CM Ferrera, R Simmons, J Purcell, S Popescu and I have written a new paper where we define and study C-Q boxes. A C-Q box is shared across many parties, each of which can enter a classical input and receive a quantum state as output. These boxes are defined to be non-signalling: they do not allow any party to send a message to any other party.

The motivation for looking at these boxes comes from the non-local correlations which Bell Inequalities show to be present in quantum mechanics. The simplest form of a Bell Inequality thought experiment has two parties who share an entangled quantum mechanical state. Each party chooses one of two measurements and get one of two classical outputs. Bell demonstrated that the correlations between these outputs cannot be reproduced by any local classical theory. The Bell experiment can be thought of as a no-signalling C-C box, which takes classical inputs and gives classical outputs, and which happens to have an entangled quantum state and measurements inside. Popescu and Rohrlich showed that no-signalling allows for C-C boxes which give even stronger non-local correlations than quantum mechanics, and several works have speculated what additional constraints may be required to explain why nature is quantum mechanical and does not have these super-strong correlations.

One of the main questions of our research is whether all C-Q boxes can be implemented using pre-shared entanglement and C-C Boxes. One of our main results is that we can implement any bi-partite C-Q box outputting pure quantum states in such a way. However we are still investigating whether there is in fact another C-Q box, perhaps outputting mixed quantum states, which cannot be reduced to C-C boxes in this way. Regardless of the answer, we believe that studying these super-strong non-local correlations will help us to understand the quantum mechanical non-local correlations themselves.