{"id":"313ce8b9-918e-4154-bfeb-e1e8c3ba1c72","arxiv_id":"2608.09667","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A prepare-and-measure correlation is non-classical for a given set of inputs exactly when no adversary can predict the outcomes with certainty; this yields SDP-based certification methods under three communication restrictions.","lead":"This paper offers a way to certify that quantum correlations are non-classical by checking whether an adversary with classical side information could predict the measurement outcomes with certainty. The approach reuses existing randomness-certification tools, making it applicable under different physical assumptions about the communication between the parties.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The bounded-overlap SDP in Eq. (8) replaces the absolute-value overlap assumption Sβ by a real Gram-entry constraint; the WLOG reduction in App. A2 is not generally valid, so infeasibility of that SDP can falsely certify classical correlations as non-classical.","rationale":"Observation 1 is essentially definitional and is not in serious doubt: if the guessing probability is one, every hidden-variable branch has a deterministic outcome, and conversely. The fixed-ensemble normalization issue noted by the reader is a local inconsistency in App. A.1 (the main-text Eq. (7) does not impose tr Γ^λ_{ρ_x,1}=1), and is likely repairable. The reader's weakest_assumption concerning concavity and permutation symmetry in App. B is a legitimate secondary concern about the analytical bound Eq. (13), but that bound is an illustration rather than the paper's central mechanism. The phase linearization in App. A.2 is more load-bearing: it affects the soundness of the bounded-overlap certification method itself, one of the paper's three headline contributions, and the reported n=5 numerical results. If a classical model can be infeasible for Eq. (8), then the method does not certify non-classicality. The proposed test would settle this directly. Since the reader already returned CONDITIONAL and this concern is a different reason for the same condition, the verdict category is unchanged, but the condition should be expanded to require either a proof of the WLOG reduction for n=5 or a corrected complex-phase-aware formulation of the bounded-overlap SDP.","tokens_in":15195,"tokens_out":32145,"duration_ms":311713,"concrete_test":"Construct a deterministic model with n=5 states, β=0.5, and two hidden values λ=1,2 with q=1/2: choose states |ψ_x^λ⟩ = e^{iφ_x^λ}|ϕ_x⟩ with |⟨ϕ_x|ϕ_x'⟩| ≥ β for each λ, and choose the phases so that Σ_λ q_λ e^{i(φ_x^λ-φ_{x'}^λ)}⟨ϕ_x|ϕ_x'⟩ < β for some pair x,x'. The deterministic assignment 'outcome 0 for all x' is realized by taking M_0=I for all λ. Feed this distribution into the SDP (8) with the predictability constraint; if the SDP is infeasible, Eq. (8) falsely certifies a classical correlation as non-classical. A direct preliminary check is to search for a complex PSD Gram matrix with off-diagonal moduli ≥ β whose entrywise absolute value is not PSD; any such matrix for n=5 refutes the WLOG reduction in App. A.2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing gap is the linearization of the bounded-overlap assumption in Appendix A.2, which underlies the certification method in Eq. (8). The assumption Sβ is Σ_λ q(λ)|⟨ψ_x^λ|ψ_x'^λ⟩| ≥ β_xx'. The SDP instead imposes G_{|Ψ_x>,|Ψ_x'⟩} ≥ β_xx', where G_{|Ψ_x>,|Ψ_x'⟩} = Σ_λ q(λ)⟨ψ_x^λ|ψ_x'^λ⟩ is the λ-averaged complex inner product. The paper asserts 'we can w.l.g. reduce the entries of the Gram matrix to be real and therefore ignore the absolute value', but |G_{x,x'}| ≤ Σ_λ q(λ)|⟨ψ_x^λ|ψ_x'^λ⟩|, so a model satisfying Sβ can have G_{x,x'} < β_xx' (even negative) due to λ-dependent phases. The feasible set of Eq. (8) is then not a superset of the exact classical set, and infeasibility no longer certifies the absence of a deterministic model. The phase-rotation freedom on the kets is not sufficient in general: making all averaged inner products real and nonnegative requires the phases to form a coboundary on the non-orthogonality graph, which fails for n>3 in typical complex realizations. The paper's bounded-overlap examples use n=5, so this is not an edge case.