{"id":"6db39f5f-fb28-46bf-8222-2164068e644a","arxiv_id":"2501.14032","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A hierarchy of absolute, relative, and qubit-specific thresholds is introduced and experimentally applied to certify non-Gaussian coherences in heralded Fock-state superpositions.","lead":"This paper introduces a hierarchical set of tests for certifying non-Gaussian coherence in quantum light states and demonstrates them on high-purity optical states. The tests use tailored absolute, relative, and qubit-specific thresholds, and the experiments show that a superposition of zero- and two-photon states can pass parts of the non-Gaussian coherence checks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Numerical threshold maxima in App. B are not certified global, so the criteria may be too lenient; a small upward shift in T^{L,G}_{0,2,l=1} would invalidate the reported pass of C0,2=0.72.","rationale":"Good-faith reading: the paper builds a coherent resource theory, the experimental density matrices and error bars are carefully reported, and the logic would be valid if each threshold were the true maximum over the free set. The load-bearing condition is exactness of those maxima. The reader's weakest assumption identifies exactly this condition, and I agree it is the least secure link. The manuscript states only that the function is 'numerically optimized' (App. B) and gives no global-optimality certificate. Because the thresholds are used as if they were exact separators, a local maximum would make the criteria unsound. The closeness of C0,2=0.72 to the free-set boundary (squeezed vacuum at 0.707) makes the issue concrete rather than hypothetical. This is a correctness risk, not a disagreement with consensus. A single global re-optimization would settle it. I therefore keep the reader's CONDITIONAL verdict: no change.","tokens_in":15482,"tokens_out":9239,"duration_ms":86149,"concrete_test":"Independently recompute T^{L,G}_{0,2,l=1} (and, if resources allow, T^{L,G}_{Q,2}) with a certified global method. For each integer m, the exact maximization over the free-state coefficients c_j is the largest eigenvalue of the Hermitian matrix U_G^{(m,k,l)} restricted to H_{m,l}; only the four continuous Gaussian parameters (ξ, α, ϕα, ϕξ) and the integer m remain. Use a branch-and-bound or interval-arithmetic global optimizer (or an extremely dense multistart with Lipschitz bounds) on those parameters. If the certified supremum exceeds the quoted threshold, or exceeds the experimental C0,2=0.72, the criterion is too lenient and the reported certification fails. If the certified value reproduces the quoted threshold within numerical tolerance, the concern is settled and the thresholds can be trusted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central certificates rest on thresholds T^{L,G}_{k,l}=max_{ρ_l∈H_l,ξ,α,...} C_{k,l}(U_G ρ_l U_G†) (App. B, Eq. B9) and the analogous qubit threshold T^{L,G}_{Q,l} (Eq. B13). These maxima are obtained by 'numerically optimizing' C_{k,l}(ξ,α,ϕα,ϕξ,m) over Gaussian parameters, index m, and free-state coefficients. The paper does not state that the search is global or provide any certificate (e.g., branch-and-bound, Lipschitz bound, or interval enclosure). If the optimizer terminates at a local maximum below the true supremum, every threshold quoted is too low, so the criterion is too weak: a state could pass even though it lies in the Gaussian-free set. This is not a stylistic issue because the reported experimental value C0,2=0.72 is very close to the threshold; indeed the pure squeezed-vacuum competitor already gives C0,2=1/√2≈0.707. A small correction of a few percent in the threshold would flip the headline certification. The same unverified-maximum structure underlies the relative thresholds (Eqs. B14-B17) and the qubit thresholds, so the concern is systemic rather than isolated to one data point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a hierarchical family of witnesses for certifying non-Gaussian coherence in Fock-basis superpositions, with free-state sets of increasing power. Three types of criteria are introduced: absolute coherence criteria, relative criteria conditioned on measured Fock probabilities, and qubit-specific criteria for unbalanced superpositions. The