Separation of quantum, spatial quantum, and approximate quantum correlations

Quantum nonlocal correlations are generated by implementation of local quantum measurements on spatially separated quantum subsystems. Depending on the underlying mathematical model, various notions of sets of quantum correlations can be defined. In this paper we prove separations of such sets of quantum correlations. In particular, we show that the set of bipartite quantum correlations with four binary measurements per party becomes strictly smaller once we restrict the local Hilbert spaces to be finite dimensional, i.e., $\mathcal{C}_{q}^{(4, 4, 2,2)} \neq \mathcal{C}_{qs}^{(4, 4, 2,2)}$. We also prove non-closure of the set of bipartite quantum correlations with four ternary measurements per party, i.e., $\mathcal{C}_{qs}^{(4, 4, 3,3)} \neq \mathcal{C}_{qa}^{(4, 4, 3,3)}$.