An abstract structure determines the contextuality degree of observable-based Kochen-Specker proofs

TL;DR

This paper introduces a new hypergraph-based framework called hypergrams to analyze quantum contextuality in Kochen-Specker proofs, providing conditions and algorithms to identify and quantify their contextuality degrees.

Contribution

It defines hypergrams as an abstract structure for observable-based proofs, establishes their uniform contextuality degree, and offers an efficient method to find quantum labelings and assess contextuality.

Findings

All correct quantum labelings of a hypergram have the same contextuality degree.

A necessary and sufficient condition for the existence of quantum labelings is provided.

An efficient algorithm to find quantum labelings is developed.

Abstract

This article delves into the concept of quantum contextuality, specifically focusing on proofs of the Kochen-Specker theorem obtained by assigning Pauli observables to hypergraph vertices satisfying a given commutation relation. The abstract structure composed of this hypergraph and the graph of anticommutations is named a hypergram. Its labelings with Pauli observables generalize the well-known magic sets. A first result is that all these correct quantum labelings of a given hypergram inherently possess the same degree of contextuality. Then we provide a necessary and sufficient condition for the existence of such quantum labelings and an efficient algorithm to find one of them. We finally attach to each assignable hypergram an abstract notion of contextuality degree. By presenting the study of observable-based Kochen-Specker proofs from the perspectives of graphs and matrices, this abstraction opens the way to new methods to search for original contextual configurations.