framework is tested on heralded photonic states approximately proportional to |0>+|1> and |0>+|2>, reconstructed by homodyne tomography with bootstrap uncertainties. The central experimental claims are that the |0>+|2> state surpasses the first-rank absolute non-Gaussian threshold (C0,2 = 0.72) and that two qubit states surpass the qubit-specific non-Gaussian threshold near its minimum. The paper includes derivations in Appendices A and B, a loss model in Appendix C, and experimental details in Appendices D and E.","tokens_in":15753,"tokens_out":20394,"duration_ms":158598,"significance":"If the thresholds are rigorous, this work provides a useful application-oriented alternative to global non-Gaussianity measures such as Wigner negativity and stellar rank, and the demonstration on high-purity optical states is a valuable experimental contribution. The paper reports full density matrices with bootstrap error bars, which is a strength for reproducibility. The companion theory paper [44] supplies additional derivations. The main open question is whether the numerically computed thresholds are true maxima over the free-state sets, which is essential for the certification claim to be rigorous.","major_comments":[{"comment":"The thresholds T^{L,G}_{k,l} (Eq. B9), the qubit thresholds T^{L,G}_{Q,l} (Eq. B13), and the relative-criterion functions F_n (Eqs. B14-B17) are obtained by numerical maximization over the Gaussian parameters (ξ, α, φα, φξ), the index m, and the free-state coefficients, but the paper does not state that the search is global or provide a certificate of global optimality. A numerical maximizer generally returns a lower bound on the true supremum, and if the true maximum is missed, the thresholds are too low and the criteria become too lenient. This is load-bearing: the reported pass C0,2 = 0.72 for the |0>+|2> state exceeds the rank-l=1 threshold by a small margin, and the pure squeezed vacuum already attains C0,2 = 1/√2 ≈ 0.707, so a few-percent upward correction of the threshold would invalidate the headline certification. I request a rigorous upper bound on the free-state maxima (for example via interval methods, branch-and-bound, or an analytical result from the companion paper [44]), or an explicit statement that the thresholds are numerical estimates with the corresponding implications for the strength of the certification.","section":"Appendix B, Eqs. (9), (13)-(17)"},{"comment":"The qubit-coherence operator and the target state are defined inconsistently between the main text and the appendix. In Eq. (5) the operator is S(φ,θ) = sinθ X_{n1,n2}(φ) + cosθ(|n1⟩⟨n1|-|n2⟩⟨n2|) and the target state is written as (1/√2)(cosθ|n1⟩+e^{iφ} sinθ|n2⟩), for which the expectation value of S is 1/2 cosθ, not 1. In Eq. (B12) the observable is Gθ_{k,l}(ρ) = cosθ C_{k,l}(ρ) + sinθ(⟨k+l|ρ|k+l⟩-⟨k|ρ|k⟩), and for the stated target state cos(θ/2-π/4)|k⟩+sin(θ/2-π/4)|k+l⟩ the value is cos2θ, again not 1 except at isolated angles. Please reconcile these definitions, correct the target-state parameterization (e.g., cos(θ/2)|0⟩+e^{iφ}sin(θ/2)|1⟩ for the main-text operator), and state explicitly which observable was used to compute the experimental curves and thresholds in Fig. 5.","section":"Main text Eq. (5) and Appendix B Eq. (12)"}],"minor_comments":[{"comment":"There are several typos: 'confirmes' in the conclusion should be 'confirms', 'introduce' in the conclusion should be 'introduces', 'Rigourous' in the Principle section should be 'Rigorous', and 'arxXiv' in reference [28] should be 'arXiv'.","section":"Throughout"},{"comment":"The sentence 'To derive the maximum in Eq. (13)' appears to refer to Eq. (9) or Eq. (11), since Eq. (13) defines the qubit threshold; please correct the cross-reference.","section":"Appendix B, text after Eq. (9)"},{"comment":"Please provide the numerical values of the thresholds and the experimental Gθ values at the relevant angles, along with the bootstrap uncertainties, since the graphical presentation in the zoomed inset does not allow a quantitative assessment of the claimed margin.","section":"Fig. 5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is technically promising and the experimental execution is impressive, but the certification claim hinges on the numerical maxima in Appendix B being