{"id":"955ef6c8-f8cf-44d4-a081-5d91fdf3d13e","arxiv_id":"2607.24700","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"Almost all pure entangled states, assisted by a small local ancilla, enable unbounded two-sided CHSH nonlocality sharing with only projective measurements via Hardy’s paradox.","lead":"Any quantum correlations that show Hardy’s paradox can be turned into a protocol where arbitrarily many Alice–Bob pairs all violate CHSH at once, using only projective measurements plus a tiny local memory. Because almost every pure entangled state admits Hardy correlations, two-sided nonlocality sharing is generic rather than exceptional.","discovery_kind":"new_method","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"No significant objection identified. The reader's flagged weakest assumption (Table III's strategy-(II) row/column saturating S=3/4) is the correct place to probe, and it survives direct independent recomputation.","rationale":"The reader correctly identified the single load-bearing structural premise — the exact S=3/4 saturation of strategy (II) against all opposite strategies in Table III — and correctly noted it is established by direct table computation rather than derived from a deeper principle. My stress-test consisted of redoing that computation independently for six of the nine strategy combinations (including all three (II)-row/column entries) and auditing the two subsequent inferential steps (strategy-independence factorization in Eq. (16); the sufficient condition and μ-schedule algebra in App. E). Everything reconciles exactly. The same-basis re-measurement argument that justifies applying the static Appendix-D tables to the sequential pair-marginals is valid for projective (Lüders) measurements, and the cross-side independence needed for the product weighting in Eq. (16) follows from the separate-ancilla construction. The one caveat I can attach — exponential decay of violation magnitude with chain length — is a quantitative limitation the authors already acknowledge in spirit, not a correctness problem, and it does not touch the \"for arbitrarily many pairs\" claim, which is a for-any-finite-n statement. Given a fully checkable constructive proof whose most exposed assumption I could not make fail, the reader's ACCEPT/HIGH stands; I recommend no verdict change.","tokens_in":17046,"tokens_out":11564,"duration_ms":1000803,"concrete_test":"Independent end-to-end numerical replication at small depth: take the two-qubit family at θ=π/8 (α=β, γ from Eq. (12)), set n=m=4 and μ=6+3cot²(π/8), build the p_i schedule from Eq. (18), and compute the 4×4 matrix of pair-marginals p(a_j,b_k|x_j,y_k) exactly by enumerating all input histories and ancilla branches (no sampling), then evaluate all 16 CHSH values via Eq. (2). The proof predicts every entry strictly exceeds 3/4; any entry ≤3/4 would signal a hidden history-dependence or independence failure in Eq. (16) that the hand-check of Table III cannot see.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing premise of the whole construction is Table III: whenever either party uses strategy (II), S=3/4 exactly, for all α,β,γ,δ,ε. If any of those entries drifted above 3/4 the \"harmless default\" logic fails; if below, the compensation condition (17) would need strengthening. I independently recomputed the CHSH values from the Appendix D tables using Eq. (D1): (I,II) from Table V gives sum = (α+β+2γ+δ+ε)+(1−α−γ−δ)+(1−ε)+(1−β−γ) = 3; (II,III) from Table IX gives (α+γ+δ)+(1−δ)+1+(1−α−γ) = 3; (II,I) from Table VII gives 3; (III,I) gives 3−2α; (I,III) gives 3−2β; (III,III) gives 3−2(α+β+γ); (I,I) gives 3+2γ. All match Table III exactly. The two further structural steps also check out: (i) the factorization in Eq. (16) is legitimate because each side's ancilla evolution depends only on its own inputs and local randomness, so strategy indicators for Alice-j and Bob-k are independent; and the Appendix-D tables correctly describe sequential pair-marginals because re-measuring an already-measured system in the same projective basis deterministically reproduces the earlier outcome, so the pair statistics reduce to Table I entries plus nonsignalling marginals. (ii) The algebra in App. E2–E3 is sound: the sufficient condition (E6)–(E7) is a valid relaxation, the telescoping identity (E10) is correct, and the ratio bound (E13), q^no_i p_i/q^M1_i = (μ−2)/(1−(2/μ)^{i−1}) > μ−2, holds for μ>2, so μ > 2+3(α+β+γ)/γ suffices for all i. Eq. (19) for the two-qubit family follows from α/γ = cot²θ/2. The only genuine limitation I found is quantitative, not logical: for chain length n the margins S(j,k)−3/4 scale like (μ/2)^{i−1}μ^{−n}, i.e., they shrink exponentially with n, so \"unbounded sharing\" holds pair-by-pair for every finite n but with vanishing violation magnitude. The paper scopes this honestly (\"violation magnitudes are not optimized\") and never claims uniform-in-n violation, so it does not undermine the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The authors prove that any nonsignalling correlation exhibiting a Hardy paradox (Table I, γ>0) can be converted into a sequential