Optimal tests of quantum nonlocality

We present a general method for obtaining all Bell inequalities for a given experimental setup. Although the algorithm runs slowly, we apply it to two cases. First, the Greenberger-Horne-Zeilinger setup with three observers each performing one of two possible measurements. Second, the case of two observers each performing one of three possible experiments. In both cases we obtain hundreds of inequalities. Since this is the set of all inequalities, the one that is maximally violated in a given quantum state must be among them. We demonstrate this fact with a few examples. We also note the deep connection between the inequalities and classical logic, and their violation with quantum logic.