bibkey: bernards2023nonlocalitydepth authors: F. Bernards; O. Gühne year: 2023 title: “Bell inequalities for nonlocality depth” doi: 10.1103/PhysRevA.107.022412 url: https://arxiv.org/abs/2205.04250v2 claim: “For (k,m) models with m > 2, the classical bound F_n <= n 2^(n-2) is conjectured and proving it or finding a counterexample remains open.” strata_touched:

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

Bell inequalities for nonlocality depth

F. Bernards and O. Gühne, Bell inequalities for nonlocality depth, Phys. Rev. A 107, 022412 (2023), arXiv:2205.04250v2, DOI 10.1103/PhysRevA.107.022412.

Section III.F, Eq. (28), PDF p. 5, displays

$F_n={\sum }_{\ell =1}^n(-1{)}^{1+\lceil \ell /2\rceil }\ell (1\dots 12\dots 2)\le n{2}^{n-2}$, (28)

and states that the bound can be achieved in any $(k,m)$ model with $k,m>1$. Section V, PDF p. 8, states verbatim:

For $(k,m)$ models with $m>2$, we have a conjecture for the classical bound. Proving this bound or finding a counterexample remains an open problem.

Section II, PDF p. 2, states verbatim:

Every cell c_i of the partition is considered as one system. The measurement settings are all combinations of measurement settings that apply to each subsystem within a cell. However, no restrictions apply to the correlations between subsystems within a cell, since the cell is regarded as one system. In particular, this allows for signaling to take place between the parties within one cell.

Section II, PDF p. 3, states:

where ’party permutations’ only includes permutations that yield different terms.

The hybrid model treats each cell as one system and permits signaling inside a cell; its convex-hull description means a linear functional reaches its maximum at a deterministic point. The correlator brackets sum over distinct party permutations, and the functional is symmetric under party permutations, so the cell assignment does not change the value. The Lean encoding uses Boolean setting strings, integer unit responses, and the displayed weight and sign $(-1{)}^{1+\lceil h/2\rceil }$. The sign is evaluated by natural-number division as $(-1{)}^{1+\lfloor (h+1)/2\rfloor }$.

Verified locator

• DOI: https://doi.org/10.1103/PhysRevA.107.022412
• URL: https://arxiv.org/abs/2205.04250v2