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper introduces an adversarial operational definition of classicality in prepare-and-measure scenarios: a correlation is classical for a chosen set of inputs if a hypothetical adversary with classical side-information can perfectly predict the measurement outcomes. The central observation (Observation 1) equates the existence of a deterministic model with unit guessing probability, and the authors use it to turn non-classicality certification into an SDP infeasibility problem. They instantiate this idea for three communication restrictions: fixed ensemble, bounded overlaps, and restricted observables, giving an SDP for each. They also prove a general linear-witness bound under a connected non-orthogonality graph (Result 1) and provide numerical illustrations, including a closed-form bound for bounded-overlap state discrimination in Appendix B.","tokens_in":15566,"tokens_out":12293,"duration_ms":122100,"significance":"The conceptual framework is attractive and potentially useful: it connects non-classicality certification in prepare-and-measure scenarios to a broad body of QRNG and SDP-hierarchy techniques, and it has the virtue of not being tied to dimension assumptions. Observation 1 and Result 1 are correct under the intended reading, and the presentation is generally clear. However, several load-bearing technical points are not established: the definition of the guessing probability is not written correctly, the bounded-overlap relaxation is not a valid outer approximation in general, the fixed-ensemble block-moment normalization is inconsistent as stated, and the Appendix B analytic bound relies on unproved concavity and symmetry assumptions. These issues prevent me from endorsing the current version, but they are substantial rather than trivial and require a major revision.","major_comments":[{"comment":"The definition of p_g^{(x,y)} is not faithful to the preceding sentence. Eq. (6) writes p_g^{(x,y)} = max_{p∈Q_{S_i}} Σ_λ q(λ) max_b p_λ(b|x,y), with no constraint that the optimized realization reproduces the observed distribution p(b|x,y). As written, p_g depends only on the set Q_{S_i}, not on the observed correlation; if Q_{S_i} contains any deterministic distribution, then p_g=1 for every observed p, and the 'only if' direction of Observation 1 is false. The intended optimization is over decompositions {q(λ), ρ_x^λ, M_{b|y}^λ} compatible with S_i and with the fixed p(b|x,y). This must be written explicitly, because Observation 1 is the foundation of all three certification methods.","section":"Eq. (6) and Observation 1"},{"comment":"The replacement of the overlap assumption S_β by the constraint G_{|Ψ_x>,|Ψ_{x'}>} ≥ β_{xx'} is not a legitimate WLOG reduction. The assumption is Σ_λ q(λ)|⟨ψ_x^λ|ψ_{x'}^λ⟩| ≥ β_{xx'}, while G_{|Ψ_x>,|Ψ_{x'}>} is the λ-averaged complex inner product. A valid model can have this complex average with real part below β_{xx'}, or even negative, because of λ-dependent phases. The residual freedom to rotate each |ψ_x^λ> by a λ-independent phase does not in general make all averaged inner products real and nonnegative; for that one would need the phases to form a coboundary on the non-orthogonality graph, which fails in generic complex realizations (already for three states). Consequently the feasible set of Eq. (8) is not a superset of the exact set C^{(x,y)}_{S_β}, and infeasibility of this SDP can falsely certify classical correlations as non-classical. This affects the bounded-overlap method and the numerical illustration in Fig. 2b. Please either prove the reduction for the restricted symmetric cases actually used, or replace Eq. (8) by a genuine outer relaxation that is guaranteed to contain all S_β-compatible models.","section":"App. A2 and Eq. (8)"},{"comment":"The normalization statement for the block-moment matrix is inconsistent. The text sets Γ^λ_{ρ_x,1} = q(λ) ρ_x^λ and then asserts tr(Γ^λ_{ρ_x,1}) = 1. From the definition, tr(Γ^λ_{ρ_x,1}) = q(λ) tr(ρ_x^λ) = q(λ). The correct normalization of a valid fixed-ensemble decomposition is Σ_λ tr(Γ^λ_{ρ_x,1}) = 1, equivalently tr(Γ^λ_{ρ_x,1}) = q(λ). If the individual trace constraint were imposed as written, the SDP Eq. (7) would over-constrain the feasible set and could exclude valid fixed-ensemble models, leading to false non-classicality certification. Please correct Appendix A1 and state explicitly which trace constraints are enforced in the numerical implementation of Eq. (7).","section":"App. A1, after Eq. (A2)"},{"comment":"The derivation of the closed-form bound Eq. (13) relies on three unproved assumptions: (i) the function P(c_λ) is concave in c_λ; (ii) saturation of the averaged overlap constraint implies per-realization saturation |⟨ψ_x^λ|ψ_{x'}^λ⟩| = β for all λ; and (iii) an optimal solution exists with rank-1 projective measurements and with the cyclic label-permutation invariance used in Eqs. (B5)-(B7). These assertions are used to replace the optimization over all λ-dependent preparations by a single symmetric optimization, so