Foundations of Physics

, Volume 44, Issue 10, pp 1085–1095 | Cite as

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

  • Mordecai Waegell
  • P. K. Aravind


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.


Kochen–Specker theorem Quantum contextuality parity proofs 120-Cell 


  1. 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)Google Scholar
  2. 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)MathSciNetCrossRefzbMATHADSGoogle Scholar
  3. 3.
    Specker, E.P.: The logic of propositions which are not simultaneously decidable. Dialectica 14, 239–246 (1960)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kochen, S., Specker, E.P.: The problem of hidden variables in quantum mechanics. J. Math. Mech. 17, 59–88 (1967)MathSciNetzbMATHGoogle Scholar
  5. 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)Google Scholar
  6. 6.
    Peres, A.: Two simple proofs of the Kochen-Specker theorem. J. Phys. A 24, L175–L178 (1991)CrossRefzbMATHADSGoogle Scholar
  7. 7.
    Mermin, N.D.: Simple unified form for the major no-hidden-variables theorems. Phys. Rev. Lett. 65, 3373–3376 (1990)MathSciNetCrossRefzbMATHADSGoogle Scholar
  8. 8.
    Mermin, N.D.: Hidden variables and the two theorems of John Bell. Rev. Mod. Phys. 65, 803–815 (1993)MathSciNetCrossRefADSGoogle Scholar
  9. 9.
    Kernaghan, M.: Bell-Kochen-Specker theorem for 20 vectors. J. Phys. A 27, L829–L830 (1994)MathSciNetCrossRefADSGoogle Scholar
  10. 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)MathSciNetCrossRefzbMATHADSGoogle Scholar
  11. 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)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Pavičić, M., Megill, N.D., Merlet, J.P.: New KochenSpecker sets in four dimensions. Phys. Lett. A 374, 2122–2128 (2010)MathSciNetCrossRefzbMATHADSGoogle Scholar
  13. 13.
    Pavičić, M., Merlet, J.P., McKay, B.D., Megill, N.D.: KochenSpecker vectors. J. Phys. A 38, 1577–1592 (2005)MathSciNetCrossRefzbMATHADSGoogle Scholar
  14. 14.
    Kernaghan, M., Peres, A.: Kochen-Specker theorem for eight-dimensional space. Phys. Lett. A 198, 1–5 (1995)MathSciNetCrossRefzbMATHADSGoogle Scholar
  15. 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)MathSciNetCrossRefGoogle Scholar
  16. 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)MathSciNetCrossRefGoogle Scholar
  17. 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)CrossRefADSGoogle Scholar
  18. 18.
    Lisonĕk, P., Badzia̧g, P., Portillo, J.R., Cabello, A.: Kochen-Specker set with seven contexts. Phys. Rev. A 89(6), 042101 (2014)CrossRefADSGoogle Scholar
  19. 19.
    Cabello, A.: Experimentally testable state-independent quantum contextuality. Phys. Rev. Lett. 101(4), 210401 (2008)MathSciNetCrossRefADSGoogle Scholar
  20. 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)CrossRefADSGoogle Scholar
  21. 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)CrossRefADSGoogle Scholar
  22. 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)CrossRefADSGoogle Scholar
  23. 23.
    Amselem, E., Rådmark, M., Bourennane, M., Cabello, A.: State-independent quantum contextuality with single photons. Phys. Rev. Lett. 103(4), 160405 (2009)CrossRefADSGoogle Scholar
  24. 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)MathSciNetCrossRefADSGoogle Scholar
  25. 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)CrossRefADSGoogle Scholar
  26. 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. 27.
    Cubitt, T.S., Leung, D., Matthews, W., Winter, A.: Improving zero-error classical communication with entanglement. Phys. Rev. Lett. 104(4), 230503 (2010)CrossRefADSGoogle Scholar
  28. 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)CrossRefADSGoogle Scholar
  29. 29.
    Raussendorf, R., Briegel, H.J.: A One-Way Quantum Computer. Phys. Rev. Lett. 86, 5188–5191 (2001)CrossRefADSGoogle Scholar
  30. 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)CrossRefADSGoogle Scholar
  31. 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)Google Scholar
  32. 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

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Physics DepartmentWorcester Polytechnic InstituteWorcesterUSA

Personalised recommendations