Analysis of Bell-Type Experiments and Its Local Realism


We investigate the violation factor of the original Bell-Mermin inequality. Until now, we have used an assumption that the results of measurement are . In this case, the maximum violation factor is as follows: and . The quantum predictions by n-partite Greenberger-Horne-Zeilinger state violate the Bell-Mermin inequality by an amount that grows exponentially with n. Recently, a new measurement theory is proposed [K. Nagata and T. Nakamura, International Journal of Theoretical Physics, 49, 162 (2010)]. The values of measurement outcome are . Here we use the new measurement theory. We consider a multipartite GHZ state. We use the original Bell-Mermin inequality. It turns out that the original Bell-Mermin inequality is satisfied irrespective of the number of particles. In this case, the maximum violation factor is as follows: and . Thus the original Bell-Mermin inequality is satisfied by the new measurement theory. We propose the following conjecture: All the two-orthogonal-settings experimental correlation functions admit local realistic theories irrespective of a state if we use the new measurement theory.

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Nagata, K. and Nakamura, T. (2015) Analysis of Bell-Type Experiments and Its Local Realism. Journal of Applied Mathematics and Physics, 3, 898-902. doi: 10.4236/jamp.2015.37109.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Sakurai, J.J. (1995) Modern Quantum Mechanics. Addison-Wesley Publishing Company, Revised Edition.
[2] Peres, A. (1993) Quantum Theory: Concepts and Methods. Kluwer Academic, Dordrecht.
[3] Redhead, M. (1989) Incompleteness, Nonlocality, and Realism. 2nd Edition, Clarendon Press, Oxford.
[4] von Neumann, J. (1955) Mathematical Foundations of Quantum Mechanics. Princeton University Press, Princeton.
[5] Nielsen, M.A. and Chuang, I.L. (2000) Quantum Computation and Quantum Information. Cambridge University Press, Cambridge.
[6] Einstein, A., Podolsky, B. and Rosen, N. (1935) Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? Physical Review, 47, 777.
[7] Bell, J.S. (1964) On the Einstein Podolsky Rosen Paradox. Physics, 1, 195-200.
[8] Kochen, S. and Specker, E.P. (1967) The Problem of Hidden Variables in Quantum Mechanics. Journal of Mathematics and Mechanics, 17, 59-87.
[9] Greenberger, D.M., Horne, M.A. and Zeilinger, A. (1989) Going Beyond Bell’s Theorem. In: Kafatos, M., Ed., Bell’s Theorem, Quantum Theory and Conceptions of the Universe, Kluwer Academic, Dordrecht, 69-72.
[10] Greenberger, D.M., Horne, M.A., Shimony, A. and Zeilinger, A. (1990) Bell’s Theorem without Inequalities. American Journal of Physics, 58, 1131-1143.
[11] Pagonis, C., Redhead, M.L.G. and Clifton, R.K. (1991) The Breakdown of Quantum Non-Locality in the Classical Limit. Physics Letters A, 155, 441-444.
[12] Mermin, N.D. (1990) What’s Wrong with These Elements of Reality? Physics Today, 43, 9.
[13] Mermin, N.D. (1990) Quantum Mysteries Revisited. American Journal of Physics, 58, 731.
[14] Peres, A. (1990) Incompatible Results of Quantum Measurements. Physics Letters A, 151, 107-108.
[15] Mermin, N.D. (1990) Simple Unified Form for the Major No-Hidden-Variables Theorems. Physical Review Letters, 65, 3373.
[16] Mermin, N.D. (1990) Extreme Quantum Entanglement in a Superposition of Macroscopically Distinct States. Physical Review Letters, 65, 1838.
[17] Roy, S.M. and Singh, V. (1991) Tests of Signal Locality and Einstein-Bell Locality for Multiparticle Systems. Physical Review Letters, 67, 2761.
[18] Ardehali, M. (1992) Bell Inequalities with a Magnitude of Violation That Grows Exponentially with the Number of Particles. Physical Review A, 46, 5375.
[19] Belinskii, A.V. and Klyshko, D.N. (1993) Interference of Light and Bell’s Theorem. Physics-Uspekhi, 36, 653.
[20] Werner, R.F. and Wolf, M.M. (2000) Bell’s Inequalities for States with Positive Partial Transpose. Physical Review A, 61, Article ID: 062102.
[21] Zukowski, M. (1993) Bell Theorem Involving All Settings of Measuring Apparatus. Physics Letters A, 177, 290-296.
[22] Zukowski, M. and Kaszlikowski, D. (1997) Critical Visibility for N-Particle Greenberger-Horne-Zeilinger Correlations to Violate Local Realism. Physical Review A, 56, R1682.
[23] Zukowski, M. and Brukner, C. (2002) Bell’s Theorem for General N-Qubit States. Physical Review Letters, 88, Article ID: 210401.
[24] Werner, R.F. and Wolf, M.M. (2001) All-Multipartite Bell-Correlation Inequalities for Two Dichotomic Observables Per Site. Physical Review A, 64, Article ID: 032112.
[25] Werner, R.F. and Wolf, M.M. (2001) Bell Inequalities and Entanglement. Quantum Information & Computation, 1, 1-25.
[26] Simon, C., Brukner, C. and Zeilinger, A. (2001) Hidden-Variable Theorems for Real Experiments. Physical Review Letters, 86, 4427.
[27] Larsson, J.-Å. (2002) A Kochen-Specker Inequality. Europhysics Letters, 58, 799.
[28] Cabello, A. (2002) Finite-Precision Measurement Does Not Nullify the Kochen-Specker Theorem. Physical Review A, 65, Article ID: 052101.
[29] Nagata, K. and Math. J. (2005) Inequalities for Experimental Tests of the Kochen-Specker Theorem. Journal of Mathematical Physics, 46, Article ID: 102101.
[30] Huang, Y.F., Li, C.F., Zhang, Y.S., Pan, J.W. and Guo, G.C. (2003) Experimental Test of the Kochen-Specker Theorem with Single Photons. Physical Review Letters, 90, Article ID: 250401.
[31] Werner, R.F. (1989) Quantum States with Einstein-Podolsky-Rosen Correlations Admitting a Hidden-Variable Model. Physical Review A, 40, 4277.
[32] Leggett, A.J. (2003) Nonlocal Hidden-Variable Theories and Quantum Mechanics: An Incompatibility Theorem. Foundations of Physics, 33, 1469-1493.
[33] Gröblacher, S., Paterek, T., Kaltenbaek, R., Brukner, C., Zukowski, M., Aspelmeyer, M. and Zeilinger, A. (2007) An Experimental Test of Non-Local Realism. Nature, 446, 871-875.
[34] Paterek, T., Fedrizzi, A., Gröblacher, S., Jennewein, T., Zukowski, M., Aspelmeyer, M. and Zeilinger, A. (2007) Experimental Test of Nonlocal Realistic Theories without the Rotational Symmetry Assumption. Physical Review Letters, 99, Article ID: 210406.
[35] Branciard, C., Ling, A., Gisin, N., Kurtsiefer, C., Lamas-Linares, A. and Scarani, V. (2007) Experimental Falsification of Leggett’s Nonlocal Variable Model. Physical Review Letters, 99, Article ID: 210407.
[36] Nagata, K. and Nakamura, T. (2010) Can von Neumann’s Theory Meet the Deutsch-Jozsa Algorithm? International Journal of Theoretical Physics, 49, 162-170.
[37] Nagata, K. and Nakamura, T. (2013) An Additional Condition for Bell Experiments for Accepting Local Realistic Theories. Quantum Information Processing, 12, 3785-3789.

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