protocol in which arbitrarily many Alice–Bob pairs simultaneously violate CHSH (S(j,k)>3/4), using only projective measurements and a small local ancilla (a classical history register suffices) on each side. The construction is fully explicit: three strategies (I)–(III) per observer, the ancilla-state recursion for q_no, q_M0, q_M1 (Eq. 14, App. E1), the CHSH value per strategy pair (Table III, App. D), a sufficient condition for violation (Eq. 17, App. E2), and an explicit choice of measurement probabilities p_i = μ^i/Σ_{l≥i} μ^l that satisfies it for any μ > 2+3(α+β+γ)/γ (App. E3). Since every pure entangled state with at least two distinct nonzero Schmidt coefficients admits Hardy correlations (App. B), the result covers almost all pure entangled states. The authors also show the ancilla can be reduced to a qubit plus per-side shared randomness (App. F), and that the argument extends to GPTs admitting a Hardy paradox.","tokens_in":17588,"tokens_out":4244,"duration_ms":159240,"significance":"If correct, this closes a well-identified gap in the nonlocality-sharing literature: two-sided sequential Bell sharing with projective measurements was previously supported only in minimal (2+2) scenarios and there was strong evidence [24,25] against it for a bare qubit pair. The paper identifies precisely what additional resource removes the obstruction — a one-trit (or one-bit plus shared randomness) classical memory per branch — and gives a constructive, fully written-out proof: the nine probability tables in Appendix D, the closed-form recursions in E1, and the explicit p_i family in E3 leave essentially nothing to black-box claims. The result is parameter-explicit (Eq. 19 gives the required μ for any two-qubit state) and applies beyond quantum theory to any nonsignalling Hardy correlation, which broadens its interest. I spot-checked the load-bearing entries of Table III against the Appendix D tables (e.g., (I,II), (II,III), (III,III)) and the sufficient-condition algebra in E2–E3, including the telescoping identity (E10) and the ratio bound (E13); all are correct. The exemption of maximally entangled states is sharp and consistent with γ→0.","major_comments":[{"comment":"Sec. IVd (Experimental outlook) together with Eqs. (16)–(18): the manuscript claims the protocol is 'well suited to current photonic platforms', but it nowhere quantifies the magnitude of the violations. From (E11), p_j(I)=q_no^j p_j = (μ/2)^{j-1}(1−μ)/(1−μ^n), so the surplus S(j,k)−3/4 in Eq. (16) carries an overall factor ~1/μ^n (plus combinatorial factors): for fixed μ the violations are exponentially small in the total chain length n, and near the maximally entangled state μ must diverge as γ→0 (Eq. 19, μ>5+3cot²θ). This does not affect the theoretical claim, but the experimental-outlook paragraph is not supported without it. Please add a short quantitative discussion: scaling of min_{j,k}(S(j,k)−3/4) with n at fixed θ, and a noise-tolerance estimate (e.g., white noise on the Hardy parameter γ) under which the construction still works for a given n.","section":"Sec. IVd / App. E3"},{"comment":"Sec. IIIb, Eqs. (15)–(16): the product form of S(j,k) in Eq. (16) requires that the strategy indicator of Alice-j be independent of that of Bob-k. This is true (each side's ancilla evolution depends only on its own inputs and local randomness), but the text justifies it only by 'the same construction is used for both Alice and Bob', which does not state the needed independence. Since this factorization is what converts Table III into per-pair CHSH values, please add one sentence making the cross-side independence of the local randomness and ancilla states explicit.","section":"Sec. IIIb, Eq. (16)"}],"minor_comments":[{"comment":"Title/abstract vs. Appendix B: the exception is any pure state whose nonzero Schmidt coefficients are all equal (App. B requires λ_i≠λ_j, both nonzero), which in dimension d>2 is broader than 'the maximally entangled one'. The phrasing 'all pure states except the maximally entangled one' is literally correct only for d=2; please qualify it.","section":"Abstract / App. B"},{"comment":"Sec. IVa: the shared random variable λ is used per side (λ_A for the Alice chain, λ_B for the Bob chain, as in App. C). Please state explicitly that no shared randomness between the Alice and Bob sides is required, since a reader could otherwise worry the protocol uses nonlocal classical resources.","section":"Sec. IVa / App. F"},{"comment":"Table II caption: the CHSH value is described as the 'sum of the highlighted values', but no highlighting is discernible in the tables as typeset; ensure the eight CHSH cells are visually marked in Tables II and IV–XII, or reference Eq. (D1) instead.","section":"Table II / App. D"},{"comment":"Eq. (12) gives only α=β and γ for the two-qubit family; please state the corresponding values of δ and ε (presumably zero) so that the five-parameter family is fully specified and the substitution leading to Eq. (19) is self-contained.","section":"Sec. IIB, Eq. (12)"},{"comment":"Acknowledgements: 'fincancial' → 'financial'. Reference [6]: the journal name 