# Bell Quantified: The Resource Theory of Nonclassicality of Common-Cause Boxes. (arXiv:1903.06311v1 [quant-ph])

A Bell experiment can be conceptualized as a box, i.e., a process taking

classical setting variables to classical outcome variables, which has a

common-cause structure, i.e., one that can be realized by implementing local

measurements on systems that are prepared by a common-cause mechanism, with no

cause-effect relations across the wings of the experiment. For such

common-cause boxes, one can define a distinction between classical and

nonclassical in terms of what type of causal model (classical or nonclassical)

is required to explain the box's input-output functionality. One can also

quantify their nonclassicality using a resource-theoretic approach, where the

free operations on a common-cause box are those that can be achieved by

embedding it in a circuit composed of classical common-cause boxes. These

circuits correspond to local operations and shared randomness. We prove that

the set of free operations forms a polytope, and we provide an efficient

algorithm for deciding the ordering relation between any pair of resources. We

define two distinct monotones and leverage these to reveal various global

properties of the pre-order of resources. Only one of these monotones is fully

characterized by the information contained in the degrees of violation of

facet-defining Bell inequalities, showing that although such inequalities are

sufficient for witnessing nonclassicality, they are not sufficient for

quantifying nonclassicality. For bipartite common-cause boxes with binary

inputs and outputs, we provide closed-form expressions for our two monotones,

and we use these to derive additional properties of the pre-order, including a

lower bound on the cardinality of any complete set of monotones. Finally, we

consider the pre-order on the subset of common-cause boxes that are quantumly

realizable and prove that every convexly extremal such box is at the top of

this pre-order.