global. Because the reported margin for the |0>+|2> state is narrow (C0,2 = 0.72 versus the squeezed-vacuum value 1/√2 ≈ 0.707), I would urge the editor to require either a rigorous global-optimality certificate or a clear statement that the thresholds are heuristic estimates before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know about arXiv:2501.14032. First, the experimental part is genuinely good: the group produces high-purity heralded states (82% fidelity for |0>-|1>, 83% for |0>+|2>), reconstructs the density matrices by homodyne tomography, and reports bootstrap error bars under 0.01. That is careful, reproducible work. Second, the threshold numbers at the center of the certification rest on a numerical maximization over Gaussian parameters and free-state coefficients (App. B, Eq. B9) that is not certified to be global. The paper just says \"we numerically optimize\" without stating the method or providing a certificate. This is not a pedantic point: the headline pass, C0,2=0.72 for the |0>+|2> state against the l=1 absolute non-Gaussian threshold, is only about a percent above the value 1/sqrt(2) ≈ 0.707 that pure squeezed vacuum already achieves. If the true maximum over the free set is a few percent higher, that pass disappears. The same unverified-maximum structure underlies the relative and qubit thresholds, so the concern is systemic.\n\nWhat is actually new here? The relative (2D and 3D) criteria and the qubit-specific measure are extensions beyond the companion theory paper [44], and the adaptive weighting of the relative criteria is a legitimate idea. The witness constructions themselves—convex witnesses with recomputed thresholds—are mathematically sound, and the experimental comparisons are consistent with the quoted thresholds. I do not think the central framework is wrong; I just think the numerical threshold computation needs more care.\n\nThe soft spot is real but maybe not fatal. The companion paper [44], published in PRA, presumably contains the rigorous derivation of the hierarchy and possibly a more careful optimization. The present paper should either replicate that analysis for the specific thresholds used or explicitly state that the thresholds carry over from [44] and cite the method. As it stands, an independent reader cannot verify the global optimality from this manuscript alone.\n\nWho is this for? Anyone working on non-Gaussianity certification in continuous-variable systems, especially experimental groups. It deserves a serious referee, but the referee should ask for a clear statement about the optimization method (global or not) and, ideally, a sensitivity analysis of the thresholds. The experimental data and the framework are worth engaging with; the threshold exactness is the one thing that needs to be nailed down before relying on the absolute certification.","headline":"Careful experiment and a useful hierarchy, but the headline certification rests on unproven numerical thresholds with a margin too small to ignore.","tokens_in":16302,"tokens_out":5092,"would_cite":false,"duration_ms":43206,"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 hierarchy of thresholds certifies when a Fock-superposition coherence is genuinely non-Gaussian, and an optical |0>+|2> state clears the first rung.","keywords":["non-Gaussian coherence","Fock-state superpositions","coherence witness","hierarchical certification","bosonic quantum states","heralded optical states","homodyne tomography","quantum resource theories"],"falsifier":"Exhibit a concrete free state of rank $l=1$, of the form $S(\\xi)D(\\alpha)\\sum c_f|f\\rangle$ with the sum spanning one Fock level, whose coherence $C_{0,2}$ equals or exceeds the value the paper reports as $T^{L,G}_{0,1}$ while the experimental state's $C_{0,2}=0.72$ stays below it; then the absolute criterion collapses. The same examination applies to the qubit threshold $T^{L,G}_{Q,2}$ at $\\theta=0.4\\pi/2$, by finding a free state whose $G^\\theta$ reaches the reported threshold.","tokens_in":15273,"feed_emoji":"⚛️","tokens_out":9235,"duration_ms":80485,"temperature":0.7,"pith_summary":"The paper builds a hierarchy of thresholds for deciding when the coherence between two Fock states (photon-number states) of a bosonic mode cannot be produced by Gaussian operations