A complete set of multidimensional Bell inequalities

TL;DR

This paper generalizes Bell inequalities to multidimensional systems with multiple parties and observables, providing a complete set of inequalities that define a polytope and are violated by quantum mechanics.

Contribution

It introduces a comprehensive set of multidimensional Bell inequalities for n parties and d-valued observables, extending previous two-dimensional results.

Findings

Inequalities define facets of a complex polytope.

Quantum mechanics violates these inequalities.

Examples provided for the three-dimensional case.

Abstract

We give a multidimensional generalisation of the complete set of Bell-correlation inequalities given by Werner and Wolf, and by Zukowski and Brukner, for the two-dimensional case. Our construction applies for the n parties, two-observables case, where each observable is d-valued. The d^{d^n} inequalities obtained involve homogeneous polynomials. They define the facets of a polytope in a complex vector space of dimension d^n. We also show that these inequalities are violated by Quantum Mechanics. We exhibit examples in the three-dimensional case.