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International Journal of Theoretical Physics

, Volume 58, Issue 4, pp 1311–1314 | Cite as

The Kochen-Specker Theorem Based on the Kronecker Delta

  • Koji NagataEmail author
  • Santanu Kumar Patro
  • Tadao Nakamura
Article

Abstract

We propose the Kochen-Specker theorem that relies on the properties of the Kronecker delta. We introduce the following value \(\sum _{l = 1}^{m} r_{l}(\langle \sigma _{z}\rangle )= 0\). The notation rl(〈σz〉) means the lth “hidden” outcome of quantum measurements when we would measure the expected value 〈σz〉 = 0 in a thoughtful experiment. Surprisingly, we cannot define the value as zero when we accept the Kronecker delta. We cannot determine the “hidden” results for the expected value.

Keywords

Quantum information theory Quantum measurement theory Quantum computing 

Notes

References

  1. 1.
    Einstein, A., Podolsky, B., Rosen, N.: Phys. Rev. 47, 777 (1935)ADSCrossRefGoogle Scholar
  2. 2.
    Redhead, M.: Incompleteness, Nonlocality, and Realism, 2nd edn. Clarendon Press, Oxford (1989)zbMATHGoogle Scholar
  3. 3.
    Peres, A.: Quantum Theory: Concepts and Methods. Kluwer Academic, Dordrecht (1993)zbMATHGoogle Scholar
  4. 4.
    Bell, J.S.: Physics 1, 195 (1964)CrossRefGoogle Scholar
  5. 5.
    Kochen, S., Specker, E.P.: J. Math. Mech. 17, 59 (1967)MathSciNetGoogle Scholar
  6. 6.
    Mermin, N.D.: Phys. Rev. Lett. 65, 1838 (1990)ADSMathSciNetCrossRefGoogle Scholar
  7. 7.
    Roy, S.M., Singh, V.: Phys. Rev. Lett. 67, 2761 (1991)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Ardehali, M.: Phys. Rev. A 46, 5375 (1992)ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    Belinskii, A.V., Klyshko, D.N.: Phys. Usp. 36, 653 (1993)ADSCrossRefGoogle Scholar
  10. 10.
    Werner, R.F., Wolf, M.M.: Phys. Rev. A 61, 062102 (2000)ADSMathSciNetCrossRefGoogle Scholar
  11. 11.
    Żukowski, M.: Phys. Lett. A 177, 290 (1993)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Żukowski, M., Kaszlikowski, D.: Phys. Rev. A 56, R1682 (1997)ADSCrossRefGoogle Scholar
  13. 13.
    Żukowski, M., Brukner, Č.: Phys. Rev. Lett. 88, 210401 (2002)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Werner, R.F., Wolf, M.M.: Phys. Rev. A 64, 032112 (2001)ADSCrossRefGoogle Scholar
  15. 15.
    Werner, R.F., Wolf, M.M.: Quantum Inf. Comput. 1, 1 (2001)MathSciNetGoogle Scholar
  16. 16.
    Greenberger, D.M., Horne, M.A., Zeilinger, A.: In: Kafatos, M. (ed.) Bell’s Theorem, Quantum Theory and Conceptions of the Universe, pp. 69–72. Kluwer Academic, Dordrecht (1989)Google Scholar
  17. 17.
    Greenberger, D.M., Horne, M.A., Shimony, A., Zeilinger, A.: Am. J. Phys. 58, 1131 (1990)ADSCrossRefGoogle Scholar
  18. 18.
    Pagonis, C., Redhead, M.L.G., Clifton, R.K.: Phys. Lett. A 155, 441 (1991)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    Mermin, N.D.: Phys. Today 43(6), 9 (1990)CrossRefGoogle Scholar
  20. 20.
    Mermin, N.D.: Am. J. Phys. 58, 731 (1990)ADSCrossRefGoogle Scholar
  21. 21.
    Peres, A.: Phys. Lett. A 151, 107 (1990)ADSMathSciNetCrossRefGoogle Scholar
  22. 22.
    Mermin, N.D.: Phys. Rev. Lett. 65, 3373 (1990)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Leggett, A.J.: Found. Phys. 33, 1469 (2003)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Gröblacher, S., Paterek, T., Kaltenbaek, R., Brukner, Č., Żukowski, M., Aspelmeyer, M., Zeilinger, A.: Nature (London) 446, 871 (2007)ADSCrossRefGoogle Scholar
  25. 25.
    Paterek, T., Fedrizzi, A., Gröblacher, S., Jennewein, T., Żukowski, M., Aspelmeyer, M., Zeilinger, A.: Phys. Rev. Lett. 99, 210406 (2007)ADSCrossRefGoogle Scholar
  26. 26.
    Branciard, C., Ling, A., Gisin, N., Kurtsiefer, C., Lamas-Linares, A., Scarani, V.: Phys. Rev. Lett. 99, 210407 (2007)ADSCrossRefGoogle Scholar
  27. 27.
    Suarez, A.: Found. Phys. 38, 583 (2008)ADSCrossRefGoogle Scholar
  28. 28.
    Żukowski, M.: Found. Phys. 38, 1070 (2008)ADSCrossRefGoogle Scholar
  29. 29.
    Suarez, A.: Found. Phys. 39, 156 (2009)ADSMathSciNetCrossRefGoogle Scholar
  30. 30.
    Nagata, K., Nakamura, T.: Int. J. Theor. Phys. 48, 3287 (2009)CrossRefGoogle Scholar
  31. 31.
    Nagata, K.: Int. J. Theor. Phys. 48, 3532 (2009)CrossRefGoogle Scholar
  32. 32.
    Nagata, K., Nakamura, T.: Int. J. Theor. Phys. 49, 162 (2010)CrossRefGoogle Scholar
  33. 33.
    Nagata, K.: Eur. Phys. J. D 56, 441 (2010)ADSCrossRefGoogle Scholar
  34. 34.
    Nagata, K., Nakamura, T., Farouk, A.: Asian J. Math. Phys. 1(1), 15 (2017)Google Scholar
  35. 35.
    Nagata, K., Nakamura, T.: Phys. J. 1(3), 183 (2015)Google Scholar
  36. 36.
    Nagata, K.: Phys. Rev. A 72, 012325 (2005)ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Nagata, K., Nakamura, T.: Int. J. Theor. Phys. 55, 5193 (2016)CrossRefGoogle Scholar
  38. 38.
    Clapham, C., Nicholson, J.: The Concise Oxford Dictionary of Mathematics, 5th edn. Oxford (2014)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Department of PhysicsKorea Advanced Institute of Science and TechnologyDaejeonKorea
  2. 2.Department of MathematicsBerhampur University OdishaIndia
  3. 3.Department of Information and Computer ScienceKeio University YokohamaJapan

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