State-independent quantum contextuality with projectors of nonunit rank

TL;DR

This paper extends the study of quantum contextuality beyond rank-one projectors, demonstrating that state-independent contextuality can occur with higher-rank projectors and identifying minimal numbers of projectors needed.

Contribution

It shows that state-independent quantum contextuality can involve nonunit rank projectors, reducing the number of projectors needed compared to the rank-one case.

Findings

State-independent contextuality can require fewer than 13 projectors with nonunit rank.

At least 9 projectors are necessary for any rank.

13 projectors are minimal for uniform rank two or three projectors.

Abstract

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