{"id":"5037f836-9965-4ab7-a78c-cb636c4a4bdf","arxiv_id":"2412.14356","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Witness thresholds for the stellar rank of light states can be computed by maximizing a single finite-dimensional eigenvalue over Gaussian unitary parameters, with the optimization dimension independent of the rank.","lead":"This paper gives a fast numerical method for building witnesses that certify how many photons are needed to prepare a quantum state of light. The method keeps the number of optimized parameters fixed regardless of the certified rank, which makes high-rank certification much easier.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The exact identity in Eq. (12) is sound, but the practical validity of every witness threshold depends on the numerical optimization over Gaussian unitaries reaching the global supremum; the paper provides no such guarantee.","rationale":"The reader's weakest assumption—that the numerical optimization over Gaussian unitaries reaches the global supremum defining a valid threshold—is precisely the load-bearing concern. The mathematical derivation of Eq. (12) is exact and does not itself require a global-optimization proof; the issue is in the transition from the exact supremum to a computed numerical threshold used for certification. Because the paper gives no convexity or global-convergence argument and no reproducibility artifacts, the numerical thresholds in the examples cannot be verified as true upper bounds. This does not undermine the theoretical identity, but it does affect the practical correctness of the proposed witness-construction method. The reader's CONDITIONAL verdict already accounts for this gap, so I recommend no change. The concrete test would settle the concern by checking whether a more thorough global search can beat the reported thresholds; if it cannot for the tested cases, confidence in the method increases, though a general guarantee would still be desirable.","tokens_in":11447,"tokens_out":5246,"duration_ms":48904,"concrete_test":"For the witness W_{02} = cos(pi/4)|0><0| + sin(pi/4)|2><2| and n=2, recompute the objective in Eq. (12) over the three real parameters (Re alpha, Im alpha, r) using a global optimization method such as differential evolution with 10^5 evaluations followed by local refinement, and compare the resulting maximum with the threshold implied by Fig. 1(a). If a larger objective value is found, the reported threshold is too low. As a stronger check, solve the stationarity conditions for the objective and verify that the reported value corresponds to the global maximum among all stationary points.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Equation (12) is mathematically exact: for each Gaussian unitary the core-state maximization is performed by the largest eigenvalue, and the remaining supremum over the Gaussian parameters is well-defined. The load-bearing gap is that turning this identity into a concrete witness threshold requires evaluating that supremum over four (or, for Fock-diagonal witnesses, three) real parameters. The paper reports numerical optimization without proving convexity, providing a global-optimization certificate, or supplying code or data. If the optimizer settles at a local maximum, the computed W_n is below the true threshold, and inequality (4) can falsely certify a state of lower stellar rank. This is not a purely formal issue: the boundaries in Figs. 1 and 2 are drawn from these numerics, and Sec. 4 explicitly states that for higher stellar ranks fully numerical approaches seem unavoidable. Consequently, the central practical claim—that stellar-rank witness thresholds can be efficiently computed with a rank-independent number of optimization parameters—is only as reliable as the global optimization step, which the paper does not secure.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents a method for constructing witnesses of the stellar rank of nonclassical states of light. For a Hermitian witness operator W, the threshold W_n is expressed as a supremum over Gaussian unitaries of the largest eigenvalue of a finite-dimensional operator: Eq. (12) for single-mode states and Eq. (18) for multimode states. This reduces the optimization from a search over core-state coefficients (whose number grows with n) to a fixed-dimensional search over Gaussian parameters (4 for one mode, O(N^2) for N modes). The method is illustrated with witnesses based on pairs of Fock-state probabilities and pairs of fidelities with even/odd coherent states, with threshold boundaries shown in Figs. 1 and 2. The paper also gives a lower bound on the trace distance to lower-rank states in Eq. (22).","tokens_in":11700,"tokens_out":10056,"duration_ms":91292,"significance":"The central derivation is exact: Eq. (5) to Eq. (12) follows from the definition of W_n as a supremum, with the saturation of the eigenvalue inequality by the core-state eigenvector. The multimode extension via Bloch-Messiah decomposition is well justified, and the recovery of the fidelity-witness results of Ref. [25] provides a useful consistency check. If the numerical suprema displayed in Figs. 1 and 2 are correct, the method is a significant practical advance: it turns stellar-rank witness construction into a fixed-parameter optimization problem whose dimension does not grow with the targeted rank. The trace-distance bound in Eq. (22) adds further value. The main uncertainty is whether the reported numerical thresholds are