Exceptional and Non-crystallographic Root Systems and the Kochen-Specker Theorem

Abstract

The Kochen-Specker theorem states that a 3-dimensional complex Euclidean space admits a finite configuration of projective lines such that the corresponding quantum observables (the orthogonal projectors) cannot be assigned with 0 and 1 values in a classically consistent way. This paper shows that the irreducible root systems of exceptional and of non-crystallographic types are useful in constructing such configurations in other dimensions. The cases $E_6$ and $E_7$ lead to new examples, while $F_4$, $E_8$, and $H_4$, yield a new interpretation of the known ones. The described configurations have an additional property: they are saturated, i.e. the tuples of mutually orthogonal lines, being partially ordered by inclusion, yield a poset with all maximal elements having the same cardinality (the dimension of space).

References in corpus (1)

• Hidden Variables and the Two Theorems of John Bell

Cited by in corpus (5)

• Kochen-Specker Contextuality
• Entanglement can increase asymptotic rates of zero-error classical communication over classical channels
• Parity proofs of the Kochen-Specker theorem based on the Lie algebra E8
• Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem
• New Examples of Kochen-Specker Type Configurations on Three Qubits