Geometry of quantum correlations

We consider the set $\mathcal{Q}$ of quantum correlation vectors for two observers, each with two possible binary measurements. Quadric (hyperbolic) inequalities which are satisfied by every $q∊\mathcal{Q}$ are proved, and equality holds on a two-dimensional manifold consisting of the local boxes and all quantum correlation vectors that maximally violate the Clauser-Horne-Shimony-Holt (CHSH) inequality. The quadric inequalities are tightly related to the CHSH inequality; they are their iterated versions. Consequently, it is proved that $\mathcal{Q}$ is contained in a hyperbolic cube whose axes lie along the nonlocal (Popescu-Rohrlich) boxes. As an application, a tight constraint on the rate of local boxes that must be present in every quantum correlation is derived. The inequalities allow one to test the validity of quantum mechanics on the basis of data available from experiments which test the violation of the CHSH inequality. It is noted how these results can be generalized to the case of $n$ sites, each with two possible binary measurements.