Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem

TL;DR

This paper refines the Bell-Kochen-Specker theorem proof by identifying smaller, critical subsets of directions in the 600-cell that cannot be colored, demonstrating more economical and symmetry-related proofs of quantum contextuality.

Contribution

It identifies critical subsets of directions within the 600-cell that provide more efficient proofs of the Bell-Kochen-Specker theorem, highlighting their symmetry properties and significance.

Findings

Critical subsets of size 36 and 30 directions are identified.

These subsets are critical; removing any direction invalidates the proof.

The critical sets belong to orbits of 200 and 240 members under polytope symmetries.

Abstract

Aravind and Lee-Elkin (1997) gave a proof of the Bell-Kochen-Specker theorem by showing that it is impossible to color the 60 directions from the center of a 600-cell to its vertices in a certain way. This paper refines that result by showing that the 60 directions contain many subsets of 36 and 30 directions that cannot be similarly colored, and so provide more economical demonstrations of the theorem. Further, these subsets are shown to be critical in the sense that deleting even a single direction from any of them causes the proof to fail. The critical sets of size 36 and 30 are shown to belong to orbits of 200 and 240 members, respectively, under the symmetries of the polytope. A comparison is made between these critical sets and other such sets in four dimensions, and the significance of these results is discussed.