Almost quantum correlations

There have been a number of attempts to derive the set of quantum non-local correlations from reasonable physical principles. Here we introduce $tilde{Q}$, a set of multipartite supra-quantum correlations that has appeared under different names in fields as diverse as graph theory, quantum gravity and quantum information science. We argue that $tilde{Q}$ may correspond to the set of correlations of a reasonable physical theory, in which case the research program to reconstruct quantum theory from device-independent principles is met with strong obstacles. In support of this conjecture, we prove that $tilde{Q}$ is closed under classical operations and satisfies the physical principles of Non-Trivial Communication Complexity, No Advantage for Nonlocal Computation, Macroscopic Locality and Local Orthogonality. We also review numerical evidence that almost quantum correlations satisfy Information Causality.

Maximally nonlocal theories cannot be maximally random

Correlations that violate a Bell Inequality are said to be nonlocal, i.e. they do not admit a local and deterministic explanation. Great effort has been devoted to study how the amount of nonlocality (as measured by a Bell inequality violation) serves to quantify the amount of randomness present in observed correlations. In this work we reverse this research program and ask what do the randomness certification capabilities of a theory tell us about the nonlocality of that theory. We find that, contrary to initial intuition, maximally nonlocal theories cannot allow maximal randomness certification. We go on and show that quantum theory, in contrast, permits certification of maximal randomness in all dichotomic scenarios. We hence pose the question of whether quantum theory is optimal for randomness, i.e. is it the most nonlocal theory that allows maximal randomness certification? We answer this question in the negative by identifying a larger-than-quantum set of correlations capable of this feat. Not only are these results relevant to understanding quantum mechanics fundamental features, but also put fundamental restrictions on device-independent protocols based on the no-signaling principle.