Isomorphism between the Peres and Penrose proofs of the BKS theorem in three dimensions

TL;DR

This paper demonstrates that the Peres and Penrose proofs of the Bell-Kochen-Specker theorem in three dimensions are isomorphic, and introduces a continuous family of such proofs.

Contribution

It reveals the isomorphism between Peres and Penrose rays and identifies a three-parameter family of inequivalent proofs of the theorem.

Findings

Peres and Penrose rays have identical orthogonality relations.

They form two members of a continuous family of proofs.

The family includes unitarily inequivalent rays proving the theorem.

Abstract

It is shown that the 33 complex rays in three dimensions used by Penrose to prove the Bell-Kochen-Specker theorem have the same orthogonality relations as the 33 real rays of Peres, and therefore provide an isomorphic proof of the theorem. It is further shown that the Peres and Penrose rays are just two members of a continuous three-parameter family of unitarily inequivalent rays that prove the theorem.