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

We present a proof for a conjecture previously formulated by Dzhafarov et al. (Found Phys 7:762–782, 2015). 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 $$q_{1},\ldots ,q_{n}$$q1,…,qn are measured in pairs $$\left( q_{i},q_{j}\right) $$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 $$q_{i}$$qi in the two pairs it enters by $$V_{i}$$Vi and $$W_{i}$$Wi, the pair of measurement outcomes for $$\left( q_{i},q_{j}\right) $$qi,qj is $$\left( V_{i},W_{j}\right) $$Vi,Wj. Contextuality is defined as follows: one computes the minimal possible value $$\Delta _{0}$$Δ0 for the sum of $$\Pr \left[ V_{i}\not =W_{i}\right] $$PrVi≠Wi (over $$i=1,\ldots ,n$$i=1,…,n) that is allowed by the individual distributions of $$V_{i}$$Vi and $$W_{i}$$Wi; one computes the minimal possible value $$\Delta _{\min }$$Δmin for the sum of $$\Pr \left[ V_{i}\not =W_{i}\right] $$PrVi≠Wi across all possible couplings of (i.e., joint distributions imposed on) the entire set of random variables $$V_{1},W_{1},\ldots ,V_{n},W_{n}$$V1,W1,…,Vn,Wn in the system; and the system is considered contextual if $$\Delta _{\min }>\Delta _{0}$$Δmin>Δ0 (otherwise $$\Delta _{\min }=\Delta _{0}$$Δmin=Δ0). 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 $$\Delta _{\min }-\Delta _{0}$$Δmin-Δ0 in terms of the distributions of the measurement outcomes $$\left( V_{i},W_{j}\right) $$Vi,Wj.