Computability limits non-local correlations

If the no-signalling principle was the only limit to the strength of non-local correlations, we would expect that any form of no-signalling correlation can indeed be realized. That is, there exists a state and measurements that remote parties can implement to obtain any such correlation. Here, we show that in any theory in which some functions cannot be computed, there must be further limits to non-local correlations than the no-signalling principle alone. We proceed to argue that even in a theory such as quantum mechanics in which non-local correlations are already weaker, the question of computability imposes such limits.