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Virtually all of the analysis of quantum contextuality is restricted to thecase where events are represented by rank-one projectors. This restriction isarbitrary and not motivated by physical considerations. We show here thatloosening the rank constraint opens a new realm of quantum contextuality and wedemonstrate that state-independent contextuality can even require projectors ofnonunit rank. This enables the possibility of state-independent contextualitywith less than 13 projectors, which is the established minimum for the case ofrank one. We prove that for any rank, at least 9 projectors are required.Furthermore, in an exhaustive numerical search we find that 13 projectors arealso minimal for the cases where all projectors are uniformly of rank two oruniformly of rank three.

State-independent quantum contextuality with projectors of nonunit rank