{"id":"840b3646-62d0-41df-9305-f282b7ca32c9","arxiv_id":"2501.14033","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A Fock-basis coherence measure is used to build two context-dependent hierarchies of quantum non-Gaussian coherence, with thresholds computed under Gaussian dynamics and evaluated under loss and thermal noise.","lead":"This paper introduces a framework for certifying quantum non-Gaussian coherence in bosonic states by targeting individual Fock-state off-diagonal elements instead of using global coherence measures. It defines two context-dependent hierarchies of thresholds and studies how loss and thermal noise affect the certification depth.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The thresholds rest on an unproved conjecture about the maximizing parameters and on random sampling; if the true maxima lie outside the searched families, the criteria falsely certify free states as quantum non-Gaussian.","rationale":"The reader's weakest-assumption analysis identifies exactly the same load-bearing concern: the thresholds that define the hierarchies are global maxima computed under an explicit conjecture and random sampling rather than rigorous optimization. This concern lands because the paper's central purpose is certification, and certification requires thresholds to be true upper bounds on the free-state set. If a threshold is only a lower bound on the true maximum, the criterion is invalid in the sense that it can generate false positives. The paper's framework and definitions are internally coherent, and the proposed hierarchies are plausible extensions of earlier stellar-rank work, but the quantitative output is not established to the standard needed for a witness. I would keep the CONDITIONAL verdict: the concern is addressable by a more careful global optimization or by presenting the thresholds as certified bounds, and it does not nullify the conceptual framework. No independent code or machine-checked proof is provided, so this remains the central risk.","tokens_in":15347,"tokens_out":3142,"duration_ms":32227,"concrete_test":"Recompute a representative threshold, e.g., T^{N,2}_{0,4}=0.55, with a global optimizer that does not impose the conjectured reductions: full complex ξ and α, all phases φ and θ, and all core-state coefficients, or alternatively use Lasserre-style moment relaxations to certify an upper bound on Eq. (6). If any free state exceeds the reported value by more than 1e-3, the reported thresholds are not valid witnesses. Publish the code and data for full reproduction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The certification thresholds are only as reliable as the global maximizations that define them. In Section IV B the authors explicitly state that they guess where the maximum occurs, restrict the search to φ=0, θ∈{0,π}, and two specific core-state families, and verify only by random sampling. Appendix A similarly obtains maxima over ξ and α by random sampling and taking the maximal attempt. Because each threshold T^{N,k}_{m,n} or T^{L,r}_{k,l} is used as an upper bound on all free states, any missed global maximum makes the threshold too low, so a state that is free by the paper's own definition could be falsely certified as having quantum non-Gaussian coherence. This is not a peripheral numerical issue: the comparative hierarchy claims, the relative criteria, and the depth analysis all inherit these values. The paper provides no independent proof, code, or data that establishes the global optima, so the central claim is conditional on an unverified maximization.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces a framework for certifying quantum non-Gaussian (QNG) coherence in the Fock-state basis by comparing individual off-diagonal coherence elements C_{m,n}(ρ) against thresholds derived from Gaussian dynamics acting on restricted 'core states.' Two context-dependent hierarchies are constructed: an N-hierarchy ordered by the larger Fock index n, and an L-hierarchy ordered by the Fock-number difference l = n−m. The paper further develops 'relative criteria' that combine C_{m,n} with diagonal probabilities to relax experimental requirements, and it analyzes the robustness of these criteria under loss and thermal noise using a first-order perturbative model. The central claim is that exceeding a hierarchy-dependent threshold certifies that the tested coherence cannot be produced by any free state of the corresponding order, yielding a structured, application-oriented classification of QNG coherence.","tokens_in":15538,"tokens_out":5916,"duration_ms":55991,"significance":"If the numerical thresholds are valid, the framework is a useful and operational complement to global coherence measures and to stellar-rank classification: it isolates individual off-diagonal elements accessible via interferometric measurements, provides two distinct orderings with clear operational interpretations (macroscopic