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Minimal Scenarios Witnessing Genuine Multipartite Nonlocality Possibilistically

Soumyadip Patra, Peter Bierhorst

arXiv:2610.11377v1quant-phcs.ITmath.IT

TL;DR

The paper asks whether perfect quantum behavior can rule out models built from smaller multipartite resources. It applies inflation to the five-party 5-cycle game and establishes a robust separation from pairwise-resource models, then constructs near-perfect tripartite games with a similar separation.

  • Problem

    The paper asks whether quantum behaviors can provide strong genuine multipartite nonlocality against models using only smaller multipartite resources.

    Source: p. 1

  • Method

    The authors apply inflation to the five-party 5-cycle game and construct a family of tripartite games with near-perfect quantum strategies.

    Source: pp. 2, 4

  • Results

    The pairwise-resource model’s average winning probability in the 5-cycle game is at most 121/126.

    Source: p. 3

  • Takeaways & Limitations

    For the tripartite family, game and trial choices can make honest all-win outcomes experimentally indistinguishable from true probability-one winning to a desired precision.

    Source: pp. 2, 5

    The pairwise-resource model considered permits independent bipartite no-signaling resources conditional on shared randomness, local wirings, and no communication after inputs arrive.

    Source: §3 · p. 9

Abstract

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We study a strong form of genuine multipartite nonlocality (GMNL), where an $n$-partite scenario admits a quantum strategy generating outcomes belonging to a restricted set with certainty, whereas every strategy built from nonclassical resources shared among at most $k<n$ parties fails this criterion with positive probability. This generalizes what is referred to as strong contextuality/nonlocality, or pseudotelepathy, beyond ruling out local models. For the simplest five-party Bell scenario of two binary-outcome measurement settings per party, we use the inflation technique to demonstrate that a game with a perfect quantum strategy witnesses a robust separation from models built from bipartite resources under local operations and shared randomness (LOSR), establishing the behavior is LOSR-GMNL. We provide broader context for this result by investigating its simulability with resources shared among three or four parties under different resource models and alternative definitions of GMNL. A central open question is whether $n$-party quantum resources can achieve such a separation from models built from $(n-1)$-party resources, for $n\ge 3$. We make progress here by constructing a family of robustly LOSR-GMNL games with $\epsilon$-optimal quantum strategies experimentally indistinguishable in the limit from true probability-one-winning strategies for any desired level of precision.

2. Proof of Proposition 1

The 5-cycle game’s deterministic-output obstruction bounds pairwise predicate winning, while a five-qubit strategy satisfies the game’s predicates with certainty.

1. A Perfect Quantum Strategy

  • A five-qubit state measured locally with Pauli X and Z observables gives a quantum strategy satisfying all predicates with certainty.The state’s specified eigenvalue relations ensure the cyclic three-player parities, while the all-Z predicate is checked separately.Source: §1 · p. 7

2. Proof of Proposition 1

  • 5/6 is the maximum winning probability when two players respond deterministically on input X.The proof establishes this by showing the summed predicate success is at most five across six predicates.Source: §2 · pp. 8–9
  • The proof combines parity constraints through a binary-variable lemma and bounds on the probabilities of predicate failures.It treats adjacent and non-adjacent pairs of players separately before concluding the failure sum is at least one.Source: §2 · p. 8

3. Proof of Proposition 2

The inflation proof gives a quantitative bound for pairwise-resource strategies in the 5-cycle game. It uses independence and marginal constraints to turn near-perfect parities into a deterministic-output contradiction.

3. Proof of Proposition 2

  • The pairwise-resource model allows independent bipartite no-signaling resources conditional on shared randomness, with local wirings and no communication after inputs arrive.The model also permits quantum entangled measurements, subject to the stated compatibility assumptions.Source: §3 · p. 9
  • Inflation’s marginal-isomorphism and ancestral-independence constraints make near-perfect parities imply outputs close to deterministic signs.The resulting deterministic parity contradiction forces a nonzero total failure probability.Source: §3 · p. 9
  • 121/126 is the upper bound on average winning probability for every behavior in the pairwise-resource network model.The bound follows by combining a lower bound on total failure probability with uniform sampling of the six questions.Source: §3 · pp. 9, 11

Appendix B: A Four-Player No-signaling Simulation

The section locates four-party no-signaling resources in the 5-cycle game’s resource hierarchy and specifies the game’s six questions and winning parities.

Appendix B: A Four-Player No-signaling Simulation

  • Freezing one player’s response tests whether a four-way no-signaling resource can supply the remaining correlations.This addresses whether five-player irreducibility is necessary after ruling out bipartite resources.Source: Appendix B · p. 11
  • Each cyclic question activates one player on Z and two cyclic neighbors on X, requiring odd output parity; the other two outputs are inactive.The caption identifies the active outputs as the set I(Qi).Source: Table II · p. 12
  • The all-Z question activates all five players and requires even output parity.Source: Table II · p. 12

1. One Deterministic Player

Fixing one player’s two outputs reduces perfect play to a feasibility test for the other four parties. The tested four-party behaviors exist for every fixed-output choice.

