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

Abstract

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 $q_{1},\ldots ,q_{n}$ are measured in pairs $\left( q_{i},q_{j}\right)$; every property enters in precisely two such pairs; and each measurement outcome is a binary random variable. Denoting the measurement outcomes for a property $q_{i}$ in the two pairs it enters by $V_{i}$ and $W_{i}$, the pair of measurement outcomes for $\left( q_{i},q_{j}\right)$ is $\left( V_{i},W_{j}\right)$. Contextuality is defined as follows: one computes the minimal possible value $\Delta _{0}$ for the sum of $\Pr \left[ V_{i}\not =W_{i}\right]$ (over $i=1,\ldots ,n$) that is allowed by the individual distributions of $V_{i}$ and $W_{i}$; one computes the minimal possible value $\Delta _{\min }$ for the sum of $\Pr \left[ V_{i}\not =W_{i}\right]$ across all possible couplings of (i.e., joint distributions imposed on) the entire set of random variables $V_{1},W_{1},\ldots ,V_{n},W_{n}$ in the system; and the system is considered contextual if $\Delta _{\min }>\Delta _{0}$ (otherwise $\Delta _{\min }=\Delta _{0}$). 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 $\Delta _{\min }-\Delta _{0}$ in terms of the distributions of the measurement outcomes $\left( V_{i},W_{j}\right)$.