Fock number versus Fock separation for sensing and error correction), and offers relaxed relative criteria that are experimentally attractive. The paper is clearly written and builds on established techniques such as the Fiurášek eigenvector method. Its main quantitative predictions, however, depend on global maxima of nonconvex functions that are not rigorously established; this makes the thresholds, and every hierarchy comparison and depth estimate built on them, conditional on an unverified conjecture.","major_comments":[{"comment":"The thresholds T^{N,k}_{m,n} and T^{L,r}_{k,l} are defined as global maxima over all Gaussian evolutions S(ξ)D(α) of all core states in a specified family. The paper explicitly states in Sec. IV B: 'we guess for which parameters the maximum occurs and, then, we use random sampling of the parameters ξ, α, φ, l and θ to verify the estimated maximum,' and Appendix A states that the maximum over ξ and α is obtained 'by random sampling and taking the maximal attempt.' This does not establish a global maximum. Because each threshold is used as an upper bound on all free states, any missed global maximum makes the threshold too low and would cause a Gaussian-generated free state to be falsely certified as having QNG coherence. This is not a peripheral numerical issue: the hierarchy comparisons, the relative criteria in Eqs. (10)–(16), and the depth analyses all inherit these threshold values. The authors should either provide a rigorous global-optimization certificate (e.g., an analytic bound, interval/branch-and-bound method, or a convex relaxation) for each reported threshold, or clearly state the maximizing-family conjecture as such and provide a systematic, dense numerical search with convergence evidence. As written, the central certification claim is conditional on an unverified maximization.","section":"Sec. IV B, Eqs. (6)–(7); Appendix A"},{"comment":"The claim that thresholds T^{(N)}_{m,n} tend to unity for large N is supported only by selected numerical examples, primarily for m=0 and m=n−1, with N up to 20. This convergence is used to justify the argument that employing thresholds with N≫1 imposes the unattainable condition C_{m,n}=1, which in turn motivates the finite-order hierarchies. Since no proof of generic convergence is given, and the numerical evidence covers a restricted set of (m,n) pairs, the assertion 'we numerically verified this tendency' does not establish the general behavior. If for some (m,n) the limit is below 1, the statement that 'employing the thresholds T^{(N)}_{m,n} with N≫1 for benchmarking implies the condition C_{m,n}=1' would be false, affecting the conceptual foundation of the hierarchy construction. The authors should either provide a proof or clearly state this as a conjecture and delineate the numerical evidence for it.","section":"Sec. III A, Eq. (4); Appendix A, Fig. 6"},{"comment":"The relative criteria are derived by computing F_{m,n}(λ) = max_{ξ,α,|ψ̃_k⟩} [C_{m,n} + λ P] and then forming the Legendre-type expression min_λ [F_{m,n}(λ) − λ P]. The validity of the resulting witness requires F_{m,n}(λ) to be the exact global maximum over the entire free-state family. Because the maximization is the same unverified one criticized above, the relative criteria inherit the same false-positive risk. Additionally, the final minimization over λ (and λ1,λ2 in Sec. IV C) is performed numerically without reporting error bars or convergence tests, so the relaxed thresholds in Figs. 4 and 5 are as conditional as the absolute thresholds. The authors should clarify how the numerical minimization is certified and, ideally, provide a rigorous bounding argument for F_{m,n}(λ) before presenting the relative criteria as definitive witnesses.","section":"Sec. IV B–C, Eqs. (10)–(16)"}],"minor_comments":[{"comment":"The model in Eq. (8) is stated to be valid only in the limits 1−η ≪ 1 and n̄ ≪ 1, but the plots in Figs. 3(e)–(h) and 4(d) display parameter ranges extending to 1−η = 0.5. Please clarify whether the reported depth values remain within the model's validity regime or whether some curves are extrapolations beyond it.","section":"Eq. (8) and Figs. 3, 4"},{"comment":"There are several apparent typos: in Sec. III D, 'T^{L,1}_{3,1}=0.8' should presumably read 'T^{L,1}_{3,4}=0.8'; the Fig. 3 caption contains 'repectively' and 'T^{l,k}_{3,4}' which should likely be 'T^{L,k}_{3,4}'; and Appendix A's title has 'Hibert space' instead of 'Hilbert space.'","section":"Sec. III D and Fig. 3 caption"},{"comment":"The sentence 'As basis states, no Fock state can obey this criterion despite their non-Gaussian nature [6]' is confusing: Fock states have zero coherence C_{m,n}=0, so they trivially fail any positive threshold. Please rephrase to clarify that the statement concerns the inadequacy of using Fock states alone as reference states.","section":"Sec. II B"},{"comment":"The notation for thresholds (t̃_{m,n}, t_{m,n}, T̃_{m,n}, T^{(N)}_{m,n}, T^{N,k}_{m,n}, T^{L,r}_{k,l}) is introduced quickly and used in multiple places. A table summarizing each quantity, its defining free-state family, and its role in the hierarchies would significantly improve readability.","section":"General notation"}],"recommendation":"major_revision","confidential_remarks":"The paper is a theoretical companion to an experimentally validated manuscript [11], and the conceptual framework is appealing. However, the central numerical thresholds are supported only by a guessed maximizing family and random sampling, as explicitly acknowledged in Sec. IV B and Appendix A. This is a load-bearing gap: if the true maxima lie outside the searched families, the certification criteria are invalid. I recommend requiring the authors to either supply a rigorous global-optimization certificate or to state the maximizing-family conjecture explicitly in the main text with comprehensive numerical evidence (e.g., dense grid scans, multiple random seeds, and convergence checks). The convergence claim T^{(N)}→1 should also be either proven or clearly flagged as a conjecture. If the authors can provide those, the paper would be publishable; without them, the quantitative claims remain conditional."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Punchline: this is a real extension of the non-Gaussianity hierarchy program from diagonal Fock probabilities to individual off-diagonal coherences, and the two context-dependent orderings (by Fock number and by Fock separation) are a useful organizing idea. The quantitative thresholds, however, are only as solid as the maximizations that define them, and those rest on a stated conjecture plus random sampling. So the certification numbers should be treated as provisional until the global maxima are proved or the code is released.\n\nWhat is actually new: prior work by this group (and others) built hierarchical witnesses for Fock-state non-Gaussianity from diagonal elements. This paper generalizes that to individual coherence elements C_{m,n}, defines two explicit hierarchies (N and L), introduces relative criteria that mix C_{m,n} with diagonal probabilities to relax experimental requirements, and analyzes loss/thermal depth. The definitions are coherent, the notation is mostly clear, and the distinction between the two hierarchies is demonstrated with concrete examples like C0,4 and C3,4. The paper also directly acknowledges the conjecture in Section IV B and the random-sampling verification in Appendix A; it does not try to hide the weak spot. That honesty counts.\n\nSoft spots, in proportion: the central one is load-bearing. Every threshold T^{N,k}_{m,n} and T^{L,r}_{k,l} is used as an upper bound on all free states. The paper guesses the maximizing parameters (phi=0, theta in {0,pi}, two specific core-state families) and then verifies by random sampling. Random sampling cannot certify a global maximum. If the true maximum lies outside the searched family, the threshold is too low and a genuinely free state gets falsely certified as quantum non-Gaussian. That is exactly the failure mode you do not want in a certification criterion. The paper provides no code or data to reproduce the maxima, which is fixable. The depth analysis in Eq. (8) is a small-noise expansion but is plotted for loss up to ~0.2 and thermal numbers up to ~0.05, outside the strict validity limit; minor since the qualitative ordering likely survives. There are a few notation slips (e.g., T^{L,1}_{3,1} should probably be T^{L,1}_{3,4}) but they are easy to fix.\n\nWho it is for: groups doing bosonic coherence experiments—optical, trapped ion, superconducting—who want a context-dependent way to certify specific Fock coherences. The companion experimental paper (PRL 134, 233604) suggests the framework is testable. The paper deserves a serious referee; the referee should push for a rigorous treatment of the maximization problem, or at least a public numerical package and a statement that the thresholds are lower bounds (which would make the criteria conservative in the other direction, though that changes the hierarchy claims). My own verdict: conditional, but conditionally publishable in a good journal.