1. One Deterministic Player

  • The reduction prescribes A1’s outputs and checks whether the other four parties have a distribution satisfying the transformed predicates and no-signaling conditions.The existence test is a linear feasibility program over a (4, 2, 2) behavior.Source: §1 · p. 12
  • For every r and s, a four-party no-signaling behavior lets the 5-cycle game be won with certainty when A1 outputs r on X and 1 ⊕ s on Z.Source: §1 · p. 12
  • For r = 1, local Z-conditioned output flips preserve the three-player parities and cancel in the four-player Z-parity.Substituting A1’s prescribed strategy then verifies all six predicates.Source: §1 · p. 12

2. Reproducing the Five-Player Quantum behavior

The five-player quantum behavior can be reproduced by mixing four four-way no-signaling boxes with two shared random bits. The boxes also yield strongly nonlocal four-player distributions that win the reduced game with certainty.

2. Reproducing the Five-Player Quantum behavior

  • Perfect play constrains only the six predicates, but a convex mixture of the same boxes reproduces every input-conditional probability of the five-player quantum behavior.The table gives the two base boxes, Qs for s=0 and s=1, generated by linear programming and checked for predicates, no-signaling, and normalization.Source: §2 · p. 12, Table III · p. 13
  • Two shared random bits and four four-way no-signaling boxes simulate the five-player quantum-permissible behavior that wins the game with certainty.The construction averages four branches; player A1 is deterministic only after conditioning on the shared bits.Source: §2 · p. 13
  • Qs is strongly nonlocal and wins the reduced 4-party nonlocal game, defined by the predicates in (2), with certainty; the passage attributes this equivalence to Ref..Qs is thus a face no-signaling correlation of the (4, 2, 2) no-signaling polytope.Source: §2 · p. 14

1. Membership via Outer-Approximation

The section tests whether a reduced four-player behavior belongs to the quantum set by maximizing a Bell functional over an outer approximation. The second-level NPA bound excludes a perfect quantum win but does not settle whether quantum mechanics beats the local bound.

1. Membership via Outer-Approximation

  • The test maximizes a Bell functional algebraically saturated by Q0 over an outer approximation of quantum-achievable distributions.The four-player functional is formed from a five-player Bell functional by making one player deterministic on both inputs.Source: §1 · p. 14
  • The second-level NPA moment matrix uses projector words of length at most k, enforcing idempotence and commutation between different parties’ operators.Operators belonging to the same party are not assumed to commute.Source: §1 · p. 15
  • 5.03 is the second-level NPA upper bound, above the local bound of 4 but below the algebraic maximum of 6 required for strong contextuality.Because NPA bounds an outer approximation, this does not itself establish a quantum violation of the local bound.Source: §1 · p. 15

2. Membership via Numerical Procedure

A direct numerical search over parameterized four-qubit states and measurements finds an explicit quantum strategy that beats the reduced game's local bound. The search is not guaranteed to find a global optimum, but a score above the local bound certifies a violation.

2. Membership via Numerical Procedure

  • Jordan’s lemma restricts the shared state to (C2)^⊗4 without loss of generality, and local basis choices parameterize each player’s observables using σX and σZ.The optimization varies measurement angles while bounding the Bell-functional value by the largest eigenvalue of its Bell operator.Source: §2 · pp. 15–16
  • Although direct numerical optimization is not guaranteed to reach a global optimum, it yielded quantum value 4.82842712474, above the local bound of four and demonstrating an explicit strategy beating it.Source: §2 · p. 16
  • The optimized measurement angles are close to π/2, from which the authors infer an exact strategy with each player’s second observable equal to σX.The reported angle vector has all four entries close to π/2.Source: §2 · p. 16

Appendix D: Impossibility of Svetlichny-type 2|3 No-Signaling Models for the 5-Cycle Game

For the 5-cycle game, Svetlichny-type models using arbitrary bipartite and tripartite no-signaling resources across 2|3 partitions do no better than local deterministic strategies. The appendix also derives chained-Bell monogamy relations by iteratively replacing probability terms.

Appendix D: Impossibility of Svetlichny-type 2|3 No-Signaling Models for the 5-Cycle Game

  • Svetlichny-type 2|3 models with arbitrary bipartite and tripartite no-signaling resources provide no advantage over local deterministic strategies in the 5-cycle game.The argument extends the bound across convex mixtures and shared randomness.Source: Appendix D · p. 17
  • The resulting win probability is at most 5/6 for these resource models.The proof treats adjacent and non-adjacent party pairs, with cyclic symmetry covering the distinct cases.Source: Appendix D · p. 17

1. Chained-Bell Quantities and Monogamy Relationships

  • Repeated applications of chained-Bell probability inequalities replace alternating mismatch terms and generate the terms of the chained quantity.The iteration proceeds through the first 2N−1 terms before the final term is handled to obtain the monogamy relation.Source: §1 · p. 18

2. A Chained-Bell Inspired Family of Arbitrarily Near-Pseudotelepathic Games

The GHZ-chained-Bell game combines an alignment task with chained-Bell tests, exploiting the trade-off faced by strategies using at-most-bipartite nonlocal resources. An inflation argument bounds such strategies, while the quantum winning probability approaches one as the game size grows.

