Degrees, Levels, and Profiles of Contextuality

We introduce a new notion, that of a contextuality profile of a system of random variables. Rather than characterizing a system's contextuality by a single number, its overall degree of contextuality, we show how it can be characterized by a curve relating the degree of contextuality (including nonlocality, as a special case) to the level at which the system is considered, level1⋯n-1n>1n+1⋯Ndegree0⋯0dn>0dn+1≥dn⋯dN≥dN-1, where N is the maximum number of variables per system's context. A system is represented at level n if one only considers the joint distributions with k≤n variables, ignoring higher-order joint distributions. We show that the level-wise contextuality analysis can be used in conjunction with any well-constructed measure of contextuality. We present a method of concatenated systems to explore contextuality profiles systematically, and we apply it to the contextuality profiles for three major measures of contextuality proposed in the literature.