indeed global suprema, a point that is not secured in the manuscript.","major_comments":[{"comment":"The practical validity of the constructed witnesses depends on computing the supremum over Gaussian unitaries in Eq. (12) and its multimode analogue (18). The paper reports numerical optimization over three or four real parameters but provides no evidence that the global supremum is reached. The objective is not shown to be convex or unimodal, the parameter domain is non-compact (the squeezing parameter r can diverge), and the manuscript does not specify the algorithm, the number of starting points, or the tolerances used. If the optimizer returns a local maximum, the computed W_n is below the true threshold and inequality (4) can falsely certify states of lower stellar rank. This is load-bearing for the central claim that witness thresholds can be efficiently constructed. The authors should either supply a global-optimization procedure or a rigorous upper bound on W_n, or explicitly present the thresholds as candidate values requiring independent validation. Making the numerical code and data behind Figs. 1 and 2 available would considerably strengthen the paper.","section":"Sec. 3, Eq. (12) and Figs. 1-2"},{"comment":"The statement in Sec. 4 that 'for higher stellar ranks fully numerical approaches seem unavoidable' is in tension with the presentation of the thresholds as exact witness values. The paper should clarify that the thresholds are exact only if the numerical supremum is exact, and should discuss the consequences of an approximate optimization for the soundness of the witness inequality (4). A concrete remedy would be to cross-check the reported thresholds with an independent global optimization method (e.g., differential evolution or a grid-based multistart) and to report the best values found and any observed plateaus.","section":"Sec. 4"}],"minor_comments":[{"comment":"There are several typos: 'probabilitiies' in the Introduction, 'characterisitc' in Sec. 2, 'qantum' in Sec. 3.1, and 'Gussian' twice in Sec. 4. These should be corrected.","section":"Introduction; Sec. 2; Sec. 3.1; Sec. 4"},{"comment":"The claim that 'the number of parameters that need to be optimized' is independent of the stellar rank is precise, but it can be misread as a claim of rank-independent overall complexity. The cost of evaluating the objective function—the largest eigenvalue of the truncated operator (11)—grows with n (matrix dimension n for single mode, and combinatorially with n and N for multimode). A sentence clarifying that the fixed scaling applies only to the number of optimization parameters would avoid overstatement.","section":"Abstract and Sec. 2"},{"comment":"The label 'n=0' in Fig. 1 is somewhat ambiguous; it may be helpful to state explicitly that n=0 corresponds to Gaussian states (no non-Gaussianity certified) and that the areas indicate the minimal stellar rank certified by the witness.","section":"Fig. 1"}],"recommendation":"major_revision","confidential_remarks":"The paper is likely of interest to the quantum optics and quantum information community. The mathematical derivation is sound, but the numerical global-optimization issue should be resolved before publication, either by providing a rigorous procedure or by clearly delimiting the scope of the claims. Asking the author to share the computational code and the numerical data underlying Figs. 1 and 2 would be appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper does something real: it generalizes the fidelity-based stellar-rank witnesses of Chabaud et al. to arbitrary Hermitian witnesses, and reduces the threshold computation to a max-eigenvalue problem over a finite-dimensional subspace plus a low-dimensional optimization over Gaussian unitaries. The derivation from Eq. (5) to Eq. (12) is exact, and the multimode version (18) follows cleanly. That's a genuine advance over Ref. 25, and the two example families—Fock-diagonal probability pairs and cat-state fidelity pairs—are well chosen and the figures are informative.\n\nThe main soft spot is the numerical optimization over the Gaussian parameters. The threshold is defined as a supremum, and the paper does not prove that the reported optimization reaches the global maximum. If it only finds a local maximum, the computed W_n is below the true value, and the witness inequality could accept states of lower stellar rank. This is not a purely formal worry; the boundaries in Figs. 1 and 2 depend on the numerics. The paper even notes that fully numerical approaches seem unavoidable for higher ranks. To make the method fully reliable, the author should either provide a global-optimization certificate (e.g., interval methods, branch-and-bound, or a proof that the objective has no other local maxima) or supply code and data so others can check the thresholds. In practice, the parameter space is only four (or three) dimensional for single mode, so a careful Monte Carlo plus local refinement is likely fine, but that's not the same as a guarantee.\n\nThere is also a minor issue: the paper says 'efficient' and claims the number of optimized parameters doesn't depend on the stellar rank, which is true. But for multimode states the quadratic scaling in mode number could still be steep; that's acknowledged. The trace-distance interpretation is a nice addition, though not central.