2. A Chained-Bell Inspired Family of Arbitrarily Near-Pseudotelepathic Games

  • Alice’s A0 setting serves as both an alignment and chained-Bell measurement, while Bob receives a separate alignment setting alongside his chained-Bell settings.Charlie’s CZ measurement projects Alice and Bob onto computational-basis states; CX steers them to one of two Bell states.Source: §2 · p. 18
  • At-most-bipartite strategies cannot perform well simultaneously on the alignment and chained-Bell tasks, motivating a modified inflation argument.The modification accommodates the fact that the same chained-Bell settings do not serve both Bell states while also including Alice’s computational-basis setting.Source: §2 · p. 19
  • The inflation proof uses cut variables and isomorphic marginal distributions to relate the sameness term to pairwise agreement probabilities, yielding an upper bound of 4N + 1.The argument applies the bound (E2) after using no-signaling to relate the relevant probabilities.Source: §2 · p. 20
  • (3N + 2 + N cos(π/2N))/(4N + 2) is the game’s quantum winning probability, approaching one and the LOSR-GMNL bound (4N + 1)/(4N + 2) as N grows.The construction therefore gives games arbitrarily close to pseudotelepathy while retaining a gap from at-most-bipartite-resource networks.Source: §2 · p. 20

3. Calculation of quantum value for Chained Bell Quantity CB and CB′

The calculation uses Pauli measurements in the X/Z plane to evaluate the chained-Bell quantities for the two Bell states. These values are then combined to obtain the GHZ-chained-Bell game’s quantum value.

3. Calculation of quantum value for Chained Bell Quantity CB and CB′

  • For Bell states measured with Pauli observables in the X/Z plane, the correlator is cos(α ∓ β), and measurement angles determine the equality and inequality probabilities.The selected measurement bases yield the chained-Bell contributions for |Φ+⟩ and |Φ−⟩; for |Φ−⟩, the relevant inequality probabilities equal (1 + cos(π/2N))/2.Source: §3 · p. 21
  • The GHZ-state value combines four alignment terms with chained-Bell probabilities that remain equal to (1 + cos(π/2N))/2 when outcomes are not conditioned on CX.The calculation uses the law of total probability for the remaining chained-Bell terms.Source: §3 · p. 21
  • 3N + 2 + N cos(π/2N) is the overall quantum value of the game’s expression (E8), obtained by summing its terms.The resulting winning probability is given by expression (E9).Source: §3 · p. 21

4. Limiting Behavior of the 3-party GHZ-Chained-Bell Game as N, t →∞

The section turns the single-trial quantum and foil-theory bounds into a finite-trial test. By choosing the game index and trial count appropriately, the quantum strategy can yield all wins with high probability while the bipartite-only foil receives a small p-value.

4. Limiting Behavior of the 3-party GHZ-Chained-Bell Game as N, t →∞

  • The single-trial quantum winning probability is raised to the t-th power to calculate the probability of observing t successes.The resulting all-success event is the protocol’s pseudotelepathic behavior.Source: §4 · pp. 21–22
  • For any desired completeness and soundness parameters, choices of N and t make t wins likely under the noiseless quantum prediction and yield a p-value below the soundness parameter against a bipartite-only network.The probability of t quantum successes tends to one when t grows more slowly than N², while the foil’s p-value can be made small when t grows more quickly than N.Source: §4 · p. 22

5. Generalization to an n|(n −1) Separation

The construction extends the GHZ-chained-Bell approach to n parties by distributing Charlie’s role across n − 2 parties. The appendix also supplies lemmas used in inflation arguments, including exact and approximate agreement results for independent binary variables.

5. Generalization to an n|(n −1) Separation

  • Replacing Charlie with n − 2 parties measuring Z and X on an n-party GHZ state gives behaviors with ε-close-to-perfect winning probabilities that cannot be replicated by underlying (n − 1)-partite resources under LOSR-GMNL.The proof uses an n-party cut inflation and iterated applications of the pairwise bound to connect agreement across parties.Source: §5 · p. 22
  • When all Charlie-like parties measure X, their outcomes’ parity determines whether Alice and Bob are steered to |Φ+⟩ or |Φ−⟩.The generalized proof conditions on this collective parity and extends the sameness terms across the Charlie parties.Source: §5 · p. 22

Appendix F: Useful Lemmas

  • If independent binary variables have fixed parity with certainty, then for each shared-randomness value they are deterministic and their values have that parity.Averaging over shared randomness can nevertheless leave their marginal distributions nondeterministic.Source: Appendix F · p. 23
  • If independent, identically distributed binary variables disagree with probability at most η, one of the two deterministic values differs from either variable with probability at most η.This exact-agreement lemma has no loss in the error parameter and converts an inflated disagreement probability into distance from a deterministic response.Source: Appendix F · p. 23
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