they are load-bearing for Eq. (13). For example, concavity is not evident because increasing c_λ shrinks the feasible set, and equality in a Jensen step would require additional strictness arguments. Please supply proofs for these steps or explicitly label Eq. (13) as a conjectured bound rather than a derived one.","section":"Appendix B, Eqs. (B2)-(B10)"}],"minor_comments":[{"comment":"The notation D_λ(b|x,y) ∈ Q_{S_i} is unclear, because D_λ is a deterministic probability table while Q_{S_i} is a set of averaged correlations. It would be clearer to say that the deterministic table D_λ arises from a realization compatible with S_i.","section":"Eq. (4)"},{"comment":"The phrase 'under the assumption that at most |X*| = n* inputs contribute deterministically' is confusing: the selected inputs in X* are deterministic by construction of the predictability constraint. Please rephrase to describe the cardinality of the selected input subset.","section":"Section V.B"},{"comment":"The Gram matrix in Eq. (A8) is presented as a block matrix, but the block indexing and the precise definition of the vector set S are not given in the displayed equation; a short explanation of how blocks correspond to elements of S would improve readability.","section":"App. A2, Eq. (A8)"},{"comment":"The symbol λ is used both for the hidden variable and for the deterministic assignment {λ_xy}; in equations such as Γ^λ_{ρ_x, M_{λ_{xy}|y}} this double use is confusing. A distinct notation for the assignment, e.g. a bold lambda index, would help.","section":"Eq. (7) and App. A1"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the bounded-overlap linearization in App. A2/Eq. (8); the skeptic's concern that infeasibility can be a false positive is well founded and needs a genuine outer relaxation or a carefully restricted claim. The other issues (Eq. (6), trace normalization, Appendix B assumptions) are fixable by rewriting and adding proofs. If the authors can repair the bounded-overlap relaxation or clearly restrict its scope, I would be willing to reconsider the paper for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThe core idea is a neat reframing: classicality in prepare-and-measure scenarios is equivalent to perfect predictability by a classical adversary (Observation 1). That equivalence is correct, and it lets the authors adapt existing QRNG SDP tools as non-classicality witnesses. The fixed-ensemble and restricted-observable SDPs are sensible adaptations, and Result 1 gives a clean analytic bound for connected non-orthogonality graphs. The MUB witness example is a nice illustration.\n\nThe soft spots: first, the bounded-overlap SDP in Eq. (8) has a load-bearing gap. The assumption Sβ involves the average of absolute values of inner products, but the Gram entry G_{|Ψ_x⟩,|Ψ_x'⟩} is the average of the complex inner products. The paper claims WLOG we can make these entries real and nonnegative, but that is not generally true: the phases of the λ-dependent states need to be compatible across the whole graph. For n>3, typical complex realizations don't allow this. So the feasible set of Eq. (8) is not a superset of the actual classical set, and infeasibility can falsely certify non-classicality. This is not an edge case; the n=5 examples rely on it.\n\nSecond, Appendix A1 has an inconsistency: Γ^λ_{ρ_x,1} is defined as q_λ ρ_x^λ, so its trace is q_λ, not 1. The constraint tr(Γ^λ_{ρ_x,1})=1 conflicts with Σ_λ Γ^λ_{ρ_x,1}=ρ_x. Likely a typo, but it needs fixing.\n\nThird, the Appendix B derivation of Eq. (13) assumes concavity of P(c_λ) and a permutation symmetry that are asserted, not proved. That is a minor gap if the numerics are correct, but there is no code or data to check.\n\nOverall, the conceptual framing and the fixed-ensemble/restricted-observable methods are likely salvageable, and the paper deserves a serious referee. But the bounded-overlap method as written is unsound; the authors need to either prove the real-reduction or revise the SDP to handle absolute values explicitly.