alone. Three kinds of criteria are constructed: absolute thresholds, which compare a phase-scanned coherence contrast against the best any free state can do; relative thresholds, which add measured Fock-population probabilities as extra conditions; and qubit-specific thresholds, which handle unbalanced superpositions. Tested on heralded optical states, the $|0\\rangle+|2\\rangle$ state exceeds the first-rank absolute non-Gaussian threshold and the qubit-specific threshold near its minimum, while the $|0\\rangle-|1\\rangle$ state needs the relative criteria to be certified. The hierarchy matters because global non-Gaussianity measures such as Wigner negativity or stellar rank do not say whether a state has the particular coherence structure that error correction or sensing requires.","feed_headline":"Three-tier test certifies non-Gaussian coherence in photonic states","feed_subtitle":"An optical |0>+|2> state clears the first absolute threshold and the qubit test, unlike Wigner-negativity checks alone.","key_machinery":"The load-bearing construction is the hierarchy of free-state sets $F^G_l$: at rank $l$, any superposition of at most $l$ Fock levels is free, and Gaussian dynamics $S(\\xi)D(\\alpha)$ may act on it. The carried measure is the interferometric coherence contrast $C_{n_1,n_2}(\\rho)$, defined through the phase-scanned projectors $X_{n_1,n_2}(\\phi)$ on the equator of the corresponding Bloch sphere, plus its qubit generalization $G^\\theta_{n_1,n_2}(\\rho)$, which adds a population-imbalance term for arbitrary splitting angles. Thresholds are computed as maxima of these measures over the free sets, using an analytical parametrization of displaced-squeezed Fock overlaps via Hermite polynomials and numerical optimization over the squeezing, displacement, phases, initial Fock index, and free-state coefficients. This turns non-Gaussian coherence into a ladder of application-tailored witnesses rather than a single global resource count.","core_discovery":"On its own terms, the paper claims that non-Gaussian coherence can be certified as a hierarchy indexed by the Fock-space distance $l=n_2-n_1$ of the targeted superposition. The free states at rank $l$ are Gaussian squeezed-displaced versions of arbitrary superpositions spanning at most $l$ adjacent Fock levels, and the threshold $T^{L,G}_{k,l}$ is the maximum of the coherence contrast $C_{k,l}(\\rho)=\\frac{1}{2}(\\max_\\phi \\mathrm{Tr}[X_{k,l}(\\phi)\\rho]-\\min_\\phi\\mathrm{Tr}[X_{k,l}(\\phi)\\rho])$ over that free set. The experimental $|0\\rangle+|2\\rangle$ state, with measured $C_{0,2}=0.72$, passes the first-rank absolute non-Gaussian threshold, and both tested qubit states pass the qubit-specific threshold near $\\theta=0.4\\,\\pi/2$; the $|0\\rangle-|1\\rangle$ state, with $C_{0,1}=0.66$, does not pass the absolute non-classical or non-Gaussian thresholds and is certified only through the relative criteria.","pith_inferences":["An implication the authors leave implicit is that the highest rank a state passes defines a discrete coherence rank, which could serve as an application-oriented resource measure complementing stellar rank.","Because the thresholds come from numerical maximization, a natural follow-up is to certify global optimality for the lowest ranks, for instance by interval methods or by deriving closed-form bounds for the $l=1$ case.","The same hierarchy could be applied directly to coherent-state superpositions and Gottesman-Kitaev-Preskill states, as the conclusion anticipates, with the thresholds then compared against stellar-rank witnesses on identical density matrices."],"forward_implications":["Any state whose measured coherence $C_{k,l}$ or $G^\\theta_{k,l}$ exceeds the corresponding threshold is certified to require non-Gaussian resources for that coherence, not merely to have a negative Wigner function.","The relative criteria let experimentalists certify states that fail the absolute test by conditioning on measured Fock probabilities, lowering the bar without weakening the definition of free states.","The qubit-specific threshold supplies a minimal pass/fail requirement for a qubit encoded in an unbalanced two-Fock-level superposition, directly usable in trapped-ion, circuit, and photonic qubit platforms.","The same threshold machinery extends