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

TL;DR

This paper proves a conjecture that provides a precise measure and criterion for contextuality in broad classes of quantum systems with binary measurements, encompassing well-known quantum phenomena.

Contribution

It offers a rigorous proof of a conjecture linking the degree of contextuality to measurement outcome distributions in cyclic systems.

Findings

Established a formula for the difference between minimal and baseline probabilities of outcome mismatches.

Unified criteria for contextuality across various quantum system types.

Validated the measure's applicability to systems with measurement errors and signaling.

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 $\Pr[V_{i}\not=W_{i}]$ (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 $\Pr[V_{i}\not=W_{i}]$ 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 $(V_{i},W_{j})$.