bibkey: grandjean2012mermincglmpfacet 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: 10.48550/arXiv.1204.3829 url: https://arxiv.org/abs/1204.3829v2 claim: “The inequality (1), Mermin-CGLMP, defines a facet of the full local polytope for every K >= 2.” strata_touched:

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

Verified locator

DOI: 10.48550/arXiv.1204.3829

Source: https://arxiv.org/abs/1204.3829v2

Archive DOI URL: https://doi.org/10.48550/arXiv.1204.3829

Journal: Phys. Rev. A 85, 052113. Journal DOI URL: https://doi.org/10.1103/PhysRevA.85.052113

The Mermin-CGLMP facet conjecture

Section II, PDF p. 1, states:

We consider a scenario involving three spatially separated parties (henceforth referred as Alice, Bob and Charlie), and with each of them performing 2 alternative K-outcome measurements.

The full joint probabilities have two inputs and K outputs per party. Input 0 in Lean is source setting 1; input 1 is source setting 2. A deterministic behaviour assigns both outputs of each party in advance, and the local polytope is the convex hull of all these behaviours.

Equation (1), PDF p. 1, is

$$\begin{array}{rl}{S}^{(K)}=& ⟨[A_2-B_1+C_1{]}_K⟩+⟨[A_1+B_2-C_1{]}_K⟩\\ & +⟨[-A_1+B_1+C_2{]}_K⟩+⟨[-A_2-B_2-C_2-1{]}_K⟩\ge K-1.\end{array}$$

The same page states:

where [X]_K stands for X modulo K

Equation (2) defines the bracket expectation as

$$⟨[X{]}_K⟩=\sum _{j=0}^{K-1}jP(X=j\mathrm{mod}K).$$

Section II, PDF p. 2, states:

It suffices to consider deterministic classical strategies for determining the minimal value of S^(K) allowed in a local theory

The conjecture sentence, Section II, PDF p. 2, is:

We conjecture that inequality (1) is indeed facet-defining for all K ≥ 2.

The paper verifies the facet property through K=8. The Lean statement MerminCglmpFacet.claim includes validity and says that the affine dimension of its saturating face plus one equals the affine dimension of the full local polytope, for every K ≥ 2. Its result proves that statement.

The decisive mechanism is a family of relabelled CGLMP slices indexed by Charlie’s two outputs. Bipartite rigidity gives a multiplier on each slice; the rectangular identity forces the multiplier to be constant. The general FacetRigidityBridge.rigidity_facet_bridge turns this rigidity into the affine-dimension equation. The proof uses a short-arc derivation of Masanes’s bipartite facet result, cited separately as masanes2003tightbell.

The facet clause of Appendix B inequality (B1), the fully symmetric generalisation, and extension to more parties are separate questions. The local-bound clause of (B1) belongs to grandjean2012threesystems; this note concerns inequality (1).