Purdue University, USA, ehtibar@purdue.edu
(joint work with Janne V. Kujala and Víctor H. Cervantes)
Cyclic systems of dichotomous random variables have played a prominent role in contextuality research, describing such experimental paradigms as the KCBS, EPR/Bell, and Leggett-Garg ones. We present a theoretical analysis of the degree of contextuality in cyclic systems (if they are contextual) and the degree of noncontextuality in them (if they are not). The Contextuality-by-Default (CbD) theory allows us to drop the commonly made assumption that systems of random variables are consistently connected (i.e., we allow for all possible forms of “disturbance” or “signaling” in them). By contrast, all previously proposed measures of contextuality are confined to consistently connected systems, and most of them cannot be extended to measures of noncontextuality. Our measures of contextuality and noncontextuality are defined by the L1-distance between a point representing a cyclic system and the surface of the polytope representing all possible noncontextual cyclic systems with the same single-variable marginals. We completely characterize this polytope, and establish that, in relation to the maximally tight Bell-type CbD inequality for cyclic systems, the measure of contextuality is proportional to the absolute value of the difference between its two sides. For noncontextual cyclic systems, the measure of noncontextuality is shown to be proportional to the smaller of the same difference and the L1-distance to the surface of the hyperbox circumscribing the noncontextuality polytope. These simple relations, however, do not generally hold beyond the class of cyclic systems, and noncontextuality of a system does not follow from noncontextuality of its cyclic subsystems. We also compute the volumes of the noncontextuality polytope and the circumscribing hyperbox to answer the following question: if one chooses a cyclic system “at random” (i.e., uniformly within the hyperbox), what are the odds that it will be (non)contextual? We find that the odds of contextuality rapidly tend to zero as the size of the system increases.
[1] Dzhafarov, E.N., Kujala, J.V., & Cervantes, V.H. Contextuality and noncontextuality measures and generalized Bell inequalities for cyclic systems. Physical Review A 101:042119. + Erratum Note in Physical Review A 101:06990 (arXiv:1907.03328)
[2] Dzhafarov, E.N., Kujala, J.V., & Cervantes, V.H. (2021). Epistemic odds of contextuality in cyclic systems. European Physics Journal - Special Topics. (arXiv:2002.07755.)