\n\nOn the whole, this is a solid, clearly written method paper. The central identity is proven exactly, the examples are useful, and the caveats are honestly stated. The lack of a global-optimization guarantee is a real limitation, but it is a common one in this literature and does not invalidate the approach. I would send it to a competent referee, with the request that they either verify a few thresholds independently or ask the author to share code.\n\nRecommendation: engage with the paper, cite it if you work on stellar rank or non-Gaussian witnesses, and read it with the optimization caveat in mind.","headline":"A clean, exact reformulation of stellar-rank witness thresholds that is useful for the quantum-optics community, with a genuine caveat about the reliability of the numerical supremum.","tokens_in":12165,"tokens_out":2731,"would_cite":true,"duration_ms":23904,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P16","81V80"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper shows that every stellar-rank witness threshold reduces to a supremum over Gaussian unitaries of a largest eigenvalue of a projected witness, computable with a rank-independent number of parameters.","keywords":["stellar rank","quantum non-Gaussian states","witness operators","Gaussian unitary operations","Fock state probabilities","Schrödinger cat states","photon-number-resolving detection","trace distance lower bound"],"falsifier":"Take a specific witness, compute its threshold by Eq. (12) with a dense multistart or grid search over the Gaussian parameters, then evaluate the same witness on a known state of stellar rank $n-1$, such as a squeezed Fock state. If the expectation value ever comes out larger than the computed threshold, the optimization missed the global supremum and the published threshold is invalid.","tokens_in":11272,"feed_emoji":"💡","tokens_out":6738,"duration_ms":57268,"temperature":0.7,"pith_summary":"The paper establishes a general recipe for witnessing the stellar rank of optical quantum states. Instead of optimizing directly over the coefficients of the core Fock superposition, one optimizes over the Gaussian unitary that dresses the state, and for each Gaussian operation the best core state is simply the dominant eigenvector of the projected witness operator. The result is Eq. (12), which expresses the threshold as a maximum over Gaussian unitaries of the maximal eigenvalue of the projected witness. Because the initial phase shift commutes with the Fock-subspace projector, the effective optimization for a single mode is over four real parameters no matter how large the targeted stellar rank is, and over $O(N^2)$ parameters for $N$ modes. This matters because stellar rank counts the Fock resources needed to build a state together with Gaussian operations, so the thresholds certify genuine $n$-photon quantum non-Gaussianity in experiments.","feed_headline":"Four parameters certify any stellar rank of light","feed_subtitle":"A new formula turns arbitrary measurements into witnesses of genuine n-photon non-Gaussianity.","key_machinery":"The central object is the projected, Gaussian-conjugated witness operator $\\hat W_{n,\\hat U_G} = \\hat\\Pi_{n-1}\\hat U_G \\hat W \\hat U_G^\\dagger \\hat\\Pi_{n-1}$, a Hermitian operator on the $n$-dimensional Fock subspace spanned by $|0\\rangle,\\dots,|n-1\\rangle$. Eq. (12) says that its maximal eigenvalue, maximized over $\\hat U_G$, is the witness threshold. The argument is carried by the observation that the initial phase shift commutes with $\\hat\\Pi_{n-1}$, so the Gaussian optimization is over squeezing $r$, one phase $\\vartheta$, and the complex displacement $\\alpha$; for Fock-diagonal witnesses the phase drops out and only three parameters remain. Numerical evaluation rests on the analytic Fock-basis matrix elements of Gaussian unitaries expressed as finite sums of Hermite polynomials. In the multimode case the same construction uses the Bloch-Messiah decomposition, and the output interferometer drops out because it commutes with the total-photon-number projector.","core_discovery":"The paper claims that for every Hermitian witness operator $\\hat W$ and every target stellar rank $n$, the threshold separating rank at least $n$ from lower ranks is exactly $$W_n = \\sup_{\\hat U_G} \\max \\mathrm{eig}\\left(\\hat\\Pi_{n-1}\\hat U_G \\hat W \\hat U_G^\\dagger \\hat\\Pi_{n-1}\\right),$$ where $\\hat\\Pi_{n-1}$ projects onto photon numbers $0,\\dots,n-1$. The supremum over all pure lower-rank states is absorbed in two steps: the Gaussian unitary is separated from the core state, and for a fixed unitary the optimal core state is the largest-eigenvalue eigenstate of the projected operator. Thus the number of optimized parameters is fixed by the Gaussian operation alone, and for $\\hat W = |w\\rangle\\langle w|$ the formula reduces to the earlier fidelity-based witnesses. The new generality allows witnesses built from arbitrary linear combinations of data, such as pairs of Fock-state probabilities or fidelities with even and odd coherent states, and the paper demonstrates these examples numerically.","pith_inferences":["Beyond the paper: the formula can be inverted for witness design, optimizing $\\hat W$ to maximize the gap $\\mathrm{Tr}[\\hat W\\hat\\rho]-W_n$ for a target state $\\hat\\rho$, which would trade the fixed parameter count for a witness-design search.","Beyond the paper: the same spectral construction, with $\\mathrm{SU}(2)$ rotations in place of Gaussian unitaries, would yield a stellar-rank hierarchy for spin states, a direction the paper sketches but leaves open.","Beyond the paper: