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

Abstract

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), 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). 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.

version to appear in European Physical Journal Plus

References in corpus (3)

• Logical Bell Inequalities
• Five-Qubit Contextuality, Noise-Like Distribution of Distances Between Maximal Bases and Finite Geometry
• New Examples of Kochen-Specker Type Configurations on Three Qubits

Cited by in corpus (2)

• Five-Qubit Contextuality, Noise-Like Distribution of Distances Between Maximal Bases and Finite Geometry
• New Examples of Kochen-Specker Type Configurations on Three Qubits