Quantifying nonlocality: a resource theory of nonclassicality of common-cause processes

We introduce a resource theory of non-signalling black-box correlations in nonlocality scenarios. We imagine a number of parties under a physical restriction: they cannot signal each other, and the common causes they share are classical. The parties are able to freely perform local processings of classical inputs and outputs, potentially conditioned on their shared classical randomness. These free operations impose a partial order on the black-box correlations—the resources—in the theory. Correlations that are higher in the order are more nonclassical than those that are lower in the order. We discuss various monotones and conversion witnesses, and we show that standard Bell inequalities do not form a complete set of monotones. That is, we find that one correlation can be more nonclassical than another, even if the two differ only in their marginals. Additionally, we provide an algorithm for deducing, for any pair of correlations in this setting, which is more nonclassical (if either).