bibkey: grandjean2012threesystems authors: B. Grandjean; Y.-C. Liang; J.-D. Bancal; N. Brunner; N. Gisin year: 2012 title: “Bell inequalities for three systems and arbitrarily many measurement outcomes” doi: 10.1103/PhysRevA.85.052113 url: https://arxiv.org/abs/1204.3829v2 claim: “The local-bound clause of the conjecture on (B1): the deterministic minimum is 6(K-1) for every K >= 2. The facet clause is separate.” strata_touched:
- D5/S3/QuantumBounds/SliwaSevenArbitraryOutcomeLocalBound license: citation-only triage: anchor
Bell inequalities for three systems and arbitrarily many measurement outcomes
Appendix B, PDF p. 5, states:
A possible symmetric generalization of Sliwa’s 7th inequality [46] to an arbitrary number of outputs reads as:
2⟨[A_1 + B_1 + C_1]_K⟩ + 2⟨[−A_1 − B_1 − C_1 − 1]_K⟩
- ⟨[−A_1 − B_1 − C_1]_K⟩ + 3⟨[−A_2 − B_2 − C_2 − 1]_K⟩
- ⟨[A_2 + B_2 + C_2 − 1]_K⟩ + ⟨[A_2 + B_2 + C_2]_K⟩
- ⟨[−A_2 + B_1 + C_1]_K⟩ + ⟨[−A_1 + B_2 + C_2]_K⟩ + ⋄ ≥ 6(K − 1), (B1)
The conjecture sentence continues on PDF p. 6:
We conjecture that both the local bound and the facet-defining property of inequality (B1) hold for general K.
Section II, PDF p. 2, states fact 1:
It suffices to consider deterministic classical strategies for determining the minimal value of S^{(K)} allowed in a local theory
In a deterministic strategy, the expectation of a bracket is its least nonnegative residue. Appendix A defines the permutation symbol through:
missing terms which must be added to ensure that the inequality is symmetrical with respect to arbitrary permutation of parties.
With a,b,c representing first-setting outputs and A,B,C second-setting
outputs, the six mixed terms are [-A+b+c]_K, [-B+a+c]_K,
[-C+a+b]_K, [-a+B+C]_K, [-b+A+C]_K and [-c+A+B]_K.
The Lean definition J preserves the six pure terms and these six mixed
terms using ZMod K and ZMod.val; claim asserts their sharp minimum
for every K >= 2. Only the local-bound clause is settled. The
facet-defining property is a separate open question.
Verified locator
- DOI: https://doi.org/10.1103/PhysRevA.85.052113
- URL: https://arxiv.org/abs/1204.3829v2
- PDF: https://arxiv.org/pdf/1204.3829v2, Section II p. 2 and Appendix B pp. 5–6.