Marco T. Quintino talked about one of the most annoying open problems in quantum information theory. Please, don’t get me wrong, the talk was really great, but I do find quite embarrassing that we still don’t know how to relate entanglement and non-locality.
The whole thing goes all the way back to the founding fathers of quantum mechanics like Schrödinger and Einstein-Podolsky-Rosen. They realized that whenever you describe the state of two (or more) systems as vectors living in the tensor product of the individual vector spaces, you are bound to have states that cannot be written as the product of local states. That is the underling math behind entangled states. It took almost 30 years for entanglement to pass from a puzzling mathematical feature to a property with physical consequences, as demonstrated by John Bell. Bell created inequalities, called, guess what, Bell inequalities, that are violated by some entangled states.
The catch is the “some”, in the previous sentence. Despite the fact that all pure entangled states violate a Bell-like inequality, and no separable states (pure or mixed) do, in 1989 Reinhard Werner showed that there exist entangled states that don’t violate any Bell inequality. The result by Werner is for the case where Alice and Bob are only allowed to make projective measurements, but this was fixed by Barret that showed that the problem remains even when the parties are allowed to make the most general kind of quantum measurements, known as POVMs.
What Marco and coauthors showed, following the steps of Popescu, was that if one allows Alice and Bob to perform general local filtering operations and then measure their pre-selected state with local POVMs, some entangled states that would return local probability distributions now show some non-locality. As such, these states now violate some Bell inequality, and one says that their non-locality was just hidden.
I think that a very interesting tool that Marco&Co developed was a procedure to transform a state that leads to local correlations under projective measurements into a state that is non-local for POVM measurements after suitable filtering operations. I’m not sure how to exploit it even further, but it does look powerful.
Well, but the question still remains: is there an alternative scenario for Bell inequalities in which every entangled state leads to a violation? Until this day comes, we remain annoyed.