Logical Difficulty from Combining Counterfactuals in the GHZ-Bell Theorems

Author(s)
Louis Sica

Affiliation(s)

Institute for Quantum Studies, Chapman University, Orange, CA & Burtonsville, MD, USA Inspire Institute Inc., Alexandria, VA, USA.

Institute for Quantum Studies, Chapman University, Orange, CA & Burtonsville, MD, USA Inspire Institute Inc., Alexandria, VA, USA.

ABSTRACT

In eliminating the fair sampling assumption, the
Greenberger, Horne, Zeilinger (GHZ) theorem is believed to confirm Bell’s
historic conclusion that local hidden variables are inconsistent with the
results of quantum mechanics. The GHZ theorem depends on predicting the results
of sets of measurements of which only one may be performed. In the present
paper, the noncommutative aspects of these unperformed measurements are
critically examined. Classical examples and the logic of the GHZ construction
are analyzed to demonstrate that combined counterfactual results of
noncommuting operations are in general logically inconsistent with performed
measurement sequences whose results depend on noncommutation. The
Bell theorem is also revisited in the light of this result. It is concluded
that negative conclusions regarding local hidden variables do not follow
from the GHZ and

KEYWORDS

HZ-Theorem; Bell-Theorem; Noncommutation; Counterfactual; Hidden Variables; Locality; Nonlocality

HZ-Theorem; Bell-Theorem; Noncommutation; Counterfactual; Hidden Variables; Locality; Nonlocality

Cite this paper

L. Sica, "Logical Difficulty from Combining Counterfactuals in the GHZ-Bell Theorems,"*Applied Mathematics*, Vol. 4 No. 10, 2013, pp. 90-94. doi: 10.4236/am.2013.410A3012.

L. Sica, "Logical Difficulty from Combining Counterfactuals in the GHZ-Bell Theorems,"

References

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http://dx.doi.org/10.1007/978-1-4757-9808-1

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http://dx.doi.org/10.1007/978-3-540-70626-7_40

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[10] L. Sica, “Bell’s Inequalities I: An Explanation for Their Experimental Violation,” Optics Communications, Vol. 170, No. 1-3, 1999, pp. 55-60.

http://dx.doi.org/10.1016/S0030-4018(99)00417-4

[11] L. Sica, “Bell’s Inequalities II: Logical Loophole in Their Interpretation,” Optics Communications, Vol. 170, No. 1-3, 1999, pp. 61-66.

http://dx.doi.org/10.1016/S0030-4018(99)00418-6

[12] L. Sica, “Correlations for a New Bell’s Inequality Experiment,” Foundations of Physics Letters, Vol. 15, No. 5, 2002, pp. 473-486.

http://dx.doi.org/10.1023/A:1023920230595

[13] L. Sica, “Bell’s Inequality Violation Due to Misidentification of Spatially Non-Stationary Random Processes,” Journal of Modern Optics, Vol. 50, No. 15-17, 2003, pp. 2465-2474.

[14] J. Malley, “All Quantum Observables in a Hidden-Variable Model Must Commute Simultaneously,” Physical Review A, Vol. 69, No. 2, 2004, pp. 022118-1-022118-3.

http://dx.doi.org/10.1103/PhysRevA.69.022118

[15] A. Papoulis and S. U. Pillai, “Probability, Random Variables, and Stochastic Processes,” McGraw-Hill, New York, 2002, p. 387.

[16] J. S. Bell, “Speakable and Unspeakable in Quantum Mechanics,” Cambridge University Press, Cambridge, 1987, p. 65.

[17] L. Sica, “Logical Difficulty from Combining Counterfactuals in the GHZ-Bell Theorems,” 2013.

http://meetings.aps.org/link/BAPS.2013.MAR.R26.10

[18] L. Sica, “Logical Difficulty from Combining Counterfactuals in the GHZ-Bell Theorems,” arxiv:quant-ph/1202. 0841.

