bibkey: aloy2024threeoutcomepibi authors: A. Aloy; G. Müller-Rigat; J. Tura; M. Fadel year: 2024 title: “Deriving three-outcome permutationally invariant Bell inequalities” doi: 10.3390/e26100816 url: https://arxiv.org/abs/2406.11792v1 claim: “Section III.A proposes five Table III Bell inequalities for every N > 3 and asks for proofs of their validity for arbitrary party number.” strata_touched:

• D5/S3/Quantum/Entanglement/ThreeOutcomePermutationallyInvariantBell license: citation-only triage: anchor

Verified locator

DOI: https://doi.org/10.3390/e26100816

Source: https://arxiv.org/abs/2406.11792v1

Three-outcome permutationally invariant Bell inequalities

A. Aloy, G. Müller-Rigat, J. Tura and M. Fadel, Deriving three-outcome permutationally invariant Bell inequalities, Entropy 26 (2024), 816. The locators below refer to arXiv:2406.11792v1.

The scenario has N parties, two inputs per party and three outcomes. Equation (3), p. 2, defines the PI one-party observable as the sum of marginal probabilities and the PI two-party observable as the sum over ordered distinct parties. Equation (19), p. 5, gives the five symmetrized observables; Eq. (20) gives their Bell expression. Table II gives the deterministic outcome-pair count formulas.

Section III.A, p. 5, states verbatim:

To give a concrete example, we propose for any $N>3$ the five 3PIBIs shown in Tab. III.

At this point, for each conjectured inequality we have to prove that it is indeed valid for arbitrary number of parties $N$, or at least for all $N$ larger than a minimum number.

Table III, p. 6, lists the coefficient rows in the order $({\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_4,{\alpha }_5,{\beta }_c)$:

The Lean definitions retain the literal observable sums and coefficient rows. The hidden variable has a finite type L, with normalized nonnegative real weights w and local response probabilities p. Parties, inputs and outcomes use Fin N, Fin 2 and Fin 3. The two-party observable excludes the equal-party term. The finite model is the finite convex-hull formulation; the module does not prove an integral representation theorem for arbitrary measurable hidden variables. The claim asks for all five inequalities when N > 3. Their new proof is assessed as repository-derived with this note acknowledging the source of the proposal. Validity does not by itself establish facetness or a quantum violation.