\n\nBest,\n\n[Your name]","headline":"A sound reframing of classicality as predictability, but the bounded-overlap SDP has a load-bearing linearization error that needs fixing.","tokens_in":16060,"tokens_out":3619,"would_cite":true,"duration_ms":29208,"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":"In prepare-and-measure scenarios, a correlation is classical for a chosen pair of inputs exactly when an adversary with access to the shared randomness can predict the measurement outcome with certainty.","keywords":["prepare-and-measure scenarios","deterministic correlations","classicality certification","adversarial guessing probability","semidefinite programming","communication restrictions","state discrimination"],"falsifier":"For a fixed $n\\ge 5$, compute numerically the largest state-discrimination success probability compatible with the perfect-predictability constraint and pairwise overlaps at least $\\beta$, and compare it with the right-hand side of Eq. (13); any feasible configuration exceeding the bound would disprove it, and evaluating the per-realization optimal success probability as a function of the overlap at intermediate $c$ values would test the concavity step.","tokens_in":15005,"feed_emoji":"🎯","tokens_out":10446,"duration_ms":89404,"temperature":0.7,"pith_summary":"This paper proposes a unified way to certify non-classical correlations in prepare-and-measure experiments: instead of asking whether the data fits a classical message model, ask whether an adversary holding the classical side-information could predict Bob's outcome perfectly. The paper establishes that, for any fixed choice of Alice's and Bob's settings, these two questions are the same—perfect predictability is exactly equivalent to the existence of a deterministic explanation. It then turns this equivalence into a computational tool, formulating the search for a deterministic explanation as a semidefinite-programming feasibility problem under three different communication restrictions: a fixed ensemble of prepared states, lower bounds on pairwise state overlaps, and bounds on observable expectation values. A sympathetic reader would care because the approach can inherit a large body of semidefinite-programming tools from randomness certification, and it works without assuming a fixed Hilbert-space dimension.","feed_headline":"Predictable outcomes mean classical prepare-and-measure correlations","feed_subtitle":"One equivalence turns non-classicality certification into a semidefinite-programming feasibility test.","key_machinery":"The load-bearing object is the adversarial guessing probability $p_g^{(x,y)}$ and the equivalence in Observation 1. The certification machinery consists of three SDP relaxations: a block-moment matrix for the fixed-ensemble restriction, a Gram-matrix hierarchy for bounded overlaps, and a tracial moment-matrix hierarchy for restricted observables; in each, the observed statistics, the restriction, and the perfect-predictability condition are encoded as linear constraints on a positive semidefinite matrix. The paper also uses the non-orthogonality graph of the prepared states—vertices are preparations and edges connect non-orthogonal states—to prove that, under a fixed pure-state ensemble, deterministic models force identical outcomes across the whole connected graph, yielding the linear-witness bound of Result 1.","core_discovery":"The central claim is Observation 1: correlations produced in a prepare-and-measure scenario for a concrete choice of input settings $(x,y)$ admit a classical model if and only if the optimal guessing probability $p_g^{(x,y)}=1$, where $p_g^{(x,y)}$ is the maximum, over all quantum realizations compatible with the communication restriction, of $\\sum_\\lambda q(\\lambda)\\max_b p_\\lambda(b|x,y)$. The forward direction is immediate from the definition of a deterministic model; the converse holds because a guessing probability of one forces every hidden-variable branch to have a deterministic outcome distribution. Consequently, showing that no admissible quantum realization allows an adversary to predict perfectly is logically equivalent to certifying that the correlations are non-classical. The paper uses this equivalence to certify non-classicality by proving the infeasibility of an SDP that imposes the linear constraint of perfect predictability, and it illustrates the method with three restrictions, including a closed-form bound for state discrimination with bounded overlaps.","pith_inferences":["The same SDP templates should apply to any other communication restriction expressible as linear constraints, such as energy bounds or information-content bounds, making the three worked examples a proof of principle rather than an exhaustive list.","If the size of the input subset for which perfect predictability holds is treated as a quantifier, the approach defines a graded measure of determinism that could be used to compare how non-classical different correlations are.","A numerical audit of the two auxiliary assumptions used in deriving Eq. (13)—concavity of the per-realization optimal success probability and the existence of rank-one projective measurements with cyclic symmetry—would establish how broadly that closed-form bound