Degrees, Levels, and Profiles of Contextuality

Abstract

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 degree of contextuality to level at which the system is considered. A system is represented at level n if one only considers the joint distributions with no more than 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.

Figures (15)

• Figure 1: Four possible contextuality profiles with the same final degree of contextuality at level 5.
• Figure 2: A two-dimensional projection of a vector $\mathbf{v}^{*}$ and a noncontextuality polytope $\mathbb{V}$, with the $L^{1}$-distance between them.
• Figure 3: Contextuality profiles for the method of concatenated systems, $n=2$. The boxes represent the systems being concatenated, with the numbers in them indicating their final level of contextuality. Symbols attached to the curves indicate contextuality values.
• Figure 4: Four possible types of the contextuality profiles for concatenated systems ($n=2$): superadditive (top left panel), additive (top right), subadditive (bottom left), and, as the extreme case of subadditivity, plateau (bottom right).
• Figure 5: Contextuality profiles for a selection of undisturbed concatenated systems. Symbols $\mathcal{A}$ and $\mathcal{B}$ with indices refer to $\mathcal{A}$- and $\mathcal{B}$-subsystems, respectively (as specified in Appendix). The dashed lines attached to each profile show the increment from $d_{2}$ to $d_{2}+\Delta_{3}$: if it is above the corresponding segment of the profile we have subadditivity, and when the dashed line is not seen (coincides with the segment) we have additivity.