On the minimum dimension of a Hilbert space needed to generate a quantum correlation

Consider a two-party correlation that can be generated by performing local measurements on a bipartite quantum system. A question of fundamental importance is to understand how many resources, which we quantify by the dimension of the underlying quantum system, are needed to reproduce this correlation. In this paper, we identify an easy-to-compute lower bound on the smallest Hilbert space dimension needed to generate a given two-party quantum correlation. We show that our bound is tight on the correlations generated by optimal quantum strategies for the CHSH Game and the Magic Square Game. We also identify sufficient conditions for showing that a correlation cannot be generated using finite-dimensional quantum strategies. Using this we give alternative proofs that a family of PR-boxes and all correlations corresponding to perfect strategies for the Fortnow-Feige-Lov\'asz Game cannot be realized using finitely many qubits.