 Chained modular Bell inequality

ID: chained-modular-bell-inequality

For outcomes in

{0, \dots, d - 1}

, let

[x]_{d}

be their nonnegative modular representative. A deterministic local hidden-variable model satisfies

I_{N, d} = \sum_{j = 1}^{N} E ([A_{j} - B_{j}]_{d}) + \sum_{j = 1}^{N - 1} E ([B_{j} - A_{j + 1}]_{d}) + E ([B_{N} - A_{1} - 1]_{d}) \geq d - 1.

The unreduced chain telescopes to

- 1

, so its reduced nonnegative sum is at least

d - 1

for each assignment. For binary outcomes and two settings, it is a form of the CHSH inequality:

I_{2, 2} = 2 - (E_{11} + E_{21} + E_{22} - E_{12}) /2

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