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

TL;DR

This paper explores the geometric structure of small Bell-Kochen-Specker proofs across 2 to 4 qubits, revealing symmetries and signatures in the distances between bases that challenge non-contextual realistic theories.

Contribution

It introduces new small BKS proofs for four qubits and analyzes their geometric and symmetry properties, extending prior work on smaller qubit systems.

Findings

Identified characteristic signatures in basis distances.

Discovered a new non-parity proof for four qubits.

Analyzed symmetries in the graph representations of bases.

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.