Necessary and sufficient condition for contextuality from incompatibility

TL;DR

This paper establishes that in quantum measurement scenarios, the presence of induced cycles larger than three in the incompatibility graph is both necessary and sufficient for the emergence of contextuality, linking graph structure to quantum contextuality.

Contribution

It provides a precise graph-theoretic criterion for when incompatibility leads to contextuality in quantum systems.

Findings

Induced cycles larger than three are necessary for quantum contextuality.

The criterion is both necessary and sufficient for contextuality to arise.

Graph structure directly determines the presence of contextuality in quantum measurements.

Abstract

Measurement incompatibility is the most basic resource that distinguishes quantum from classical physics. Contextuality is the critical resource behind the power of some models of quantum computation and is also a necessary ingredient for many applications in quantum information. A fundamental problem is thus identifying when incompatibility produces contextuality. Here, we show that, given a structure of incompatibility characterized by a graph in which nonadjacent vertices represent incompatible ideal measurements, the necessary and sufficient condition for the existence of a quantum realization producing contextuality is that this graph contains induced cycles of size larger than three.