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A set of quantum measurements exhibits quantum contextuality when anyconsistent value assignment to the measurement outcomes leads to acontradiction with quantum theory. In the original Kochen-Specker-type ofargument the measurement projectors are assumed to be rays, that is, of unitrank. Only recently a contextuality scenario has been identified wherestate-independent contextuality requires measurements with projectors of ranktwo. Using the disjunctive graph product, we provide a systematic method toconstruct contextuality scenarios which require non-unit rank. We constructexplicit examples requiring ranks greater than rank one up to rank five.

Systematic construction of quantum contextuality scenarios with rank advantage