to other free-state sets and to other pairs of Fock levels, so the hierarchy is not restricted to the $|0\\rangle$-$|1\\rangle$ and $|0\\rangle$-$|2\\rangle$ cases demonstrated here."],"supporting_citations":[{"why":"Defines the stellar hierarchy whose inability to distinguish application-relevant coherence motivates the paper's own hierarchy.","marker":"[25]"},{"why":"Supplies the displaced-squeezed Fock-state overlap formula used to parametrize the coherence measure and thresholds.","marker":"[41]"},{"why":"Provides the efficient method for maximizing the witness over the free-state superposition coefficients.","marker":"[42]"},{"why":"Gives the reformulation used to turn the relative multi-dimensional criteria into direct conditions on measured quantities.","marker":"[43]"},{"why":"Companion theoretical paper containing the detailed derivation of the L-hierarchy and its thresholds.","marker":"[44]"},{"why":"Describes the experimental encoding converter used to generate the $|0\\rangle-|1\\rangle$ superposition.","marker":"[37]"},{"why":"Reports the optical synthesis method used to generate the $|0\\rangle+|2\\rangle$ superposition via two-photon heralding.","marker":"[17]"},{"why":"Supplemental material with experimental details, threshold derivations, and the tomographic uncertainty analysis.","marker":"[38]"}],"fun_headline_variants":["Non-Gaussian coherence hierarchy passes first absolute threshold","Hierarchical test certifies non-Gaussian coherence in optics","|0>+|2> state passes absolute non-Gaussian threshold","Fock-space hierarchy certifies non-Gaussian coherence","Graded thresholds reveal non-Gaussian coherence in bosonic states"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the numerical maximizations used to set every threshold actually find the global maximum over the free-state manifold; if any optimization misses the true maximum, the thresholds are too low and the criteria could certify a state that Gaussian operations can produce.","fun_headline_variants_meta":{"raw":{"variants":["Non-Gaussian coherence hierarchy passes first absolute threshold","Hierarchical test certifies non-Gaussian coherence in optics","|0>+|2> state passes absolute non-Gaussian threshold","Fock-space hierarchy certifies non-Gaussian coherence","Graded thresholds reveal non-Gaussian coherence in bosonic states"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001004,"raw_usage":{"total_tokens":4236,"prompt_tokens":921,"completion_tokens":3315,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":537,"completion_tokens_details":{"reasoning_tokens":3234}},"tokens_in":537,"tokens_out":3315,"duration_ms":22591,"temperature":1.0,"reasoning_tokens":3234,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T15:27:39.391673+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a concrete free state of rank $l=1$, of the form $S(\\xi)D(\\alpha)\\sum c_f|f\\rangle$ with the sum spanning one Fock level, whose coherence $C_{0,2}$ equals or exceeds the value the paper reports as $T^{L,G}_{0,1}$ while the experimental state's $C_{0,2}=0.72$ stays below it; then the absolute criterion collapses. The same examination applies to the qubit threshold $T^{L,G}_{Q,2}$ at $\\theta=0.4\\pi/2$, by finding a free state whose $G^\\theta$ reaches the reported threshold.","supporting_citations":[{"cited_title":"Kr´ al, Displaced and Squeezed Fock States, J","cited_arxiv_id":null,"evidence_quote":"Supplies the displaced-squeezed Fock-state overlap formula used to parametrize the coherence measure and thresholds."},{"cited_title":"Filip, and L","cited_arxiv_id":null,"evidence_quote":"Gives the reformulation used to turn the relative multi-dimensional criteria into direct conditions on measured quantities."},{"cited_title":"Lachman, B","cited_arxiv_id":null,"evidence_quote":"Companion theoretical paper containing the detailed derivation of the L-hierarchy and its thresholds."},{"cited_title":"Huang, H","cited_arxiv_id":null,"evidence_quote":"Reports the optical synthesis method used to generate the $|0\\rangle+|2\\rangle$ superposition via two-photon heralding."},{"cited_title":"[39– 43]","cited_arxiv_id":null,"evidence_quote":"Supplemental material with experimental details, threshold derivations, and the tomographic uncertainty analysis."}],"review_version":1}