[01] Koji Nagata, Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon, Korea. [02] Tadao Nakamura, Department of Information and Computer Science, Keio University, Hiyoshi, Kohoku-ku, Yokohama, Japan.

We use the validity of Addition and Multiplication for a hidden variables theory. First, we provide an example that the two operations Addition and Multiplication do not commute with each other as revealed by the analyses that are performed in a finite set of numbers. Our discussion leads to an initial conclusion that Sum rule and Product rule do not commute with each other in a hidden variables theory. If we accept this conclusion, we do not get the Bell-Kochen-Specker paradox. In more detail, quantum mechanics may accept the hidden variables theory. Next, we discuss the validity of operators under an assumption that Sum rule and Product rule commute with each other. In this case, we indeed get the Bell-Kochen-Specker paradox. We got the non-classicality of macroscopic experimental data observed in the Stern-Gerlach experiment and the double-slit experiment. If we detect |↑> and then we detect |↓>, the experiments cannot accept the hidden variables theory. We considered whether we can assign the predetermined “hidden” result to numbers 1 and -1 as in results of measurements with the number of measurements finite (e.g., twice) in the experiments. It turned out that we cannot assign the predetermined hidden result to such results of measurements. The next conclusion indicates interestingly that the Stern-Gerlach experiment cannot accept classical mechanics. The double-slit experiment had led to the same situation, and they were indeed quantum mechanical phenomena.

Quantum Nonlocality, Algebra, Set Theory, Formalism

[01] J. J. Sakurai, Modern Quantum Mechanics (Addison-Wesley Publishing Company, 1995), Revised ed. [02] A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, Dordrecht, The Netherlands, 1993). [03] M. Redhead, Incompleteness, Nonlocality, and Realism (Clarendon Press, Oxford, 1989), 2nd ed. [04] J. von Neumann, Mathematical Foundations of Quantum Mechanics (Princeton University Press, Princeton, New Jersey, 1955). [05] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information (Cambridge University Press, 2000). [06] A. Einstein, B. Podolsky, and N. Rosen, Phys. Rev. 47, 777 (1935). [07] J. S. Bell, Physics 1, 195 (1964). [08] S. Kochen and E. P. Specker, J. Math. Mech. 17, 59 (1967). [09] C. Pagonis, M. L. G. Redhead, and R. K. Clifton, Phys. Lett. A 155, 441 (1991). [10] N. D. Mermin, Phys. Today 43(6), 9 (1990). [11] N. D. Mermin, Am. J. Phys. 58, 731 (1990). [12] A. Peres, Phys. Lett. A 151, 107 (1990). [13] N. D. Mermin, Phys. Rev. Lett. 65, 3373 (1990). [14] N. D. Mermin, Phys. Rev. Lett. 65, 1838 (1990). [15] D. M. Greenberger, M. A. Horne, and A. Zeilinger, in Bell’s Theorem, Quantum Theory and Conceptions of the Universe, edited by M. Kafatos (Kluwer Academic, Dordrecht, The Netherlands, 1989), pp. 69-72. [16] D. M. Greenberger, M. A. Horne, A. Shimony, and A. Zeilinger, Am. J. Phys. 58, 1131 (1990). [17] S. M. Roy and V. Singh, Phys. Rev. Lett. 67, 2761 (1991). [18] M. Ardehali, Phys. Rev. A 46, 5375 (1992). [19] A. V. Belinskii and D. N. Klyshko, Phys. Usp. 36, 653 (1993). [20] R. F. Werner and M. M. Wolf, Phys. Rev. A 61, 062102 (2000). [21] M. Zukowski, Phys. Lett. A 177, 290 (1993). [22] M. Zukowski and D. Kaszlikowski, Phys. Rev. A 56, R1682 (1997). [23] M. Zukowski and C. Brukner, Phys. Rev. Lett. 88, 210401 (2002). [24] R. F. Werner and M. M. Wolf, Phys. Rev. A 64, 032112 (2001). [25] R. F. Werner and M. M. Wolf, Quantum Inf. Comput. 1, 1 (2001). [26] C. Simon, C. Brukner, and A. Zeilinger, Phys. Rev. Lett. 86, 4427 (2001). [27] J.-A. Larsson, Europhys. Lett. 58, 799 (2002). [28] A. Cabello, Phys. Rev. A 65, 052101 (2002). [29] K. Nagata, J. Math. Phys. 46, 102101 (2005). [30] Y. - F Huang, C. -F. Li, Y. -S. Zhang, J. - W. Pan, and G. - C. Guo, Phys. Rev. Lett. 90, 250401 (2003). [31] R. F. Werner, Phys. Rev. A 40, 4277 (1989). [32] A. J. Leggett, Found. Phys. 33, 1469 (2003). [33] S. Groblacher, T. Paterek, R. Kaltenbaek, C. Brukner, M. Zukowski, M. Aspelmeyer, and A. Zeilinger, Nature (London) 446, 871 (2007). [34] T. Paterek, A. Fedrizzi, S. Groblacher, T. Jennewein, M. Zukowski, M. Aspelmeyer, and A. Zeilinger, Phys. Rev. Lett. 99, 210406 (2007). [35] C. Branciard, A. Ling, N. Gisin, C. Kurtsiefer, A. Lamas-Linares, and V. Scarani, Phys. Rev. Lett. 99, 210407 (2007). [36] M. F. Pusey, J. Barrett, and T. Rudolph, Nature Phys. 8, 475 (2012). [37] K. Nagata and T. Nakamura, International Journal of Emerging Engineering Research and Technology, Volume 3, Issue 6, 78 (2015).