Skip to main content
SpringerLink
Log in

Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell

  • Published:
Foundations of Physics Aims and scope Submit manuscript

Abstract

It is shown how the 300 rays associated with the antipodal pairs of vertices of a 120-cell (a four-dimensional regular polytope) can be used to give numerous “parity proofs” of the Kochen–Specker theorem ruling out the existence of noncontextual hidden variables theories. The symmetries of the 120-cell are exploited to give a simple construction of its Kochen–Specker diagram, which is exhibited in the form of a “basis table” showing all the orthogonalities between its rays. The basis table consists of 675 bases (a basis being a set of four mutually orthogonal rays), but all the bases can be written down from the few listed in this paper using some simple rules. The basis table is shown to contain a wide variety of parity proofs, ranging from 19 bases (or contexts) at the low end to 41 bases at the high end. Some explicit examples of these proofs are given, and their implications are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. The dual of a 24-cell is another 24-cell rotated relative to the first (about their common center), with the vertices of the dual being along the same directions as the cell centers of the original, and vice-versa.

  2. The multiplicity of a ray is the number of bases it occurs in.

  3. The Kochen–Specker diagram of a set of rays is a graph whose vertices are the rays and whose edges connect vertices corresponding to orthogonal rays.

  4. Since \(V\) and \(W\) are symmetry operations of the 120-cell, they can be described by the permutations they perform on its vertices: \(V\) replaces the ray \(i\) by the ray \((i+60)\) mod 300, while \(W\) replaces ray \(i\) by \(i+12\) if \(60n < i \le 60n+48\) for \(n=0,1,2,3,4\), or by \(i-48\) otherwise. The operator \(U\) also performs a permutation, but it cannot be described simply.

  5. The 24-cell, 600-cell and 120-cell are all centrally symmetric figures whose vertices come in antipodal pairs.

  6. See Ref. [32], p.270, where it is pointed out that the 600-cells in the rows and columns of Table 1 form a pair of enantiomorphous sets.

  7. A noncontextual value assignment to a ray is one in which the ray is assigned the same value in all the bases in which it occurs.

References

  1. Waegell, M., Aravind, P.K.: Parity proofs of the Kochen-Specker theorem based on the 24 rays of peres. Found. Phys. 41, 1786–1799 (2011)

  2. Waegell, M., Aravind, P.K., Megill, N.D., Pavičić, M.: Parity proofs of the Bell-Kochen-Specker theorem based on the 600-cell. Found. Phys. 41, 883–904 (2011)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  3. Specker, E.P.: The logic of propositions which are not simultaneously decidable. Dialectica 14, 239–246 (1960)

    Article  MathSciNet  Google Scholar 

  4. Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–88 (1967)

    MathSciNet  MATH  Google Scholar 

  5. Bell, J.S.: On the problem of hidden variables in quantum mechanics. Rev. Mod. Phys. 38, 447–452 (1966) (Reprinted in J.S. Bell, Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, Cambridge, 1987)

  6. Peres, A.: Two simple proofs of the Kochen-Specker theorem. J. Phys. A 24, L175–L178 (1991)

    Article  MATH  ADS  Google Scholar 

  7. Mermin, N.D.: Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett. 65, 3373–3376 (1990)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  8. Mermin, N.D.: Hidden variables and the two theorems of John Bell. Rev. Mod. Phys. 65, 803–815 (1993)

    Article  MathSciNet  ADS  Google Scholar 

  9. Kernaghan, M.: Bell-Kochen-Specker theorem for 20 vectors. J. Phys. A 27, L829–L830 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  10. Cabello, A., Estebaranz, J.M., García-Alcaine, G.: Bell-Kochen-Specker theorem: a proof with 18 vectors. Phys. Lett. A 212, 183–187 (1996)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  11. Aravind, P.K.: How Reye’s configuration helps in proving the BellKochenSpecker theorem: a curious geometrical tale. Found. Phys. Lett. A 13, 499–519 (2000)

    Article  MathSciNet  Google Scholar 

  12. Pavičić, M., Megill, N.D., Merlet, J.P.: New KochenSpecker sets in four dimensions. Phys. Lett. A 374, 2122–2128 (2010)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  13. Pavičić, M., Merlet, J.P., McKay, B.D., Megill, N.D.: KochenSpecker vectors. J. Phys. A 38, 1577–1592 (2005)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  14. Kernaghan, M., Peres, A.: Kochen-Specker theorem for eight-dimensional space. Phys. Lett. A 198, 1–5 (1995)

    Article  MathSciNet  MATH  ADS  Google Scholar 

  15. Waegell, M., Aravind, P.K.: Parity proofs of the KochenSpecker theorem based on 60 complex rays in four dimensions. J. Phys. A 44(15), 505303 (2011)

