{"id":"5ed24bcf-81ba-4a0b-a6d8-2607f330dc80","arxiv_id":"2505.17585","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A construction certifies maximal randomness in bipartite and tripartite scenarios directly from probability distributions, with an incompatibility trade-off that lets all but one user use nearly compatible measurements.","lead":"This paper shows how to certify maximal randomness from observed quantum correlations without assuming device dimension or relying on Bell-inequality violations. The result could make quantum random number generation simpler and more practical, because only one user's measurements need be strongly incompatible.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Certification gap: boundary does not imply indecomposability; constructed distributions are not proven extreme points of Q.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing gap: the paper treats 'boundary' as sufficient for non-decomposability, but without extremality (or a stronger uniqueness property), Eve may still decompose the distribution and guess outcomes with probability above 1/n^2. My analysis agrees with this assessment and adds that the two-qubit pure-state optimization in Eqs. (4)-(7) does not establish that the constructed points lie on the boundary of the full quantum set, since Q allows arbitrary Hilbert-space dimension and POVMs. This is the single most load-bearing concern because the central claim — maximal randomness certification — depends directly on the guessing-probability bound, which in turn depends on indecomposability. The tripartite section's admitted lack of rigorous justification is a further, but secondary, gap. The paper has genuine strengths: the direct distribution-based optimization is a useful approach, the incompatibility trade-off is an interesting qualitative finding, and the bipartite derivation is analytically plausible. However, the certification argument is incomplete, not merely missing a technical detail. The concern is addressable via a proof of extremality (or an NPA-based numerical verification), so the reader's CONDITIONAL verdict remains appropriate; my stress-test does not change it.","tokens_in":8410,"tokens_out":12901,"duration_ms":108559,"concrete_test":"For a fixed parameter set (e.g., the maximally entangled case used in Fig. 3), construct the linear functional L(P) = P(1,1|2,2) + Σ μ_i (constraint_i(P) - c_i) using the Lagrange multipliers from the paper's optimum, and solve the NPA level-2 semidefinite program maximizing L. If the maximum is strictly larger than L(P*) or the maximizer set has positive dimension, the constructed point is not certified as an extreme point of Q, confirming the gap. If the maximum equals L(P*) with a unique optimizer, extremality is established within the NPA set; repeating at level 3 gives additional confidence. Independently, maximize P(1,1|2,2) over the NPA set under the constraints of Eq. (3); if this bound exceeds the paper's value from Eqs. (5)-(7), the constructed distributions are not on the true boundary of Q.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The certification step in 'Definitions and Method' claims that Eve cannot decompose a distribution on the boundary of Q because the boundary is 'non-flat due to the non-polyhedral nature of the quantum set.' This inference is invalid: non-polyhedral convex sets can have flat faces (e.g., a cylinder), and boundary points can be convex combinations of other points. To obtain Pg = 1/n^2 in Eq. (1), one must show the constructed distribution is an extreme point of Q, or at least that every convex decomposition has a uniform target distribution. The paper proves neither. Moreover, Eq. (5) optimizes only over two-qubit pure states, so the resulting values may not correspond to the true boundary of Q (which allows arbitrary dimensions and POVMs); without an upper bound over all Q, the 'boundary positioning' condition (2) is unverified. The same gap propagates to the tripartite result, where the text explicitly admits the restriction to GHZ states 'lacks rigorous justification' and relies on a numerical check with relative error below 10^-5. Thus the central claim of maximal randomness is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a framework for certifying maximal randomness directly from observed probability distributions, bypassing the usual step of Bell-inequality violation. It states two conditions for maximal randomness—uniform output distribution and boundary positioning in the quantum set—and applies them to construct bipartite and tripartite distributions from two-qubit and GHZ states with projective measurements. The authors derive analytical bounds for the objective probability, quantify the incompatibility robustness of the users' measurements, and report a trade-off whereby sufficiently large incompatibility of one party permits arbitrarily small incompatibility of others. The central claim is that maximal randomness, with guessing probability 1/n^p and min-entropy p log2 n, is achieved by these