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 (17)

• Hidden Variables and the Two Theorems of John Bell
• Bell-Kochen-Specker theorem: A proof with 18 vectors
• Specker's Parable of the Over-protective Seer: A Road to Contextuality, Nonlocality and Complementarity (PLUS AN ERRATUM)
• Logical Bell Inequalities
• Mubs and Hadamards of Order Six
• Kochen-Specker theorem for 8-dimensional space
• Mutually unbiased bases in dimension six: The four most distant bases
• Kochen-Specker Vectors
• Proofs of the Kochen-Specker theorem based on a system of three qubits
• Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function ?
• Parity proofs of the Kochen-Specker theorem based on 60 complex rays in four dimensions
• Probabilistic Generation of Quantum Contextual Sets
• Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres
• Mermin's Pentagram as an Ovoid of PG(3,2)
• Five-Qubit Contextuality, Noise-Like Distribution of Distances Between Maximal Bases and Finite Geometry
• Three-qubit entangled embeddings of CPT and Dirac groups within E8 Weyl group
• New Examples of Kochen-Specker Type Configurations on Three Qubits

Cited by in corpus (25)

• Kochen-Specker Contextuality
• Kochen-Specker set with seven contexts
• Quantum Contextuality
• Distinguished three-qubit 'magicity' via automorphisms of the split Cayley hexagon
• Geometry of contextuality from Grothendieck's coset space
• Vector Generation of Quantum Contextual Sets in Even Dimensional Hilbert Spaces
• What makes quantum clicks special?
• Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets
• Magic informationally complete POVMs with permutations
• Contextuality with a Small Number of Observables
• Quantum contextual finite geometries from dessins d'enfants
• A proof of the Kochen-Specker theorem can always be converted to a state-independent noncontextuality inequality
• The magic of universal quantum computing with permutations
• Zoology of Atlas-groups: dessins d'enfants, finite geometries and quantum commutation
• Five-Qubit Contextuality, Noise-Like Distribution of Distances Between Maximal Bases and Finite Geometry
• Contextuality degree of quadrics in multi-qubit symplectic polar spaces
• The Poincaré half-plane for informationally complete POVMs
• Group geometrical axioms for magic states of quantum computing
• It from qubit: how to draw quantum contextuality
• New Examples of Kochen-Specker Type Configurations on Three Qubits
• Kochen-Specker Sets with a Mixture of 16 Rank-1 and 14 Rank-2 Projectors for a Three-Qubit System
• Irreducible magic sets for $n$-qubit systems
• Vector Generation of Contextual Sets
• Geometric contextuality from the Maclachlan-Martin Kleinian groups
• Two-letter words and a fundamental homomorphism ruling geometric contextuality