because multiplexed click statistics are linear in thermal-state overlaps, realistic detector response functions might be folded into $\\hat W$ analytically, making threshold computation fully analytic for common detector models.","Beyond the paper: a practical protocol could certify the highest stellar rank by sweeping $n$ and checking inequality (4) on the same measured data, without full state tomography."],"forward_implications":["Every Hermitian observable can be converted into a stellar-rank certificate, extending certification beyond projectors onto a single reference state.","Thresholds for any chosen $n$ can be computed with the same bounded parameter count, so high-rank certification avoids optimizing over the core-state coefficients.","The witness family is closed under Gaussian unitary conjugation: if $\\hat W$ certifies rank $n$, then $\\hat V_G \\hat W \\hat V_G^\\dagger$ certifies the same rank with the same threshold.","Bounded witnesses give a quantitative lower bound on the trace distance from the set of states of stellar rank at most $n-1$, so witness violations measure how far the state is from that set.","For experiments, only a few probabilities or fidelities are needed, and the examples show that pairs of Fock probabilities or cat-state fidelities already certify ranks up to at least $n=3$."],"supporting_citations":[{"why":"Defines stellar rank via the core-state representation and the Husimi-Q zero counting that underpins the hierarchy.","marker":"[19]"},{"why":"Introduced fidelity-based stellar-rank witnesses that the present method generalizes to arbitrary Hermitian witness operators.","marker":"[25]"},{"why":"Established the hierarchy of genuine n-photon quantum non-Gaussian states and the previous direct optimization over core-state coefficients that this method avoids.","marker":"[18]"},{"why":"Provides the trace-distance lemma used to turn witness violations into lower bounds on distance from lower-rank states.","marker":"[29]"},{"why":"Supplies the single-mode and Bloch-Messiah decompositions of Gaussian unitaries used to reduce the optimization parameters.","marker":"[28]"},{"why":"Gives the analytic Fock-basis matrix elements of Gaussian unitaries used to evaluate the projected witness operator.","marker":"[30]"}],"fun_headline_variants":["Fixed parameters witness any stellar rank of light","Stellar rank witnesses with quadratic mode scaling","Efficient witness construction for nonclassical light","Witness hierarchy of light states with constant parameters","Stellar rank certified by scalable witness methods"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The numerical search over the squeezing, phase, and displacement parameters is assumed to reach the true global maximum rather than a local one, and a local maximum would make the threshold too low and could falsely certify a lower-rank state.","fun_headline_variants_meta":{"raw":{"variants":["Fixed parameters witness any stellar rank of light","Stellar rank witnesses with quadratic mode scaling","Efficient witness construction for nonclassical light","Witness hierarchy of light states with constant parameters","Stellar rank certified by scalable witness methods"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000173,"raw_usage":{"total_tokens":1243,"prompt_tokens":876,"completion_tokens":367,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":492,"completion_tokens_details":{"reasoning_tokens":299}},"tokens_in":492,"tokens_out":367,"duration_ms":3774,"temperature":1.0,"reasoning_tokens":299,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:18:47.223312+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a specific witness, compute its threshold by Eq. (12) with a dense multistart or grid search over the Gaussian parameters, then evaluate the same witness on a known state of stellar rank $n-1$, such as a squeezed Fock state. If the expectation value ever comes out larger than the computed threshold, the optimization missed the global supremum and the published threshold is invalid.","supporting_citations":[{"cited_title":"Stellar Representation of Non-Gaussian Quantum States,","cited_arxiv_id":null,"evidence_quote":"Defines stellar rank via the core-state representation and the Husimi-Q zero counting that underpins the hierarchy."},{"cited_title":"Certification of Non-Gaussian States with Operational Measurements,","cited_arxiv_id":null,"evidence_quote":"Introduced fidelity-based stellar-rank witnesses that the present method generalizes to arbitrary Hermitian witness operators."},{"cited_title":"Faithful Hierarchy of Genuine n-Photon Quantum Non-Gaussian Light,","cited_arxiv_id":null,"evidence_quote":"Established the hierarchy of genuine n-photon quantum non-Gaussian states and the previous direct optimization over core-state coefficients that this method avoids."},{"cited_title":"Witnessing Wigner Negativity,","cited_arxiv_id":null,"evidence_quote":"Provides the trace-distance lemma used to turn witness violations into lower bounds on distance from lower-rank states."},{"cited_title":"Squeezing as an irreducible resource,","cited_arxiv_id":null,"evidence_quote":"Supplies the single-mode and Bloch-Messiah decompositions of Gaussian unitaries used to reduce the optimization parameters."},{"cited_title":"Displaced and Squeezed Fock States,","cited_arxiv_id":null,"evidence_quote":"Gives the analytic Fock-basis matrix elements of Gaussian unitaries used to evaluate the projected witness operator."}],"review_version":1}