Systematic construction of quantum contextuality scenarios with rank advantage

TL;DR

This paper introduces a systematic method to construct quantum contextuality scenarios that require higher-rank measurements, expanding the understanding of contextuality beyond traditional rank-one projectors.

Contribution

The authors develop a disjunctive graph product approach to systematically create contextuality scenarios with non-unit rank measurements, including explicit examples up to rank five.

Findings

Constructed explicit examples of contextuality scenarios requiring ranks greater than one.

Provided a systematic method for generating higher-rank contextuality scenarios.

Expanded the understanding of measurement ranks necessary for quantum contextuality.

Abstract

A set of quantum measurements exhibits quantum contextuality when any consistent value assignment to the measurement outcomes leads to a contradiction with quantum theory. In the original Kochen-Specker-type of argument the measurement projectors are assumed to be rays, that is, of unit rank. Only recently a contextuality scenario has been identified where state-independent contextuality requires measurements with projectors of rank two. Using the disjunctive graph product, we provide a systematic method to construct contextuality scenarios which require non-unit rank. We construct explicit examples requiring ranks greater than rank one up to rank five.