Bell inequalities in four dimensional phase space and the three marginal theorem

We address the classical and quantum marginal problems, namely the question of simultaneous realizability through a common probability density in phase space of a given set of compatible probability distributions. We consider only distributions authorized by quantum mechanics, i.e., those corresponding to complete commuting sets of observables. For four-dimensional phase space with position variables q-vector and momentum variables p-vector, we establish the two following points: (i) given four compatible probabilities for (q1,q2), (q1,p2), (p1,q2), and (p1,p2), there does not always exist a positive phase space density ρ(q-vector,p-vector) reproducing them as marginals; this settles a long standing conjecture; it is achieved by first deriving Bell-type inequalities in phase space which have their own theoretical and experimental interest. (ii) Given instead at most three compatible probabilities, there always exist an associated phase space density ρ(q-vector,p-vector); the solution is not unique and its general form is worked out. These two points constitute our 'three marginal theorem'