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

Abstract

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 $q_1,...,q_n$ are measured in pairs $(q_i,q_j)$; 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 $(q_i,q_j)$ is $(V_i,W_j)$. Contextuality is defined as follows: one computes the minimal possible value ${\Delta }_0$ for the sum of $\mathrm{Pr}[V_i\ne W_i]$ (over $i=1,...,n$) 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 $\mathrm{Pr}[V_i\ne W_i]$ across all possible couplings of (i.e., joint distributions imposed on) the entire set of random variables $V_1,W_1,...,V_n,W_n$ 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 $(V_i,W_j)$.

This version has two typos found in the published article corrected (Foundations of Physics 46, 282-299, 2016; Eq. 7 and Section 1.4.)

References in corpus (3)

• A simple test for hidden variables in spin-1 system
• Necessary and Sufficient Conditions for Extended Noncontextuality in a Broad Class of Quantum Mechanical Systems
• Contextuality in Three Types of Quantum-Mechanical Systems

Cited by in corpus (16)

• Necessary and Sufficient Conditions for Extended Noncontextuality in a Broad Class of Quantum Mechanical Systems
• True Contextuality Beats Direct Influences in Human Decision Making
• Contextuality in Canonical Systems of Random Variables
• Probabilistic Foundations of Contextuality
• Measures of Contextuality and Noncontextuality
• Contextuality and Noncontextuality Measures and Generalized Bell Inequalities for Cyclic Systems
• Contextuality Analysis of the Double Slit Experiment (With a Glimpse Into Three Slits)
• On Joint Distributions, Counterfactual Values, and Hidden Variables in Relation to Contextuality
• Graph Approach to Extended Contextuality
• Advanced Analysis of Quantum Contextuality in a Psychophysical Double-Detection Experiment
• Measuring Observable Quantum Contextuality
• Contextuality Analysis of Impossible Figures
• Epistemic Odds of Contextuality in Cyclic Systems
• On Universality of Classical Probability with Contextually Labeled Random Variables
• Contextuality and dichotomizations of random variables
• On contextuality and logic in scientific discourse: Reply to Aliakbarzadeh, Kitto, and Bruza