A complete set of multidimensional Bell inequalities

International audienceWe give a multidimensional generalization of the complete set of Bell-correlation inequalities given by Werner and Wolf (2001 Phys. Rev. A 64 032112) and by Zˆukowski and Brukner (2002 Phys. Rev. Lett. 88 210401), for the two-dimensional case. Our construction applies to the n-party, two-observable case, where each observable is d-valued. The inequalities obtained involve homogeneous polynomials. They define the facets of a polytope in a complex vector space of dimension dn. We detail the inequalities obtained in the case d = 3 and, from them, we recover known inequalities. We finally explain how the violations of our inequalities by quantum mechanics can be computed and could be observed, when using unitary observables