Parity proofs of the Kochen-Specker theorem based on the Lie algebra E8

TL;DR

This paper explores the structure of the E8 Lie algebra to find numerous parity proofs of the Kochen-Specker theorem in 8-dimensional quantum systems, revealing new, compact proofs and their underlying geometric and algebraic structures.

Contribution

It introduces a systematic method to identify parity proofs of the Kochen-Specker theorem using the E8 root system and provides the most compact proof in 8 dimensions to date.

Findings

Hundreds of parity proofs found, ranging from 9 to 35 bases.

A new, compact proof involving 34 rays and 9 bases.

The triacontagonal representation aids in identifying proof structures.

Abstract

The 240 root vectors of the Lie algebra E8 lead to a system of 120 rays in a real 8-dimensional Hilbert space that contains a large number of parity proofs of the Kochen-Specker theorem. After introducing the rays in a triacontagonal representation due to Coxeter, we present their Kochen-Specker diagram in the form of a "basis table" showing all 2025 bases (i.e., sets of eight mutually orthogonal rays) formed by the rays. Only a few of the bases are actually listed, but simple rules are given, based on the symmetries of E8, for obtaining all the other bases from the ones shown. The basis table is an object of great interest because all the parity proofs of E8 can be exhibited as subsets of it. We show how the triacontagonal representation of E8 facilitates the identification of substructures that are more easily searched for their parity proofs. We have found hundreds of different types of parity proofs, ranging from 9 bases (or contexts) at the low end to 35 bases at the high end, and involving projectors of various ranks and multiplicities. After giving an overview of the proofs we found, we present a few concrete examples of the proofs that illustrate both their generic features as well as some of their more unusual properties. In particular, we present a proof involving 34 rays and 9 bases that appears to be the most compact parity proof found to date in 8 dimensions.