Extremal quantum correlations for N parties with two dichotomic observables per site

Consider a scenario where $N$ separated quantum systems are measured, each with one among two possible dichotomic observables. Assume that the $N$ events corresponding to the choice and performance of the measurement in each site are space-like separated. In the present paper, the correlations among the measurement outcomes that arise in this scenario are analyzed. It is shown that all extreme points of this convex set are attainable by measuring $N$-qubit pure-states with projective observables. This result allows the possibility of using known algorithms in order decide whether some correlations are achievable within quantum mechanics or not. It is also proven that if an $N$-partite state $\rho$ violates a given Bell inequality, then, $\rho$ can be transformed by stochastic local operations into an $N$-qubit state that violates the same Bell inequality by an equal or larger amount.