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

Michel Planat (1)

1 FEMTO-ST - Franche-Comté Électronique Mécanique, Thermique et Optique - Sciences et Technologies (UMR 6174)

The 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 $v-l$ 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 \cite{Cabello1996}), the parity proof $36-11$ with three qubits (M. Kernaghan \& A. Peres, 1995 \cite{Kernaghan1965}) and a newly discovered non-parity proof 80-21 with four qubits (that improves work of P.~K Aravind's group in 2008 \cite{Harvey2008}). 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.

Mots clés: multiple qubits; Bell-Kochen-Specker theorem; quantum contextuality; multiple qubits.

Domaines: Physique Quantique [quant-ph]; Théorie des groupes [math.GR]

Michel Planat. On small proofs of Bell-Kochen-Specker theorem for two, three and four qubits. The European Physical Journal Plus, 2012, 127 (8), 86 (11 p.). ⟨10.1140/epjp/i2012-12086-x⟩. ⟨hal-00688962v3⟩