1204.4275 (Michel Planat)

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

Michel Planat

Bell-Kochen-Specker theorem (BKS) theorem rules out realistic {\it 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 BKS-proofs $v-l$ 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 a work at 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 universal signatures of the distances among the bases, that carry various symmetries in their graphs.

View original: http://arxiv.org/abs/1204.4275