    Article  MathSciNet  Google Scholar 

  16. Waegell, M., Aravind, P.K.: Proofs of the KochenSpecker theorem based on a system of three qubits. J. Phys. A 45(13), 405301 (2012)

    Article  MathSciNet  Google Scholar 

  17. Waegell, M., Aravind, P.K.: Proofs of the KochenSpecker theorem based on the N-qubit Pauli group. Phys. Rev. A 88(10), 012102 (2013)

    Article  ADS  Google Scholar 

  18. Lisonĕk, P., Badzia̧g, P., Portillo, J.R., Cabello, A.: Kochen-Specker set with seven contexts. Phys. Rev. A 89(6), 042101 (2014)

    Article  ADS  Google Scholar 

  19. Cabello, A.: Experimentally testable state-independent quantum contextuality. Phys. Rev. Lett. 101(4), 210401 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  20. Badzia̧g, P., Bengtsson, I., Cabello, A., Pitowsky, I.: Universality of state-independent violation of correlation inequalities for noncontextual theories. Phys. Rev. Lett. 103(4), 050401 (2009)

    Article  ADS  Google Scholar 

  21. Kirchmair, G., Zähringer, F., Gerritsma, R., Kleinmann, M., Gühne, O., Cabello, A., Blatt, R., Roos, C.F.: State-independent experimental test of quantum contextuality. Nature 460, 494–497 (2009)

    Article  ADS  Google Scholar 

  22. Bartosik, H., Klep, J., Schmitzer, C., Sponar, S., Cabello, A., Rauch, H., Hasegawa, Y.: Experimental test of quantum contextuality in neutron interferometry. Phys. Rev. Lett. 103(4), 040403 (2009)

    Article  ADS  Google Scholar 

  23. Amselem, E., Rådmark, M., Bourennane, M., Cabello, A.: State-independent quantum contextuality with single photons. Phys. Rev. Lett. 103(4), 160405 (2009)

    Article  ADS  Google Scholar 

  24. Moussa, O., Ryan, C.A., Cory, D.G., Laflamme, R.: Testing contextuality on quantum ensembles with one clean qubit. Phys. Rev. Lett. 104(4), 160501 (2010)

    Article  MathSciNet  ADS  Google Scholar 

  25. Aolita, L., Gallego, R., Acín, A., Chiuri, A., Vallone, G., Mataloni, P., Cabello, A.: Fully nonlocal quantum correlations. Phys. Rev. A 85(8), 032107 (2012)

    Article  ADS  Google Scholar 

  26. D’Ambrosio, V., Herbauts, I., Amselem, E., Nagali, E., Bourennane, M., Cabello, A.: Experimental implementation of a Kochen-Specker set of quantum tests. Phys. Rev. X 3(10), 011012 (2009)

    Google Scholar 

  27. Cubitt, T.S., Leung, D., Matthews, W., Winter, A.: Improving zero-error classical communication with entanglement. Phys. Rev. Lett. 104(4), 230503 (2010)

    Article  ADS  Google Scholar 

  28. Hu, D., Tang, W., Zhao, M., Chen, Q., Yu, S., Oh, C.H.: Graphical nonbinary quantum error-correcting codes. Phys. Rev. A 78(11), 012306 (2008)

    Article  ADS  Google Scholar 

  29. Raussendorf, R., Briegel, H.J.: A One-Way Quantum Computer. Phys. Rev. Lett. 86, 5188–5191 (2001)

    Article  ADS  Google Scholar 

  30. Gühne, O., Budroni, C., Cabello, A., Kleinmann, M., Larsson, J.-A.: Bounding the quantum dimension with contextuality. Phys. Rev. A 89(11), 062107 (2014)

    Article  ADS  Google Scholar 

  31. Abramsky, S.: In: Wong, V.T.L., Fan, L.L.W., Fourman, W.C.T.M. (eds.) In Search of Elegance in the Theory and Practice of Computation. Lecture Notes in Computer Science, vol. 8000. Springer, Berlin Heidelberg (2013)

  32. Coxeter, H.: A detailed discussion of the geometrical properties of the 120-cell can be found in. Regular Polytopes. Dover, New York (1973)

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Mordecai Waegell & P. K. Aravind

Authors
  1. Mordecai Waegell
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. K. Aravind
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. K. Aravind.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Waegell, M., Aravind, P.K. Parity Proofs of the Kochen–Specker Theorem Based on the 120-Cell. Found Phys 44, 1085–1095 (2014). https://doi.org/10.1007/s10701-014-9830-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10701-014-9830-0

Keywords

Search

Navigation