distributions.","tokens_in":8701,"tokens_out":9625,"duration_ms":78391,"significance":"If the certification step were sound, the framework would be a useful contribution to device-independent randomness: it works directly with probability distributions, avoids post-selecting on Bell-inequality violations, gives explicit two-qubit analytical formulas, and introduces a quantitative incompatibility trade-off. The manuscript is also transparent about the tripartite restriction to GHZ states and includes numerical checks. However, the foundational gap between boundary and extreme points of the quantum set means the claimed maximal-randomness certification is not established; the paper's own admission that the tripartite restriction 'lacks rigorous justification' further limits the current result. The geometric idea is interesting, but the proof needs substantial additional work before the main conclusion can be accepted.","major_comments":[{"comment":"The paper treats 'the distribution lies on the boundary of the quantum set' as sufficient for Pg = 1/n^2, with the justification that the boundary is 'non-flat due to the non-polyhedral nature of the quantum set.' This inference is invalid: a boundary point of a convex set need not be an extreme point, and non-polyhedral convex sets can have flat faces (for example, a cylinder is non-polyhedral and has non-extreme boundary points). To bound the guessing probability in Eq. (1), one must prove either that the constructed point is an extreme point of Q or that every convex decomposition of it has uniform target-conditioned distributions. The manuscript does neither, so the central certification step is unproven.","section":"Definitions and Method (conditions after Eq. (2))"},{"comment":"The optimization in Eq. (5) is restricted to two-qubit pure states and projective measurements, but the quantum set Q in Eq. (1) includes arbitrary finite dimensions and POVMs. The solution of Eq. (5) therefore gives an extremum of the objective within a restricted two-qubit slice, not necessarily a point on the boundary of the full quantum set. No upper or lower bound over all quantum realizations is supplied, so the 'boundary positioning' condition is not verified even for the bipartite examples shown in Figs. 2-4. Without that verification, the derivations in Eqs. (4)-(7) do not certify maximal randomness.","section":"Results, Eqs. (3)-(5)"},{"comment":"The sentence 'Their nonlocality guarantees the generation of maximal randomness' is not a valid inference. A CHSH violation rules out local hidden-variable models, but it does not imply that the guessing probability Pg in Eq. (1) equals 1/n^2; nonlocal quantum correlations can generically be decomposed into convex combinations of other nonlocal quantum points, allowing Eve a nontrivial guess. The manuscript needs an explicit proof—for the specific displayed distributions—that Pg = 1/4, rather than relying on nonlocality alone.","section":"Results, discussion of Fig. 3"},{"comment":"The text explicitly states that the restriction to GHZ states and identical A', B' 'lacks rigorous justification' and that correctness is only checked numerically with relative error below 10^-5. A numerical check over a restricted family does not prove that the optimized value lies on the boundary of the full tripartite quantum set, nor that the corresponding distribution certifies 3 log2 2 bits. The tripartite claim in the Conclusion is therefore stronger than what is established in the manuscript.","section":"Results, tripartite paragraph after Eq. (8)"},{"comment":"Equation (7) states sqrt(2 P(1,1|2,2)) = |alpha1 alpha2 + beta1 beta2| <= g(x,z), which is an upper bound on the objective, yet the text identifies this as the 'minimum value' of P(1,1|2,2). For the minimization problem in Eq. (3), a lower bound of the form P >= g^2/2 is needed. Please clarify the intended inequality direction and provide the corresponding derivation; if the displayed direction is correct, the subsequent identification of optimal boundary points in Figs. 3 and 4 does not follow.","section":"Results, Eq. (7)"}],"minor_comments":[{"comment":"The compatibility condition on M^eta_{a|x} is not defined in the main text; please specify that for each input x the set {M^eta_{a|x}}_a must be jointly measurable, and state the allowed range of eta (presumably eta in [0,1]).","section":"Eq. (6)"},{"comment":"The caption says 'The entanglement can be arbitrarily big (A^2 -> 0.5) or arbitrarily small (A^2 -> 0)' without defining A^2; please state explicitly that A is the Schmidt coefficient in the two-qubit state of Eq. (4), since 'entanglement' alone is ambiguous.","section":"Figure 2 caption"},{"comment":"The blank region in Fig. 3 is described only as resulting from 'failure to satisfy the constraints'; please list the explicit constraints on x, z, and g(x,z) that define the feasible region, so that the figure can be interpreted quantitatively.","section":"Results, Fig. 3 discussion"},{"comment":"The tripartite variables x and z are reused without restating