[1] D. M. Greenberger, M. Horne and A. Zeilinger, “Going beyond Bell’s Theorem,” In: M. Kafatos, Ed., Bell’s Theorem, Quantum Theory and Conceptions of the Universe, Kluwer, Dordrecht, 2010, pp. 69-72.

[2] N. David Mermin, “Simple Unified Form for the Major No-Hidden-Variables Theorems,” Physical Review Letters, Vol. 65, No. 27, 1990, pp. 3373-3376.

http://dx.doi.org/10.1103/PhysRevLett.65.3373

[3] D. Home, “Conceptual Foundations of Quantum Physics,” Plenum Press, New York, 1997, p. 234.

http://dx.doi.org/10.1007/978-1-4757-9808-1

[4] A. Afriat and F. Selleri, “The Einstein, Podolsky, and Rosen Paradox,” Plenum Press, New York, 1999, p. 121.

http://dx.doi.org/10.1007/978-1-4899-0254-2

[5] D. M. Greenberger, “GHZ (Greenberger-Horne-Zeilinger) Theorem and GHZ States,” In: D. Greenberger, K. Hentschel and F. Weinert, Eds., Compendium of Quantum Physics, Springer, Dordrecht, Heidelberg, London, New York, 2009, pp. 258-263.

[6] L. Vaidman, “Counterfactuals in Quantum Mechanics,” In: D. Greenberger, K. Hentschel and F. Weinert, Eds., Compendium of Quantum Physics, Springer, Dordrecht, Heidelberg, London, New York, 2009, pp. 132-136.

http://dx.doi.org/10.1007/978-3-540-70626-7_40

[7] H. Goldstein, “Classical Mechanics,” Addison-Wesley, Reading, 1980, p. 148.

[8] R. B. Griffiths, “Consistent Histories,” In: D. Greenberger, K. Hentschel and F. Weinert, Eds., Compendium of Quantum Physics, Springer, Dordrecht, Heidelberg, London, New York, 2009, pp. 117-122.

[9] F. Mandl, “Quantum Mechanics,” John Wiley and Sons, Chichester, England, 1992, Chapter 5.

[10] L. Sica, “Bell’s Inequalities I: An Explanation for Their Experimental Violation,” Optics Communications, Vol. 170, No. 1-3, 1999, pp. 55-60.

http://dx.doi.org/10.1016/S0030-4018(99)00417-4

[11] L. Sica, “Bell’s Inequalities II: Logical Loophole in Their Interpretation,” Optics Communications, Vol. 170, No. 1-3, 1999, pp. 61-66.

http://dx.doi.org/10.1016/S0030-4018(99)00418-6

[12] L. Sica, “Correlations for a New Bell’s Inequality Experiment,” Foundations of Physics Letters, Vol. 15, No. 5, 2002, pp. 473-486.

http://dx.doi.org/10.1023/A:1023920230595

[13] L. Sica, “Bell’s Inequality Violation Due to Misidentification of Spatially Non-Stationary Random Processes,” Journal of Modern Optics, Vol. 50, No. 15-17, 2003, pp. 2465-2474.

[14] J. Malley, “All Quantum Observables in a Hidden-Variable Model Must Commute Simultaneously,” Physical Review A, Vol. 69, No. 2, 2004, pp. 022118-1-022118-3.

http://dx.doi.org/10.1103/PhysRevA.69.022118

[15] A. Papoulis and S. U. Pillai, “Probability, Random Variables, and Stochastic Processes,” McGraw-Hill, New York, 2002, p. 387.

[16] J. S. Bell, “Speakable and Unspeakable in Quantum Mechanics,” Cambridge University Press, Cambridge, 1987, p. 65.

[17] L. Sica, “Logical Difficulty from Combining Counterfactuals in the GHZ-Bell Theorems,” 2013.

http://meetings.aps.org/link/BAPS.2013.MAR.R26.10

[18] L. Sica, “Logical Difficulty from Combining Counterfactuals in the GHZ-Bell Theorems,” arxiv:quant-ph/1202. 0841.