bibkey: masanes2003tightbell authors: Ll. Masanes year: 2003 title: “Tight Bell inequality for d-outcome measurements correlations” doi: 10.26421/QIC3.4-4 url: https://arxiv.org/abs/quant-ph/0210073v1 claim: “CGLMP is a facet of the two-party local polytope with two d-outcome measurements per party for every d >= 2.” strata_touched:

• D5/S3/QuantumBounds/CglmpFacetRigidity license: citation-only triage: anchor

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DOI: 10.26421/QIC3.4-4

Source: https://arxiv.org/abs/quant-ph/0210073v1

Journal: Quantum Information & Computation 3(4), 345–358. DOI URL: https://doi.org/10.26421/QIC3.4-4

CGLMP tightness

The abstract, PDF p. 1, states:

In this paper we prove that the inequality introduced by Collins, Gisin, Linden, Massar and Popescu [11] is tight, or in other words, it is a facet of the convex polytope generated by all local-realistic joint probabilities of d outcomes.

Section III proves the facet property for all numbers of outcomes. The Lean theorem CglmpFacetRigidity.bipartite_rigidity is its dual-functional formulation: every bipartite local function vanishing on the saturating strategies is a scalar multiple of the CGLMP slack. With a=A_1, A=A_2, b=B_1 and B=B_2, the bracket expression is

$$J_2=[A-b{]}_K+[a+B{]}_K+[-a+b{]}_K+[-A-B-1{]}_K.$$

The sign of B differs by an outcome relabelling from the usual CGLMP convention. In the theorem K is nonzero; its facet interpretation uses K ≥ 2. The conclusion is literature-attested. The proof here uses a short-arc potential and a single wrap coefficient, rather than the source’s explicit linearly independent saturating vectors.