their definitions; please either define them again in the tripartite paragraph or refer explicitly to the corresponding equation in the supplementary material.","section":"Tripartite section, Eq. (8)"},{"comment":"There are minor language issues: 'Several fundamental questions naturally arises' should be 'naturally arise', and the Conclusion contains 'with as long as' where 'as long as' is intended. Also, the ket in the maximally entangled state is mistyped as '|11)' instead of '|11>'.","section":"Introduction and Conclusion"}],"recommendation":"major_revision","confidential_remarks":"The manuscript's own admission that the tripartite restriction 'lacks rigorous justification' is a strong signal that the Conclusion overstates the result. The bipartite claim also depends on an unproven extremality condition. The authors should either prove extremality of their constructed points (or provide explicit upper bounds on Pg) and supply a rigorous tripartite treatment, or substantially weaken the maximal-randomness claims. The topic is suitable for the journal if the proof gap is closed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. The genuinely new part is the direct optimization of probability distributions for randomness certification, without Bell-inequality post-processing, and the explicit projective-measurement constructions for arbitrary bipartite entanglement. The quantitative trade-off between incompatibility robustness values (Fig. 4) is also novel and, if it holds, practically useful: one party can have nearly compatible measurements as long as the other party's incompatibility is sufficiently large. That part is worth engaging with.\n\nWhat is not established is the central claim of maximal randomness. The certification step in Definitions and Method asserts that a distribution on the boundary of the quantum set cannot be decomposed by Eve because the boundary is \"non-flat due to the non-polyhedral nature of the quantum set.\" That inference is wrong as stated. Non-polyhedral convex sets can have flat faces, and a boundary point can be a convex combination of other boundary points. To get guessing probability 1/n^2 from Eq. (1), you need the constructed distribution to be an extreme point of Q, or at least that every convex decomposition yields a uniform target distribution. The paper proves neither. Also, Eq. (5) optimizes only over two-qubit pure states and projective measurements, so the resulting point may lie on the boundary of that restricted inner approximation, not on the boundary of Q. Without an upper bound over all states and POVMs, condition (2) is unverified.\n\nThe tripartite extension is weaker still. The text explicitly says the restriction to GHZ states \"lacks rigorous justification\" and that correctness is checked numerically with relative error below 10^-5. That is an honest admission, but it means the tripartite certification claim is not a proof.\n\nThe bipartite algebra in Eqs. (4)-(7) is explicit and self-consistent, and the numerical dependence on x and z does show nonlocality in the sense of CHSH violation. So this is not a careless paper — the construction is likely correct as a set of quantum behaviors, and the incompatibility trade-off is a real result. The missing piece is the certification step, which is load-bearing for the maximal-randomness conclusion but addressable.\n\nWho gets value from this? People working on DI randomness protocols will want the construction and the trade-off, but they should not treat the certified randomness claim as proven. The paper deserves a serious referee; I would send it to review with a request to fix the extremality argument and provide a genuine upper bound over Q for the bipartite case. The tripartite section should be reframed as numerical evidence, not a theorem.\n\nMy recommendation: engage with it, but treat the maximal-randomness certification as open until the boundary-to-extreme-point step is resolved.","headline":"A promising direct-optimization construction with a real incompatibility trade-off, but the maximal-randomness claim is not proven because boundary points of the quantum set need not be extreme points.","tokens_in":9153,"tokens_out":1472,"would_cite":true,"duration_ms":13591,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["81P15","81P40","81P45","94A60"],"pacs":["03.65.Ud","03.67.