Three-Qubit-Embedded Split Cayley Hexagon is Contextuality Sensitive

TL;DR

This paper investigates two distinct embeddings of the split Cayley hexagon into a symplectic polar space, revealing that one embedding exhibits quantum contextuality sensitivity while the other does not, highlighting the geometric roots of quantum contextuality.

Contribution

It demonstrates that the classical and skew embeddings of the split Cayley hexagon into a symplectic space differ in their quantum contextuality properties, linking geometry with quantum theory.

Findings

Classical embedding's complement is non-contextual.

Skew embedding's complement is contextual.

Embedding type determines quantum contextuality sensitivity.

Abstract

It is known that there are two non-equivalent embeddings of the split Cayley hexagon of order two into $\mathcal{W}(5,2)$, the binary symplectic polar space of rank three, called classical and skew. Labelling the 63 points of $\mathcal{W}(5,2)$ by the 63 canonical observables of the three-qubit Pauli group subject to the symplectic polarity induced by the (commutation relations between the elements of the) group, the two types of embedding are found to be quantum contextuality sensitive. In particular, we show that the complement of a classically-embedded hexagon is not contextual, whereas that of a skewly-embedded one is.