'Physical Research' appears to be a typo — please verify the correct venue for the Cai et al. review.","section":"Acknowledgements / Ref. [6]"},{"comment":"Notation: p_i denotes both the per-observer strategy-(I) rate (Eq. 13) and appears inside p(λ) (App. F); while consistent, a brief reminder at the start of App. F that the p_i are those of Eq. (14)/(18) would help the reader.","section":"App. F"}],"recommendation":"minor_revision","confidential_remarks":"The result is, to my knowledge, new and the paper is well matched to a high-visibility letter journal: it resolves a question left explicitly open by the Cheng–Liu–Baker–Hall no-go results and does so with an elementary, checkable construction. The one soft spot is the experimental-outlook paragraph, which currently overstates practicality given the exponentially small violations at large n — I have asked the authors to address this quantitatively rather than qualitatively. No concerns about priority or citation practice; the related literature (Silva et al., Brown–Colbeck, Cheng et al., Steffinlongo–Tavakoli) is cited appropriately, including the first author's own prior work, which is directly relevant rather than self-promotional."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know: they close the two-sided projective sharing gap that Cheng et al. left open for bare qubits. Any nonsignalling Hardy table (γ>0) plus a three-level (or qubit+shared-randomness) history flag yields simultaneous CHSH violations for every Alice-j/Bob-k pair in arbitrarily long chains, using only projective measurements. Because almost every pure entangled state admits Hardy correlations, the result is generic once you allow that flag.\n\nWhat is actually new is the Hardy↔sharing link and the three-strategy protocol. Strategy (II) = {M0, identity} always sits exactly at the local bound against every opposite strategy; that is the structural fact that lets later observers default harmlessly and only occasionally measure. The appendices spell out all nine strategy-pair tables, the recursion for the unmeasured/measured probabilities, the sufficient condition on the pi’s, and an explicit geometric schedule that works for any finite length. I recomputed the CHSH sums from the tables; they match Table III exactly, and the algebra in E.2–E.3 checks out. The ancilla can be classical memory; no coherence is used.\n\nSoft spots are real but scoped. Maximally entangled states are excluded (no Hardy). Violation margins shrink exponentially with chain length, so “unbounded” means every finite n, not uniform-in-n strength; they say so. The memory flag is essential—without it the Cheng-type obstructions remain—and they are honest about that. Magnitudes are not optimized. None of this breaks the central claim.\n\nThis is for people working on sequential nonlocality, device-independent protocols, or GPT nonlocality. The math is elementary and fully written; a serious referee can verify it in an afternoon. I would bring it to reading group, cite it when the topic comes up, and send it to peer review without hesitation.","headline":"Clean constructive proof that Hardy correlations give unbounded two-sided projective CHSH sharing for almost all pure entangled states once you allow a tiny local memory flag.","tokens_in":18564,"tokens_out":481,"would_cite":true,"duration_ms":15881,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.5","headline":"Any Hardy-paradox correlations, and thus almost every pure entangled state with a small local memory, let arbitrarily many Alice–Bob pairs violate CHSH at once using only projective measurements.","keywords":["Hardy's paradox","nonlocality sharing","CHSH inequality","sequential measurements","projective measurements","Bell nonlocality","entangled states","local ancilla"],"falsifier":"Prepare any pure two-qubit state that is not maximally entangled, run the stated projective protocol with the geometric probabilities for two Alices and two Bobs, and check whether all four CHSH estimators simultaneously exceed 3/4; a single pair that stays at or below 3/4 while the others are above would falsify the simultaneous-sharing claim.","tokens_in":18117,"feed_emoji":"🔗","tokens_out":1084,"duration_ms":26203,"temperature":0.7,"pith_summary":"This paper shows that nonlocality need not be used up by the first measurement. If two parties can produce correlations that exhibit Hardy’s paradox—three zero probabilities plus one strictly positive joint outcome—then those same correlations can be recycled so that every pair in two long chains of observers violates the CHSH inequality simultaneously. The protocol uses only ordinary projective measurements, a one-trit (or qubit-plus-shared-randomness) local memory that records which basis was already used, and a simple schedule of when each observer is allowed to measure versus pass. Because every pure entangled state except the maximally entangled one admits a Hardy paradox, the result is generic: almost all pure entanglement, once given a tiny classical flag, supports unbounded two-sided nonlocality sharing. A sympathetic reader cares because earlier work suggested