-a","03.67.Dd"],"model":"deepseek-v4-flash","headline":"The paper aims to prove that maximal randomness can be certified by placing the observed distribution on a curved boundary point of the quantum set, and that only one user then needs highly incompatible measurements.","keywords":["maximal randomness","device-independent randomness certification","measurement incompatibility","incompatibility robustness","quantum probability distributions","Bell nonlocality","min-entropy","quantum networks"],"falsifier":"Take one of the optimized distributions from Eq. (3) with parameters in the allowed region and run a convergent hierarchy of semidefinite relaxations, or an explicit search over decompositions, to test whether the point in $Q$ can be written as $\\lambda P_1 + (1-\\lambda) P_2$ with $P_1 \\neq P_2$ both in $Q$. If such a decomposition exists, the point is not extreme, the guessing probability exceeds $1/4$, and the claimed 2-bit min-entropy is false. For the tripartite claim, the same test should be applied to the reported points, since the paper's only evidence is a single numerical optimization with error below $10^{-5}$.","tokens_in":8216,"feed_emoji":"🎲","tokens_out":10342,"duration_ms":76719,"temperature":0.7,"pith_summary":"This paper is trying to show that maximal randomness can be certified directly from observed probability distributions, without assuming the dimension of the systems or relying on Bell-inequality violations, and that the key resource is the placement of the distribution on the non-flat boundary of the quantum set. In the bipartite case with two inputs and two outputs, the authors exhibit two-qubit pure states and projective measurements whose outputs are perfectly uniform and whose point lies on this boundary, giving guessing probability $1/4$ and $2$ bits of min-entropy. They then derive a quantitative trade-off: maximal randomness is possible as long as one party's measurement incompatibility robustness parameter is sufficiently large, while the other party's parameter can approach the limiting value that corresponds to arbitrarily small incompatibility. The same asymmetric pattern is reported for three parties sharing a GHZ state, with the explicit caveat, stated in the text, that the tripartite maximally-entangled simplification is not rigorously justified and was validated only numerically.","feed_headline":"One strong measurement unlocks maximal quantum randomness","feed_subtitle":"Reaches the randomness limit without Bell-inequality violations, letting other users remain nearly compatible.","key_machinery":"The engine of the argument is the constrained optimization problem of Eq. (3): optimize one conditional probability, say $P(1,1|2,2)$, subject to the uniform-output constraints $P(a,b|1,1)=1/4$ and extra linear constraints fixing marginals $s$ and $t$, over the quantum set $Q$. Convexity of $Q$ plus the assumption that its boundary is non-flat means the optimizer lands on an extreme, non-decomposable point, which forces Eve's guessing probability to its lower bound. The optimization is made explicit by writing two-qubit pure states in Schmidt form, reducing the problem to the one-dimensional bound $f(A;s,t)$ of Eq. (5), and then to the incompatibility-robustness quantifier of Eq. (6), whose feasible region yields the trade-off curve between the two users' incompatibility parameters. The tripartite version repeats this structure with a GHZ state and a function $g_T(x,z)$ analogous to the bipartite $g(x,z)$.","core_discovery":"The central claim, stated on the paper's own terms, is that globally maximal randomness is achievable whenever the users' joint distribution is uniform on the target outputs and lies on a boundary point of the quantum set $Q$, because the quantum set has a curved (non-polyhedral) boundary and therefore does not admit the decomposition Eve needs. Concretely, the paper identifies a family of two-qubit states and projective measurements solving the constrained optimization problem of Eq. (3); for these, the guessing probability reaches $1/n^2$ in the bipartite case and $1/n^3$ in the tripartite case, i.e. min-entropy $p\\log_2 n$ bits for $p$ users. A quantitative relation to measurement incompatibility follows: for maximally entangled bipartite states, the incompatibility parameters of the two users obey a trade-off, and if one user's parameter approaches the limiting value $η \\to \\sqrt{2}/2$, then the other user's parameter can approach $η \\to 1$ while maximal randomness is retained. In the tripartite case the analogous conclusion is that one party, $C'$, needs sufficiently large incompatibility while the others need almost none.","pith_inferences":["The paper stops short of proving that its boundary points are extreme points of the quantum set; checking extremality directly, for example with a convergent hierarchy of semidefinite relaxations, would turn the boundary criterion into a fully rigorous certificate.","If the boundary-non-flatness argument transfers, a similar asymmetry between one strong and many weak measurements may hold in steering or prepare-and-measure scenarios, where the relevant sets have different geometries.","The explicit caveat that the GHZ-based tripartite solution lacks rigorous justification invites a search for three-qubit states that are not maximally entangled and could attain the same or smaller objective value, which would strengthen or invalidate the reported trade-off."],"forward_implications":["If the boundary-point criterion holds, maximal randomness can be certified without any Bell-inequality violation, using only the raw probability distribution and simple projective measurements.","Only one user in the network needs highly incompatible measurements; the