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 JAMP  Vol.3 No.7 , July 2015
Analysis of Bell-Type Experiments and Its Local Realism
Abstract: 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.
Cite this paper: 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.
References

[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. http://dx.doi.org/10.1103/PhysRev.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. http://dx.doi.org/10.1512/iumj.1968.17.17004

[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. http://dx.doi.org/10.1007/978-94-017-0849-4_10

[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. http://dx.doi.org/10.1119/1.16243

[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. http://dx.doi.org/10.1016/0375-9601(91)90643-M

[12]   Mermin, N.D. (1990) What’s Wrong with These Elements of Reality? Physics Today, 43, 9.
http://dx.doi.org/10.1063/1.2810588

[13]   Mermin, N.D. (1990) Quantum Mysteries Revisited. American Journal of Physics, 58, 731.
http://dx.doi.org/10.1119/1.16503

[14]   Peres, A. (1990) Incompatible Results of Quantum Measurements. Physics Letters A, 151, 107-108.
http://dx.doi.org/10.1016/0375-9601(90)90172-K

[15]   Mermin, N.D. (1990) Simple Unified Form for the Major No-Hidden-Variables Theorems. Physical Review Letters, 65, 3373. http://dx.doi.org/10.1103/PhysRevLett.65.3373

[16]   Mermin, N.D. (1990) Extreme Quantum Entanglement in a Superposition of Macroscopically Distinct States. Physical Review Letters, 65, 1838. http://dx.doi.org/10.1103/PhysRevLett.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. http://dx.doi.org/10.1103/PhysRevLett.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. http://dx.doi.org/10.1103/PhysRevA.46.5375

[19]   Belinskii, A.V. and Klyshko, D.N. (1993) Interference of Light and Bell’s Theorem. Physics-Uspekhi, 36, 653. http://dx.doi.org/10.1070/PU1993v036n08ABEH002299

[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. http://dx.doi.org/10.1103/PhysRevA.61.062102

[21]   Zukowski, M. (1993) Bell Theorem Involving All Settings of Measuring Apparatus. Physics Letters A, 177, 290-296. http://dx.doi.org/10.1016/0375-9601(93)90002-H

[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.
http://dx.doi.org/10.1103/PhysRevA.56.R1682

[23]   Zukowski, M. and Brukner, C. (2002) Bell’s Theorem for General N-Qubit States. Physical Review Letters, 88, Article ID: 210401. http://dx.doi.org/10.1103/PhysRevLett.88.210401

[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.
http://dx.doi.org/10.1103/PhysRevA.64.032112

[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. http://dx.doi.org/10.1103/PhysRevLett.86.4427

[27]   Larsson, J.-Å. (2002) A Kochen-Specker Inequality. Europhysics Letters, 58, 799.
http://dx.doi.org/10.1209/epl/i2002-00444-0

[28]   Cabello, A. (2002) Finite-Precision Measurement Does Not Nullify the Kochen-Specker Theorem. Physical Review A, 65, Article ID: 052101. http://dx.doi.org/10.1103/PhysRevA.65.052101

[29]   Nagata, K. and Math. J. (2005) Inequalities for Experimental Tests of the Kochen-Specker Theorem. Journal of Mathematical Physics, 46, Article ID: 102101. http://dx.doi.org/10.1063/1.2081115

[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.
http://dx.doi.org/10.1103/PhysRevLett.90.250401

[31]   Werner, R.F. (1989) Quantum States with Einstein-Podolsky-Rosen Correlations Admitting a Hidden-Variable Model. Physical Review A, 40, 4277. http://dx.doi.org/10.1103/PhysRevA.40.4277

[32]   Leggett, A.J. (2003) Nonlocal Hidden-Variable Theories and Quantum Mechanics: An Incompatibility Theorem. Foundations of Physics, 33, 1469-1493. http://dx.doi.org/10.1023/A:1026096313729

[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.
http://dx.doi.org/10.1038/nature05677

[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. http://dx.doi.org/10.1103/PhysRevLett.99.210406

[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.
http://dx.doi.org/10.1103/PhysRevLett.99.210407

[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.
http://dx.doi.org/10.1007/s10773-009-0189-5

[37]   Nagata, K. and Nakamura, T. (2013) An Additional Condition for Bell Experiments for Accepting Local Realistic Theories. Quantum Information Processing, 12, 3785-3789.
http://dx.doi.org/10.1007/s11128-013-0635-4

 
 
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