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The Kochen-Specker Theorem Based on the Kronecker Delta

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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.

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References

  1. Einstein, A., Podolsky, B., Rosen, N.: Phys. Rev. 47, 777 (1935)

    Article  ADS  Google Scholar 

  2. Redhead, M.: Incompleteness, Nonlocality, and Realism, 2nd edn. Clarendon Press, Oxford (1989)

    MATH  Google Scholar 

  3. Peres, A.: Quantum Theory: Concepts and Methods. Kluwer Academic, Dordrecht (1993)

    MATH  Google Scholar 

  4. Bell, J.S.: Physics 1, 195 (1964)

    Article  Google Scholar 

  5. Kochen, S., Specker, E.P.: J. Math. Mech. 17, 59 (1967)

    MathSciNet  Google Scholar 

  6. Mermin, N.D.: Phys. Rev. Lett. 65, 1838 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  7. Roy, S.M., Singh, V.: Phys. Rev. Lett. 67, 2761 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  8. Ardehali, M.: Phys. Rev. A 46, 5375 (1992)

    Article  ADS  MathSciNet  Google Scholar 

  9. Belinskii, A.V., Klyshko, D.N.: Phys. Usp. 36, 653 (1993)

    Article  ADS  Google Scholar 

  10. Werner, R.F., Wolf, M.M.: Phys. Rev. A 61, 062102 (2000)

    Article  ADS  MathSciNet  Google Scholar 

  11. Żukowski, M.: Phys. Lett. A 177, 290 (1993)

    Article  ADS  MathSciNet  Google Scholar 

  12. Żukowski, M., Kaszlikowski, D.: Phys. Rev. A 56, R1682 (1997)

    Article  ADS  Google Scholar 

  13. Żukowski, M., Brukner, Č.: Phys. Rev. Lett. 88, 210401 (2002)

    Article  MathSciNet  Google Scholar 

  14. Werner, R.F., Wolf, M.M.: Phys. Rev. A 64, 032112 (2001)

    Article  ADS  Google Scholar 

  15. Werner, R.F., Wolf, M.M.: Quantum Inf. Comput. 1, 1 (2001)

    MathSciNet  Google Scholar 

  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)

  17. Greenberger, D.M., Horne, M.A., Shimony, A., Zeilinger, A.: Am. J. Phys. 58, 1131 (1990)

    Article  ADS  Google Scholar 

  18. Pagonis, C., Redhead, M.L.G., Clifton, R.K.: Phys. Lett. A 155, 441 (1991)

    Article  ADS  MathSciNet  Google Scholar 

  19. Mermin, N.D.: Phys. Today 43(6), 9 (1990)

    Article  Google Scholar 

  20. Mermin, N.D.: Am. J. Phys. 58, 731 (1990)

    Article  ADS  Google Scholar 

  21. Peres, A.: Phys. Lett. A 151, 107 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  22. Mermin, N.D.: Phys. Rev. Lett. 65, 3373 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  23. Leggett, A.J.: Found. Phys. 33, 1469 (2003)

    Article  MathSciNet  Google Scholar 

  24. Gröblacher, S., Paterek, T., Kaltenbaek, R., Brukner, Č., Żukowski, M., Aspelmeyer, M., Zeilinger, A.: Nature (London) 446, 871 (2007)

    Article  ADS  Google Scholar 

  25. Paterek, T., Fedrizzi, A., Gröblacher, S., Jennewein, T., Żukowski, M., Aspelmeyer, M., Zeilinger, A.: Phys. Rev. Lett. 99, 210406 (2007)

    Article  ADS  Google Scholar 

  26. Branciard, C., Ling, A., Gisin, N., Kurtsiefer, C., Lamas-Linares, A., Scarani, V.: Phys. Rev. Lett. 99, 210407 (2007)

    Article  ADS  Google Scholar 

  27. Suarez, A.: Found. Phys. 38, 583 (2008)

    Article  ADS  Google Scholar 

  28. Żukowski, M.: Found. Phys. 38, 1070 (2008)

    Article  ADS  Google Scholar 

  29. Suarez, A.: Found. Phys. 39, 156 (2009)

    Article  ADS  MathSciNet  Google Scholar 

  30. Nagata, K., Nakamura, T.: Int. J. Theor. Phys. 48, 3287 (2009)

    Article  Google Scholar 

  31. Nagata, K.: Int. J. Theor. Phys. 48, 3532 (2009)

    Article  Google Scholar 

  32. Nagata, K., Nakamura, T.: Int. J. Theor. Phys. 49, 162 (2010)

    Article  Google Scholar 

  33. Nagata, K.: Eur. Phys. J. D 56, 441 (2010)

    Article  ADS  Google Scholar 

  34. Nagata, K., Nakamura, T., Farouk, A.: Asian J. Math. Phys. 1(1), 15 (2017)

    Google Scholar 

  35. Nagata, K., Nakamura, T.: Phys. J. 1(3), 183 (2015)

    Google Scholar 

  36. Nagata, K.: Phys. Rev. A 72, 012325 (2005)

    Article  ADS  MathSciNet  Google Scholar 

  37. Nagata, K., Nakamura, T.: Int. J. Theor. Phys. 55, 5193 (2016)

    Article  Google Scholar 

  38. Clapham, C., Nicholson, J.: The Concise Oxford Dictionary of Mathematics, 5th edn. Oxford (2014)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, 34141, Korea

    Koji Nagata

  2. Department of Mathematics, Berhampur University, Berhampur-760007, Odisha, India

    Santanu Kumar Patro

  3. Department of Information and Computer Science, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama, 223-8522, Japan

    Tadao Nakamura

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  1. Koji Nagata
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  2. Santanu Kumar Patro
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  3. Tadao Nakamura
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Correspondence to Koji Nagata.

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Nagata, K., Patro, S.K. & Nakamura, T. The Kochen-Specker Theorem Based on the Kronecker Delta. Int J Theor Phys 58, 1311–1314 (2019). https://doi.org/10.1007/s10773-019-04023-9

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  • DOI: https://doi.org/10.1007/s10773-019-04023-9

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