\n\nEnd with recommendation: engage with it, send to review, require the max issue addressed before acceptance.","headline":"Genuine extension of the non-Gaussianity hierarchy program to off-diagonal Fock coherences, but the certification thresholds rest on an unproved conjecture and random sampling, so the numbers are provisional.","tokens_in":16046,"tokens_out":2541,"would_cite":true,"duration_ms":23243,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A single Fock-basis coherence element can certify quantum non-Gaussian coherence, provided it beats hierarchy-specific thresholds defined by what Gaussian squeezing and displacement can generate from restricted core states.","keywords":["quantum non-Gaussian coherence","Fock-state basis","Gaussian operations","coherence hierarchy","bosonic systems","coherence certification","loss and thermal noise","relative criteria"],"falsifier":"Find any core state in the allowed Hilbert space and any squeezing and displacement parameters $\\xi,\\alpha$ such that $C_{m,n}(S(\\xi)D(\\alpha)|\\tilde\\psi\\rangle)$ exceeds the quoted $T^{N,k}_{m,n}$ or $T^{L,r}_{k,l}$ for a small case such as $C_{0,4}$; a global numerical optimization producing such a counterexample would refute the claimed bounds.","tokens_in":15136,"feed_emoji":"⚛️","tokens_out":4321,"duration_ms":37921,"temperature":0.7,"pith_summary":"This paper claims that the quantum non-Gaussian nature of coherence in bosonic systems can be certified one Fock-basis element at a time. Instead of summing all off-diagonal matrix elements into a global coherence quantifier, it isolates a single coherence $C_{m,n}$, the magnitude of the $|m\\rangle\\langle n|$ density-matrix element, and compares it with the largest value that Gaussian squeezing and displacement can produce from a 'core' state of a given order. Beating the corresponding threshold proves that the state's coherence cannot be mimicked by Gaussian dynamics acting on low-order superpositions of Fock states, i.e. it is genuinely quantum non-Gaussian. The authors construct two hierarchies that order the same coherences differently—one by the highest Fock level $n$, one by the separation $l=n-m$—and show that the two classifications prioritize different states and tolerate different amounts of loss and thermal noise. This matters because single-coherence certification is directly accessible to interferometric measurements and can be tailored to application contexts such as sensing and bosonic error correction.","feed_headline":"One Fock coherence element certifies non-Gaussianity","feed_subtitle":"Two hierarchies—large Fock numbers vs. Fock separation—rank coherence resources and set loss-tolerant experimental thresholds.","key_machinery":"The load-bearing object is the coherence measure $C_{m,n}(\\rho)=\\tfrac12\\left[\\max_\\phi \\mathrm{Tr}[\\rho X_{m,n}(\\phi)]-\\min_\\phi \\mathrm{Tr}[\\rho X_{m,n}(\\phi)]\\right]$ with $X_{m,n}(\\phi)=e^{i\\phi}|m\\rangle\\langle n|+e^{-i\\phi}|n\\rangle\\langle m|$, which isolates $|\\langle m|\\rho|n\\rangle|$. Around it the paper builds core Hilbert spaces $H_k$ of Fock states below a given order; free states are Gaussian images $S(\\xi)D(\\alpha)|\\tilde\\psi_k\\rangle$ of core states. The thresholds $T^{N,k}_{m,n}$ and $T^{L,r}_{k,l}$ are suprema of $C_{m,n}$ over these free states, computed via an eigenvector method for optimizing over core states and polynomial root-finding for the displacement parameter. Relative criteria replace the single threshold with $\\min_\\lambda[F_{m,n}(\\lambda)-\\lambda P]$, where $F_{m,n}(\\lambda)$ is the maximum of $C_{m,n}+\\lambda P$ over free states and $P$ is a diagonal probability; this is the mechanism that relaxes experimental requirements.","core_discovery":"The central claim is that individual Fock-basis coherence elements carry enough information to certify quantum non-Gaussian coherence in a hierarchically ordered way. For each target superposition $(|m\\rangle+|n\\rangle)/\\sqrt{2}$, the paper defines thresholds $T^{N,k}_{m,n}$ and $T^{L,r}_{k,l}$ as the maximum of $C_{m,n}$ over states $S(\\xi)D(\\alpha)|\\tilde\\psi\\rangle$ obtained by applying Gaussian squeezing and displacement to a core state of order $k$ or $r$—states whose coherence is restricted to lower Fock levels or shorter Fock separations. A measured coherence exceeding such a threshold proves the state is not a Gaussian-transformed core state of that order. The paper derives thresholds for representative cases such as $C_{0,4}$, $C_{3,4}$ and $C_{1,2}$, shows that the two hierarchies give different orderings of the same coherences, and introduces relative criteria that combine $C_{m,n}$ with diagonal probabilities to lower the experimental bar while preserving the hierarchy.","pith_inferences":["Because the thresholds rest on an unproved conjecture about where the global maximum occurs, the numerical values should be treated as certified bounds only under that conjecture; a broader optimization could raise them and would make current certifications stricter rather than looser.","The context-dependent ordering suggests an application-driven choice: experiments aimed at bosonic error correction or phase sensing should use the $L$-hierarchy, while experiments targeting macroscopic superpositions should use the $N$-hierarchy, and the two should not be compared as if they ranked the same resource.","The relative criterion involving $P_{e,n}$ tolerates arbitrarily large loss in the noiseless limit, which implies that photon-number-resolving detection of high-Fock errors can substitute for high transmission in practical certification.","A direct falsification test would be to run a