More non-locality with less entanglement

Abstract: We provide an explicit example of a Bell inequality with 3 settings and 2 outcomes per site for which the largest violation is not obtained by the maximally entangled state, even if its dimension is allowed to be arbitrarily large. This complements recent results by Junge and Palazuelos (arXiv:1007.3042) who show, employing tools from operator space theory, that such inequalities do exist. Our elementary example provides arguably the simplest setting in which it can be demonstrated that even an infinite supply of EPR pairs is not the strongest possible nonlocal resource.