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

Foundations of Physics 46 (3):282-299 (2016)
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Abstract

We present a proof for a conjecture previously formulated by Dzhafarov et al.. The conjecture specifies a measure for the degree of contextuality and a criterion 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 \ are measured in pairs \ \); every property enters in precisely two such pairs; and each measurement outcome is a binary random variable. Denoting the measurement outcomes for a property \ in the two pairs it enters by \ and \, the pair of measurement outcomes for \ \) is \ \). Contextuality is defined as follows: one computes the minimal possible value \ for the sum of \ ) that is allowed by the individual distributions of \ and \; one computes the minimal possible value \ for the sum of \ across all possible couplings of the entire set of random variables \ in the system; and the system is considered contextual if \ ). 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 \ in terms of the distributions of the measurement outcomes \ \)

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