from the extremal geometry of the projected manifold, is complete for the slow dynamics; a different representative set or a larger slow subspace could shift or remove the reported routing crossovers.","fun_headline_variants_meta":{"raw":{"variants":["Eigenvalues alone miss which modes a protocol excites and detects","Left and right eigenoperators determine which Liouvillian modes dominate","Gauge-invariant modal weights expose switching suppression and routing crossovers","Operational spectral theory: matched eigenoperators decide mode visibility"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000202,"raw_usage":{"total_tokens":1404,"prompt_tokens":988,"completion_tokens":416,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":604,"completion_tokens_details":{"reasoning_tokens":342}},"tokens_in":604,"tokens_out":416,"duration_ms":5319,"temperature":1.0,"reasoning_tokens":342,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:37:16.891057+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct test in a driven Kerr resonator: prepare a coherent state on the predicted zero contour of the switching-mode excitation map, measure the transient of an observable, and check whether the slow exponential tail at the switching rate is absent. If a tail at that rate persists, the excitation-detection factorization for that mode is wrong; equivalently, independently reconstruct left and right eigenoperators from transient fits and compare the predicted gauge-invariant modal weight with the measured amplitude.","supporting_citations":[{"cited_title":"Honda, H","cited_arxiv_id":null,"evidence_quote":"fixes the biorthonormal structure and symmetry handling for Lindblad operators"},{"cited_title":"Minganti, A","cited_arxiv_id":null,"evidence_quote":"establishes that source-and-readout residues, not only poles, control response through residues $R_{AB}^k$"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the damping-basis method for writing the Liouvillian in matched right and left eigenmodes"},{"cited_title":"Seibold, G","cited_arxiv_id":null,"evidence_quote":"gives the spectral resolution of the Lindblad master equation used for the mode expansion"},{"cited_title":"Macieszczak, M","cited_arxiv_id":null,"evidence_quote":"shows residue reorganization can change chirality-resolved response while the Liouvillian gap stays finite"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the anomalously fast relaxation result that suppressing the slowest mode accelerates approach to stationarity"},{"cited_title":"For the Kerr model of Sec","cited_arxiv_id":null,"evidence_quote":"supplies the metastable-reduction method and representative-coordinate construction used for slow sectors"},{"cited_title":"Beato and G","cited_arxiv_id":null,"evidence_quote":"supports the three-coordinate slow-sector description of the two-photon driven Kerr oscillator"}],"review_version":1}