two-sided projective sharing might be impossible for qubits; the paper shows that a minimal memory register is enough to remove that barrier.","feed_headline":"Almost all entangled states let endless observer pairs share nonlocality","feed_subtitle":"A tiny local memory and Hardy's paradox turn projective measurements into unbounded two-sided Bell sharing","key_machinery":"The Hardy-to-sharing map: three strategies (I)={M0,M1}, (II)={M0,identity}, (III)={identity,M1} together with a three-level ancilla that tracks whether the system is still unmeasured or has already been measured in basis 0 or 1. Strategy (II) always returns exactly the local CHSH value 3/4 against every opposite strategy, so observers can default to it and only rarely invoke (I); a geometric schedule of probabilities then guarantees every pair violates CHSH at once.","core_discovery":"Any nonsignalling correlations that satisfy the Hardy conditions (three vanishing joint probabilities and one positive probability γ>0) can be turned into a sequential protocol in which every Alice-j and Bob-k pair simultaneously obtains a CHSH value strictly above the local bound 3/4. The construction uses only the original two projective observables, three elementary strategies (measure both inputs, measure only input 0, or measure only input 1), and a local ancilla that prevents incompatible re-measurement. Because every pure non-maximally entangled state of any dimension admits such Hardy correlations, almost all pure entangled states enable unbounded two-sided projective nonlocality sha","pith_inferences":["The necessity of a memory flag suggests that earlier impossibility results for bare qubits are really statements about memoryless instruments, not about entanglement itself.","Because the schedule of measurement probabilities grows geometrically along the chain, the first observers must measure very rarely when many parties are present; practical demonstrations will therefore trade visibility against chain length.","Extending the same Hardy-table compensation idea to multipartite or network Bell inequalities is a direct next test the paper leaves open.","If a Hardy-like zero pattern can be found for other bipartite inequalities, the same default-to-safe-strategy logic may yield unbounded sharing for those inequalities as well."],"forward_implications":["Two-sided sequential Bell nonlocality with only projective measurements becomes a generic feature of pure entanglement rather than a special-case phenomenon.","A classical one-trit (or qubit-plus-shared-randomness) memory register is sufficient; no coherent ancilla dynamics are required.","The same construction applies outside quantum theory to any generalized probabilistic theory that admits a Hardy paradox, including PR-box correlations.","Photonic platforms that already demonstrate sequential steering can target the first experimental two-sided Bell-nonlocality sharing by adding only a local classical flag.","Maximally entangled states remain the sole pure-state exception for this particular Hardy-based route."],"fun_headline_variants":["Almost all pure entangled states enable unbounded nonlocality sharing","Hardy paradox turns non-maximal entanglement into endless CHSH sharing","Any Hardy correlations enable arbitrary sequential nonlocality pairs","Local ancilla lets almost all pure states share nonlocality unboundedly","Non-maximal pure entanglement generically allows unbounded Bell sharing"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The protocol stands only if measuring just the first observable and always outputting +1 on the second always lands exactly on the local CHSH bound, no matter what the other side does; if that saturation failed for generic Hardy tables, the compensation argument would not work.","fun_headline_variants_meta":{"raw":{"variants":["Almost all pure entangled states enable unbounded nonlocality sharing","Hardy paradox turns non-maximal entanglement into endless CHSH sharing","Any Hardy correlations enable arbitrary sequential nonlocality pairs","Local ancilla lets almost all pure states share nonlocality unboundedly","Non-maximal pure entanglement generically allows unbounded Bell sharing"]},"model":"grok-4.5","effort":"low","cost_usd":0.004236,"raw_usage":{"total_tokens":1215,"prompt_tokens":709,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":42364000,"prompt_tokens_details":{"text_tokens":709,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":441,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":709,"tokens_out":65,"duration_ms":8262,"temperature":1.0,"reasoning_tokens":441,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-31T07:28:23.757379+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Prepare any pure two-qubit state that is not maximally entangled, run the stated projective protocol with the geometric probabilities for two Alices and two Bobs, and check whether all four CHSH estimators simultaneously exceed 3/4; a single pair that stays at or below 3/4 while the others are above would falsify the simultaneous-sharing claim.","supporting_citations":[],"review_version":1}