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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.

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2026-07-31 07:28 UTC pith:WGP4DWSM

load-bearing objection 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. the 2 major comments →

arxiv 2607.24700 v1 pith:WGP4DWSM submitted 2026-07-27 quant-ph

Almost all pure entangled states enable unbounded nonlocality sharing

Anna Steffinlongo , Nicola D'Alessandro , Martin J. Renner This is my paper
classification quant-ph
keywords Hardy's paradoxnonlocality sharingCHSH inequalitysequential measurementsprojective measurementsBell nonlocalityentangled stateslocal ancilla
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

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.

Core claim

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

What carries the argument

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.

Load-bearing premise

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.

What would settle it

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.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • 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.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 6 minor

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.

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 (2)
  1. [Sec. IVd / App. E3] 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.
  2. [Sec. IIIb, Eq. (16)] 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.
minor comments (6)
  1. [Abstract / App. B] 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.
  2. [Sec. IVa / App. F] 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.
  3. [Table II / App. D] 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.
  4. [Sec. IIB, Eq. (12)] 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.
  5. [Acknowledgements / Ref. [6]] 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.
  6. [App. F] 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.

Circularity Check

0 steps flagged

No circularity: forward construction from external Hardy tables to simultaneous CHSH violations via explicit strategies and free design parameter μ.

full rationale

The paper’s derivation is a self-contained constructive protocol. It takes as input any nonsignalling Hardy table (Table I, γ>0)—a standard external notion (Hardy 1992/93, Goldstein, Chen et al.)—defines three local strategies and an ancilla update rule, computes the nine strategy-pair CHSH values by direct expansion of the tables (Appendix D → Table III), writes the pair-marginal S(j,k) as a bilinear combination of independent strategy probabilities (Eq. 16), and exhibits an explicit free choice p_i = μ^i/∑_{l≥i} μ^l (any μ > 2 + 3(α+β+γ)/γ) that forces S(j,k) > 3/4 for every pair (Appendix E). Nothing is fitted to data; μ is a design parameter, not a prediction. Hardy existence for non-maximally entangled pure states is proved by an independent Schmidt-ladder construction (Appendix B), not redefined to force the sharing claim. Self-citations (e.g. prior projective-sharing work) are background and not load-bearing for the central argument. The structural premise that strategy (II) always yields exactly S = 3/4 is verified by arithmetic on the Hardy table, not assumed by definition of the target. Hence the claimed result does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 5 axioms · 1 invented entities

The claim rests on standard quantum measurement theory, the known existence of Hardy correlations for non-maximally entangled pure states, the CHSH local bound, and nonsignalling marginals. The only paper-specific ingredients are the three observer strategies, the ancilla flag that records measurement history, and the free design parameter μ that sets how often later observers measure. No new physical entities or data-fitted constants are introduced.

free parameters (2)
  • μ (strategy-I rate schedule) = >2+3(α+β+γ)/γ (state-dependent lower bound)
    Design parameter controlling p_i = μ^i / ∑_{l≥i} μ^l. Must exceed 2+3(α+β+γ)/γ (or 5+3cot²θ for the two-qubit family). Chosen by hand to satisfy the sufficient violation condition; not fitted to external data.
  • p_i (per-observer measurement probability) = p_i = (1-μ)/(1-μ^{n-i+1}) for chosen μ
    Local randomness each observer uses when the ancilla is |no⟩ and the input is 1. Free protocol choice; the paper only needs some sequence satisfying inequality (17).
axioms (5)
  • domain assumption Quantum Born rule and projective measurements on pure bipartite states produce the outcome probabilities used in Hardy tables and CHSH evaluations.
    Standard QM; used throughout §§II–III and Apps. B–D.
  • domain assumption Every pure bipartite state that is not maximally entangled admits measurements yielding Hardy conditions with γ>0 (Hardy, Goldstein, Chen et al.).
    Cited classical results; App. B gives an explicit Schmidt-basis construction the paper relies on for the ‘almost all’ claim.
  • standard math CHSH local bound S≤3/4 for binary-input binary-output correlations admitting an LHV model.
    Standard Bell theory; Eq. (2).
  • domain assumption Nonsignalling constraints on bipartite correlations (Eqs. 7–8), used to complete probability tables when one party outputs a fixed value.
    Invoked to fill Tables II and IV–XII from Hardy zeros and fixed-output strategies.
  • ad hoc to paper A local classical or quantum flag (ancilla) that stores whether and in which basis the subsystem was already measured is an allowed resource in the sequential scenario.
    Scenario definition in §II.A and Discussion; without this memory the protocol does not run, and Cheng et al. evidence suggests bare two-sided projective sharing fails.
invented entities (1)
  • Three-strategy sequential protocol with history ancilla (|no⟩,|0⟩,|1⟩) no independent evidence
    purpose: Prevent incompatible re-measurement while letting each observer occasionally apply full Hardy measurements so every pair can violate CHSH.
    Core construction of §III; not a new particle or force, but a new operational gadget relative to prior sharing protocols.

pith-pipeline@v1.2.0-grok45-kimik3 · 21218 in / 3261 out tokens · 82001 ms · 2026-07-31T07:28:23.757379+00:00 · methodology

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read the original abstract

We establish a connection between Hardy's paradox and nonlocality sharing in sequential bipartite scenarios, where each subsystem is measured in turn by a chain of observers. We show that any correlations exhibiting a Hardy paradox in the two-input two-output scenario enable sequential violations of the CHSH inequality between arbitrarily many pairs of observers, using only projective measurements and the assistance of a small local ancilla system. Since almost all pure entangled states, with the only exception of the maximally entangled one, admit Hardy correlations, our protocol applies generically: almost all pure entangled states, if assisted by a local ancilla, allow for nonlocality sharing between arbitrarily many observer pairs using only projective measurements.

Figures

Figures reproduced from arXiv: 2607.24700 by Anna Steffinlongo, Martin J. Renner, Nicola D'Alessandro.

Figure 1
Figure 1. Figure 1: FIG. 1. Sequential scenario considered in this work. A source distributes a pure entangled state together with two quantum [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

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Reference graph

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    Explicit probabilities We report the explicit form of the probabilities defined in Eq. (14). We start with the probability that the state reaches partykwithout having been measured. The state is not measured if all previous parties received input 1 and chose not to measure. Thus qno k = 1 2 (1−p 1)· 1 2 (1−p 2)· · ·1 2 (1−p k−1) = k−1Y i=1 1 2 (1−p i) = 1...

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    Proof of the CHSH condition In the main text we established that S(j,k) = 3 4 + γ 2 pj(I)pk(I)− α 2 pj(III)pk(I)− β 2 pj(I)pk(III)− α+β+γ 2 pj(III)pk(III) = 3 4 + γ 2 qno j pj qno k pk − α 2 qM1 j qno k pk − β 2 qno j pjqM1 k − α+β+γ 2 qM1 j qM1 k . Here, we show thatS (j,k) >3/4 for all pairs (j, k) whenever qno i pi >3 α+β+γ γ qM1 i fori=j, k .(E4) Sett...

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    Choice of probabilities Consider the probabilities pi = µi Pn l=i µl = 1−µ 1−µ n−i+1 .(E8) We verify that this choice satisfies condition (17) for anyµ >2 + 3(α+β+γ) γ . Note that by the definition ofp i we obtain 1−p i = µ(1−µ n−i) 1−µ n−i+1 .(E9) Furthermore, the following identity is useful: i−1Y l=1 (1−p l) ! ·p i = µ(1−µ n−1) 1−µ n · µ(1−µ n−2) 1−µ n...