A Kochen-Specker System Has at Least 22 Vectors

Abstract

At the heart of the Conway-Kochen Free Will Theorem and Kochen and Specker’s argument against non-contextual hidden variable theories is the existence of a Kochen-Specker (KS) system: a set of points on the sphere that has no {0,1}-coloring such that at most one of two orthogonal points are colored 1 and of three pairwise orthogonal points exactly one is colored 1. In public lectures, Conway encouraged the search for small KS systems. At the time of writing, the smallest known KS system has 31 vectors.

Arends, Ouaknine and Wampler have shown that a KS system has at least 18 vectors, by reducing the problem to the existence of graphs with a topological embeddability and non-colorability property. The bottleneck in their search proved to be the sheer number of graphs on more than 17 vertices and deciding embeddability.

Continuing their effort, we prove a restriction on the class of graphs we need to consider and develop a more practical decision procedure for embeddability to improve the lower bound to 22.

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References

  1. 1.

    Arends, F., “A lower bound on the size of the smallest kochen-specker vector system in three dimensions,” Master’s thesis, University of Oxford, 2009. http://www.cs.ox.ac.uk/people/joel.ouaknine/download/arends09.pdf.

  2. 2.

    Arends, F., Ouaknine, J. and Wampler, C. W., “On searching for small kochen-specker vector systems,” in Proc of the 37th international conference on Graph-Theoretic Concepts in Computer Science, pp. 23–34, Springer-Verlag, 2011.

  3. 3.

    Cabello A.: “Kochen-specker theorem and experimental test on hidden variables,”. International Journal of Modern Physics A 15(18), 2813–2820 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Collins, G. E., “Quantifier elimination for real closed fields by cylindrical algebraic decomposition,” in Quantifier elimination and cylindrical algebraic decomposition, pp. 85–121 Springer, 1998.

  5. 5.

    Conway J. H., Kochen S.: “The strong free will theorem,”. Notices of the AMS, 56(2), 226–232 (2009)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Dolzmann A., Sturm T.: “Redlog: Computer algebra meets computer logic,”. Acm Sigsam Bulletin 31(2), 2–9 (1997)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Kochen, S. and Specker, E. P., “The problem of hidden variables in quantum mechanics,” in The Logico-Algebraic Approach to Quantum Mechanics, pp. 293–328, Springer, 1975.

  8. 8.

    McKay B. D.: “Isomorph-free exhaustive generation,”. Journal of Algorithms 26(2), 306–324 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Ouaknine, J., personal communication. Attended such a lecture of Conway at the Oxford Mathematical Institute in 2005.

  10. 10.

    Pavičć M., Merlet J-P., McKay B., Megill N. D.: “Kochen-specker vectors,”. Journal of Physics A: Mathematical and General 38(7), 1577–1592 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  11. 11.

    Peres, A., “Two simple proofs of the kochen-specker theorem,” Journal of Physics A: Mathematical and General 24, 4, L175, 1991.

  12. 12.

    Peres, A., Quantum theory: concepts and methods, Springer, 1995.

  13. 13.

    Sloane, N. J. A., The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A000088 (Number of graphs on n unlabeled nodes.).

  14. 14.

    Sloane, N. J. A., The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A006786 (Squarefree graphs on n vertices.).

  15. 15.

    Tarski, A., “A decision method for elementary algebra and geometry,” RAND report R-109, RAND Corp., Santa Monica, CA.

  16. 16.

    Uijlen, S. and Westerbaan, B., Code and data for “a kochen-specker system has at least 22 vectors,” https://github.com/bwesterb/ks

  17. 17.

    Uijlen, S. and Westerbaan, B., “A kochen-specker system has at least 22 vectors (extended abstract),” in Proc. of the 11th workshop on Quantum Physics and Logic, 172, pp. 154–164, 2014.

  18. 18.

    Weisstein, E. W., Square-free graph, http://mathworld.wolfram.com/Square-FreeGraph.html (Last visited on may 6th 2014.).

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Correspondence to Sander Uijlen.

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Uijlen, S., Westerbaan, B. A Kochen-Specker System Has at Least 22 Vectors. New Gener. Comput. 34, 3–23 (2016). https://doi.org/10.1007/s00354-016-0202-5

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Keywords

  • North Pole
  • Antipodal Point
  • Embeddable Graph
  • Topological Embeddability
  • Cylindrical Algebraic Decomposition