Proof of a Conjecture on Contextuality in Cyclic Systems with Binary Variables

We present a proof for a conjecture previously formulated by Dzhafarov, Kujala, and Larsson (Foundations of Physics, in press, arXiv:1411.2244). The conjecture specifies a measure for the degree of contextuality and a criterion (necessary and sufficient condition) for contextuality in a broad class of quantum systems. This class includes Leggett-Garg, EPR/Bell, and Klyachko-Can-Binicioglu-Shumovsky type systems as special cases. In a system of this class certain physical properties q1, . . . , qn are measured in pairs (qi, qj); every property enters in precisely two such pairs; and each measurement outcome is a binary random variable. Denoting the measurement outcomes for a property qi in the two pairs it enters by Vi and Wi, the pair of measurement outcomes for (qi, qj) is (Vi,Wj). Contextuality is defined as follows: one computes the minimal possible value ∆0 for the sum of Pr [Vi 6= Wi] (over i = 1, . . . , n) that is allowed by the individual distributions of Vi and Wi; one computes the minimal possible value ∆min for the sum of Pr [Vi 6= Wi] across all possible couplings of (i.e., joint distributions imposed on) the entire set of random variables V1,W1, . . . , Vn,Wn in the system; and the system is considered contextual if ∆min > ∆0 (otherwise ∆min = ∆0). This definition has its justification in the general approach dubbed Contextuality-by-Default, and it allows for measurement errors and signaling among the measured properties. The conjecture proved in this paper specifies the value of ∆min − ∆0 in terms of the distributions of the measurement outcomes (Vi,Wj).