Parity proofs of the Bell-Kochen-Specker theorem based on the 600-cell

Abstract

The set of 60 real rays in four dimensions derived from the vertices of a 600-cell is shown to possess numerous subsets of rays and bases that provide basis-critical parity proofs of the Bell-Kochen-Specker (BKS) theorem (a basis-critical proof is one that fails if even a single basis is deleted from it). The proofs vary considerably in size, with the smallest having 26 rays and 13 bases and the largest 60 rays and 41 bases. There are at least 90 basic types of proofs, with each coming in a number of geometrically distinct varieties. The replicas of all the proofs under the symmetries of the 600-cell yield a total of almost a hundred million parity proofs of the BKS theorem. The proofs are all very transparent and take no more than simple counting to verify. A few of the proofs are exhibited, both in tabular form as well as in the form of MMP hypergraphs that assist in their visualization. A survey of the proofs is given, simple procedures for generating some of them are described and their applications are discussed. It is shown that all four-dimensional parity proofs of the BKS theorem can be turned into experimental disproofs of noncontextuality.

19 pages, 11 tables, 3 figures. Email address of first author has been corrected. Ref.[5] has been corrected, as has an error in Fig.3. Formatting error in Sec.4 has been corrected and the placement of tables and figures has been improved. A new paragraph has been added to Sec.4 and another new paragraph to the end of the Appendix

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Cited by in corpus (20)

• Kochen-Specker Contextuality
• Kochen-Specker set with seven contexts
• Proofs of the Kochen-Specker theorem based on the N-qubit Pauli group
• Parity proofs of the Kochen-Specker theorem based on 60 complex rays in four dimensions
• Quantum Contextuality
• Parity proofs of the Kochen-Specker theorem based on the Lie algebra E8
• Probabilistic Generation of Quantum Contextual Sets
• Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres
• 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
• Parity proofs of the Kochen-Specker theorem based on the 120-cell
• GHZ paradoxes based on an even number of qubits
• Automated generation of Kochen-Specker sets
• The Minimum Complexity of Kochen-Specker Sets Does Not Scale with Dimension
• Hypergraph Contextuality
• The Golay codes and Quantum Contextuality
• Kochen-Specker Sets with a Mixture of 16 Rank-1 and 14 Rank-2 Projectors for a Three-Qubit System
• Vector Generation of Contextual Sets
• Generalized Householder transformations