Bounding the Set of Quantum Correlations

We introduce a hierarchy of conditions necessarily satisfied by any distribution P_αβ representing the probabilities for two separate observers to obtain outcomes α and β when making local measurements on a shared quantum state. Each condition in this hierarchy is formulated as a semidefinite program. Among other applications, our approach can be used to obtain upper bounds on the quantum violation of an arbitrary Bell inequality. It yields, for instance, tight bounds for the violations of the Collins et al. inequalities.