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Parity Proofs of the Bell-Kochen-Specker Theorem Based on the 600-cell

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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.

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Authors and Affiliations

  1. Physics Department, Worcester Polytechnic Institute, Worcester, MA, 01609, USA

    Mordecai Waegell & P. K. Aravind

  2. Boston Information Group, 19 Locke Ln., Lexington, MA, 02420, USA

    Norman D. Megill

  3. Institute for Theoretical Atomic, Molecular and Optical Physics at Physics Department at Harvard University and Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, 02138, USA

    Mladen Pavičić

  4. Chair of Physics, Faculty of Civil Engineering, University of Zagreb, Zagreb, Croatia

    Mladen Pavičić

Authors
  1. Mordecai Waegell
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  2. P. K. Aravind
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  3. Norman D. Megill
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  4. Mladen Pavičić
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Corresponding author

Correspondence to P. K. Aravind.

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Waegell, M., Aravind, P.K., Megill, N.D. et al. Parity Proofs of the Bell-Kochen-Specker Theorem Based on the 600-cell. Found Phys 41, 883–904 (2011). https://doi.org/10.1007/s10701-011-9534-7

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  • DOI: https://doi.org/10.1007/s10701-011-9534-7

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