global numerical optimizer over core states and Gaussian parameters for a small case such as $C_{0,4}$ and check whether any free state exceeds the quoted $T^{N,k}_{m,n}$ or $T^{L,r}_{k,l}$."],"forward_implications":["A measured coherence above $T^{N,k}_{m,n}$ certifies that the state is not reachable by squeezing and displacing any order-$k$ core state whose coherence is confined to Fock levels below $n$.","The two hierarchies classify the same coherence differently: for $C_{3,4}$ the $N$-hierarchy supports thresholds up to third order while the $L$-hierarchy admits only first order, whereas for $C_{0,4}$ the $L$-hierarchy's second-order threshold is stricter than the $N$-hierarchy's.","Adding diagonal probabilities such as $P_n$ or the multi-photon error probability $P_{e,n}$ lowers the coherence value needed for certification while keeping the hierarchical ordering intact.","Loss and thermal depths quantify how much imperfection each threshold tolerates; for $C_{3,4}$ the highest $N$-hierarchy threshold permits about seven times smaller loss and thermal noise than the highest $L$-hierarchy threshold.","The same construction extends to unbalanced superpositions, multi-state coherences, and multi-mode coherences, which the authors link to dual-rail binomial codes and sensing protocols."],"supporting_citations":[{"why":"Supplies the prior hierarchy of genuine n-photon quantum non-Gaussian light that this work extends from Fock probabilities to coherence elements.","marker":"[6]"},{"why":"Provides the stellar-rank formalism that motivates ordering core states by the Fock levels they may occupy.","marker":"[8]"},{"why":"The companion experimental paper that applies this coherence certification framework to optical non-Gaussian states.","marker":"[11]"},{"why":"Gives the eigenvector and polynomial-root method used to maximize over core states and Gaussian parameters when computing thresholds.","marker":"[24]"},{"why":"Introduces binomial bosonic error-correcting codes whose dependence on Fock separation motivates the L-hierarchy.","marker":"[17]"},{"why":"Supplies the convex linear combination construction behind the relative criteria that combine coherence with diagonal probabilities.","marker":"[2]"}],"fun_headline_variants":["Single Fock coherence element certifies non-Gaussianity","Hierarchies of quantum non-Gaussian coherence revealed","Loss-tolerant certification of non-Gaussian coherence","One coherence value, two resource hierarchies"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The thresholds are computed by assuming that the maximum coherence occurs for a specific narrow family of core states and Gaussian parameters; if a state outside that family achieves more coherence, every threshold in the paper is too low.","fun_headline_variants_meta":{"raw":{"variants":["Single Fock coherence element certifies non-Gaussianity","Hierarchies of quantum non-Gaussian coherence revealed","Loss-tolerant certification of non-Gaussian coherence","One coherence value, two resource hierarchies"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000193,"raw_usage":{"total_tokens":1326,"prompt_tokens":896,"completion_tokens":430,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":512,"completion_tokens_details":{"reasoning_tokens":371}},"tokens_in":512,"tokens_out":430,"duration_ms":4224,"temperature":1.0,"reasoning_tokens":371,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:26:36.305720+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find any core state in the allowed Hilbert space and any squeezing and displacement parameters $\\xi,\\alpha$ such that $C_{m,n}(S(\\xi)D(\\alpha)|\\tilde\\psi\\rangle)$ exceeds the quoted $T^{N,k}_{m,n}$ or $T^{L,r}_{k,l}$ for a small case such as $C_{0,4}$; a global numerical optimization producing such a counterexample would refute the claimed bounds.","supporting_citations":[{"cited_title":"Lachman, I","cited_arxiv_id":null,"evidence_quote":"Supplies the prior hierarchy of genuine n-photon quantum non-Gaussian light that this work extends from Fock probabilities to coherence elements."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The companion experimental paper that applies this coherence certification framework to optical non-Gaussian states."},{"cited_title":"Fiur´ aˇ sek, Efficient construction of witnesses of the stel- lar rank of nonclassical states of light, Opt","cited_arxiv_id":null,"evidence_quote":"Gives the eigenvector and polynomial-root method used to maximize over core states and Gaussian parameters when computing thresholds."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces binomial bosonic error-correcting codes whose dependence on Fock separation motivates the L-hierarchy."},{"cited_title":"Filip and L","cited_arxiv_id":null,"evidence_quote":"Supplies the convex linear combination construction behind the relative criteria that combine coherence with diagonal probabilities."}],"review_version":1}