Contextuality with a Small Number of Observables

TL;DR

This paper analyzes small geometric configurations that provide minimal observable-based proofs of the Kochen-Specker theorem, identifying the most economical proofs and introducing a new configuration with 14 observables.

Contribution

It proves the minimal number of observables needed for such proofs and introduces a new 14-observable proof using a 'magic' heptagram.

Findings

Mermin-Peres square with 9 observables is the most economical proof.

Mermin pentagram with 10 observables is also minimal.

A new 14-observable proof with a 'magic' heptagram is proposed.

Abstract

We investigate small geometric configurations that furnish observable-based proofs of the Kochen-Specker theorem. Assuming that each context consists of the same number of observables and each observable is shared by two contexts, it is proved that the most economical proofs are the famous Mermin-Peres square and the Mermin pentagram featuring, respectively, $9$ and $10$ observables, there being no proofs using less than $9$ observables. We also propose a new proof with $14$ observables forming a `magic' heptagram. On the other hand, some other prominent small-size finite geometries, like the Pasch configuration and the prism, are shown not to be contextual.