On small proofs of the Bell-Kochen-Specker theorem for two, three and four qubits

Abstract

The Bell-Kochen-Specker (BKS) theorem rules out realistic non-contextual theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases. We investigate the geometrical structure of small v-l BKS proofs involving v real rays and l 2n-dimensional bases of n-qubits (1 < n 5). Specifically, we look at the parity proof 18-9 with two qubits (A. Cabello, 1996), the parity proof 36-11 with three qubits (M. Kernaghan, A. Peres, 1995) and a newly discovered non-parity proof 80-21 with four qubits (that improves work of P.K. Aravind’s group in 2008). The rays in question arise as real eigenstates shared by some maximal commuting sets (bases) of operators in the n-qubit Pauli group. One finds characteristic signatures of the distances between the bases, which carry various symmetries in their graphs.