holds."],"forward_implications":["Any communication restriction that can be written as linear constraints on the moments fits the same certification template, so the three SDP programs are instances of a general method rather than isolated tests.","For a fixed pure-state ensemble whose non-orthogonality graph is connected, every linear witness $W=\\sum_{b,x,y} c_{bxy}p(b|x,y)$ satisfies the deterministic bound $W\\le \\sum_y \\max_b\\sum_x c_{bxy}$.","For the mutually-unbiased-bases witness $W_{n,d}$, the quantum maximum is $nd$ and the deterministic bound is $n$, so any measured value above $n$ certifies non-classicality with detector efficiency above $1/d$.","For $n$ states with pairwise overlaps at least $\\beta$ and a single deterministic input, the success probability is bounded by $\\frac{1}{n}(1+\\Delta^2/(n-1))$, and for $n=2$ this coincides with a known non-contextual bound.","Classicality becomes relative to a chosen subset of inputs, producing a nested hierarchy in which a correlation can be deterministic for some settings and non-classical for others."],"supporting_citations":[{"why":"supplies the semidefinite-programming relaxation toolbox that the paper adapts from randomness certification to classicality feasibility tests.","marker":"[14]"},{"why":"provides the block-moment matrix method used for the fixed-ensemble restriction in program (7).","marker":"[27]"},{"why":"provides the Gram-matrix hierarchy used for the bounded-overlap restriction in program (8).","marker":"[28]"},{"why":"provides the tracial moment-matrix framework and photon-number model used for the restricted-observable restriction in program (9).","marker":"[22]"},{"why":"gives the non-contextual state-discrimination bound that the closed-form expression Eq. (13) reproduces for n=2.","marker":"[30]"},{"why":"represents the dimension-restricted classicality certification approach that the paper generalizes to arbitrary communication restrictions.","marker":"[31]"}],"fun_headline_variants":["Perfect prediction flags classical prepare-and-measure correlations","Non-classical correlations resist perfect prediction by classical adversary","SDP feasibility test detects non-classical prepare-and-measure correlations","Adversary's perfect guess marks classical correlations in PM scenarios","Quantum correlations certified by impossible perfect prediction"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analytical bounded-overlap bound assumes, without proof, that the best per-realization discrimination probability is concave in the overlap and that an optimal solution can be chosen with rank-one projective measurements invariant under cyclic relabeling of the states; if either assumption fails, Eq. (13) and its numerical illustration do not follow.","fun_headline_variants_meta":{"raw":{"variants":["Perfect prediction flags classical prepare-and-measure correlations","Non-classical correlations resist perfect prediction by classical adversary","SDP feasibility test detects non-classical prepare-and-measure correlations","Adversary's perfect guess marks classical correlations in PM scenarios","Quantum correlations certified by impossible perfect prediction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000413,"raw_usage":{"total_tokens":2085,"prompt_tokens":846,"completion_tokens":1239,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":462,"completion_tokens_details":{"reasoning_tokens":1162}},"tokens_in":462,"tokens_out":1239,"duration_ms":12136,"temperature":1.0,"reasoning_tokens":1162,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T13:03:34.274899+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a fixed $n\\ge 5$, compute numerically the largest state-discrimination success probability compatible with the perfect-predictability constraint and pairwise overlaps at least $\\beta$, and compare it with the right-hand side of Eq. (13); any feasible configuration exceeding the bound would disprove it, and evaluating the per-realization optimal success probability as a function of the overlap at intermediate $c$ values would test the concavity step.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the block-moment matrix method used for the fixed-ensemble restriction in program (7)."},{"cited_title":"Pauwels, S","cited_arxiv_id":null,"evidence_quote":"gives the non-contextual state-discrimination bound that the closed-form expression Eq. (13) reproduces for n=2."},{"cited_title":"Roch i Carceller and J","cited_arxiv_id":null,"evidence_quote":"represents the dimension-restricted classicality certification approach that the paper generalizes to arbitrary communication restrictions."}],"review_version":1}