Characterization of quantum correlations with local dimension constraints and its device-independent applications

We characterize quantum nonlocality under local dimension constraints via a complete hierarchy of semidefinite programming relaxations. We apply the new method to a number of bipartite Bell inequalities, and find that the first level of the hierarchy returns non-trivial bounds in all cases considered. The method allows to attack nonlocality scenarios with four measurement settings on one side and twelve (12) on the other with a normal desktop. In the tripartite setting, we use the hierarchy to derive a Bell-type inequality that can only be violated when each of the three parties has local dimension greater than two. Finally, we apply the new method to certify entangling measurements in two-qubit scenarios.