remaining users can use nearly compatible measurements, which relaxes experimental demands in multi-user protocols.","The method generalizes to arbitrary numbers of users $p$ and outputs $n$, yielding the global min-entropy bound $p \\log_2 n$ bits whenever the analogous optimization can be solved.","The trade-off between the users' incompatibility parameters gives a quantitative resource-allocation rule for distributing measurement incompatibility across a quantum network.","The explicit numerical validation of the tripartite case (relative error below $10^{-5}$) suggests, but does not prove, that the same asymmetry extends beyond bipartite systems."],"supporting_citations":[{"why":"Defines the guessing probability used for randomness certification and shows that direct distribution analysis can beat inequality-based extraction.","marker":"[5]"},{"why":"Supplies the two conditions, uniform outputs and boundary of the quantum set, that the paper elevates into its optimization criterion.","marker":"[15]"},{"why":"Establishes the uniqueness and boundary conditions for maximal quantum randomness in Bell tests that the authors generalize.","marker":"[17]"},{"why":"Justifies restricting to pure states by showing that boundary distributions come from pure states.","marker":"[29]"},{"why":"Provides the incompatibility-robustness quantifier used to state the trade-off between users' measurements.","marker":"[30]"},{"why":"Gives the necessary-and-sufficient link between randomness certification and measurement incompatibility that the paper makes quantitative.","marker":"[31]"},{"why":"Cited to explain why the tripartite maximally-entangled simplification lacks rigorous justification, since tripartite states have no Schmidt decomposition.","marker":"[41]"}],"fun_headline_variants":["Maximal randomness certified without Bell inequality violations","One strong measurement lets others stay nearly compatible","Trade-off: one user's incompatibility, others need almost none","Directly certify maximal randomness from observed distributions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole certificate rests on the premise that a point on the boundary of the quantum set is automatically non-decomposable because that boundary is curved; if any constructed distribution sits on a flat face or can be written as a mixture of two other quantum distributions, Eve's guessing probability rises above the uniform value and the claimed maximal randomness fails.","fun_headline_variants_meta":{"raw":{"variants":["Maximal randomness certified without Bell inequality violations","One strong measurement lets others stay nearly compatible","Trade-off: one user's incompatibility, others need almost none","Directly certify maximal randomness from observed distributions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001006,"raw_usage":{"total_tokens":4241,"prompt_tokens":920,"completion_tokens":3321,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":3261}},"tokens_in":536,"tokens_out":3321,"duration_ms":27367,"temperature":1.0,"reasoning_tokens":3261,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T14:45:11.164108+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take one of the optimized distributions from Eq. (3) with parameters in the allowed region and run a convergent hierarchy of semidefinite relaxations, or an explicit search over decompositions, to test whether the point in $Q$ can be written as $\\lambda P_1 + (1-\\lambda) P_2$ with $P_1 \\neq P_2$ both in $Q$. If such a decomposition exists, the point is not extreme, the guessing probability exceeds $1/4$, and the claimed 2-bit min-entropy is false. For the tripartite claim, the same test should be applied to the reported points, since the paper's only evidence is a single numerical optimization with error below $10^{-5}$.","supporting_citations":[{"cited_title":"Bancal, L","cited_arxiv_id":null,"evidence_quote":"Defines the guessing probability used for randomness certification and shows that direct distribution analysis can beat inequality-based extraction."},{"cited_title":"Ac ´ ın, S","cited_arxiv_id":null,"evidence_quote":"Supplies the two conditions, uniform outputs and boundary of the quantum set, that the paper elevates into its optimization criterion."},{"cited_title":"Dhara, G","cited_arxiv_id":null,"evidence_quote":"Establishes the uniqueness and boundary conditions for maximal quantum randomness in Bell tests that the authors generalize."},{"cited_title":"Senno, T","cited_arxiv_id":null,"evidence_quote":"Justifies restricting to pure states by showing that boundary distributions come from pure states."},{"cited_title":"Designolle, M","cited_arxiv_id":null,"evidence_quote":"Provides the incompatibility-robustness quantifier used to state the trade-off between users' measurements."},{"cited_title":"Ac ´ ın, A","cited_arxiv_id":null,"evidence_quote":"Cited to explain why the tripartite maximally-entangled simplification lacks rigorous justification, since tripartite states have no Schmidt decomposition."}],"review_version":1}