Imagine you have a box with some switches and knobs, which displays a message on its screen every time you press the big red button that says "press me".
Being curious, you keep pressing it a lot of times and playing with the settings and eventually you figure out the types of possible messages and the frequency with which they appear, so you have the conditional probability distribution P(x|a)Pxa where xx is the message and aa represents all the settings that you chose. Unbeknownst to you, I am also conducting the same sort of 'experiment' in my laboratory and I am producing my own local probability distribution P(y|b)Pyb for my box.
Later in the day we meet and we compare our measurement results, discovering that they were actually correlated, which means that the joint probability is not separable, i.e., P(x,y|a,b)≠P(x|a)P(y|b)PxyabPxaPyb. Okay, this is plausible, we both think. But then we begin to wonder how the boxes did the trick: we clearly know about quantum mechanics and entanglement, but did the boxes need any of it at all? If they did, was that sufficient or did they need to supplement it with shared classical information? And how large would the quantum systems need to be?
John Donohue from the Institute for Quantum Computing (IQC) and Elie Wolfe of the Perimeter Institute for Theoretical Physics (PI) took a stab at these questions in a recent PRA paper ([PRA 92, 062120 (2016)] or (arXiv:1506.01119). In order to visualize all the possible conditional probability distributions (known as 'boxes'), a geometrical approach can be used. The set of boxes that can be built using certain resources can be thought of as a shape in a high-dimensional space, with extreme points defined by boxes with a constant output (for example, "always output one"). The set of boxes which can be built with unlimited classical resources is the convex hull of these extreme points, which means any point that can be made by drawing a line between other points in the space. Using some quantum resources, it is possible to build boxes outside of this set, which cannot be thought of as mixing different classical boxes together; the probability distribution that violates the CHSH-Bell inequality is a well-known example of a box that requires quantum resources.
There are different sets depending on the resources available to the box: dLd are those that allow only for sharing classical random variables of dimension dd and dQd those that allow for sharing quantum systems of dimension dd. Clearly, dLd is contained in d+1Ld1 and dQd is contained in d+1Qd1(simply by ignoring part of the resources), but what is the relation between L's and Q's and what happens when you have a box with a bit of both kinds of resource? For example, suppose you want a balanced mixture of the trivial box that always outputs 0-0 (in A and in B) and the trivial box that always outputs 1-1 (in A and in B): now you have a box that randomly outputs 0-0 or 1-1, which you clearly need some extra non-trivial resources to make! (like shared randomness or entanglement). What is interesting is to bound exactly how much you need. (by the way, this example alone shows that 1Q1 is not convex).
John and Elie showed that the amount of supplementary classical information that you need to convexify the set QdQd can be bounded by examining the geometry of the set. If you have access to the vertices of a square, you can make any point inside it as a mixture of only three of them; if you have the extremal points of a three-dimensional object, you can find any point inside the object as a mixture of four. In general, for objects of dimension D, any point in their convex hull can be written as a combination of at most D+1 points; this is known as Caratheodory's theorem. You can find the maximum number of mixtures needed to flesh out the convex hull of a box by knowing the dimension of the space it lives in. In particular, for the traditional Bell scenario, where two parties have two settings with two possible outcomes, the full set of achievable quantum boxes, ∞Q∞, can be built by supplementing two-dimensional quantum systems with at most eight-dimensional (or three bits of) shared classical randomness, i.e., Q∞⊆Q2∪L8Q∞Q2L8. This upper bound is likely not tight, but is also notably not infinity, and can be calculated for other, more complicated scenarios as well.
In addition to showing upper bounds on the randomness needed to convexify quantum correlations, John and Elie show some specific regions of quantum correlations with nonconvex features, and demonstrate some odd features of the overlap between the classical and quantum sets. They find certain boxes which cannot be built within 2Q2, but can be built trivially within 3L3 (weird huh?). However, they also find boxes which require at least two bits of classical shared randomness, i.e., 4L4, to be built classically, but exist within 2Q2, implying that quantum "shortcuts" can sometimes be used even for building classical boxes! The geometry of these interlacing sets is rich, and mostly unexplored, but may be interesting for practical reasons as well as fundamental ones. Goh, Bancal, and Scarani have recently used similar concepts to build semi-device independent entanglement verification, which can be used to verify entanglement if the dimension of the state is trusted [NJP 18, 045022 (2016)]. Additionally, if you know how much shared randomness or how large a quantum system should be to have full access to the desired quantum correlations, you can determine the resources to make these boxes in the real world.