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Journal of Mathematical Physics / Volume 51 / Issue 7 / ARTICLES / Quantum Mechanics (General and Nonrelativistic)

J. Math. Phys. 51, 072105 (2010); http://dx.doi.org/10.1063/1.3447736 (16 pages)

Maximal violation of Bell inequalities by position measurements

J. Kiukas and R. F. Werner

Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstr. 2, 30167 Hannover, Germany

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(Received 23 December 2009; accepted 14 May 2010; published online 15 July 2010)

We show that it is possible to find maximal violations of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality using only position measurements on a pair of entangled nonrelativistic free particles. The device settings required in the CHSH inequality are done by choosing one of two times at which position is measured. For different assignments of the “+” outcome to positions, namely, to an interval, to a half-line, or to a periodic set, we determine violations of the inequalities and states where they are attained. These results have consequences for the hidden variable theories of Bohm and Nelson, in which the two-time correlations between distant particle trajectories have a joint distribution, and hence cannot violate any Bell inequality.

© 2010 American Institute of Physics

Article Outline

  1. INTRODUCTION
  2. THE BOHM–NELSON THEORY
  3. GENERAL STRUCTURE OF CHSH VIOLATIONS
    1. The algebra generated by two projections
    2. Attained maximal violations
  4. POSITION MEASUREMENTS AT DIFFERENT TIMES
    1. Compact intervals: Partially commutative case
    2. Half-lines: Totally noncommutative case
    3. Periodic sets: Commutative case

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KEYWORDS and PACS

Keywords

Bell theorem, quantum entanglement

PACS

  • 03.65.Ud

    Entanglement and quantum nonlocality (e.g. EPR paradox, Bell's inequalities, GHZ states, etc.)

  • 03.65.Fd

    Algebraic methods

ARTICLE DATA

Digital Object Identifier
http://dx.doi.org/10.1063/1.3447736

PUBLICATION DATA

ISSN

0022-2488 (print)  
1089-7658 (online)

Publisher

$abstract.society.value is a Member of CrossRef American Institute of Physics

For access to fully linked references, you need to log in.
    Blanchard, P., Golin, S., and Serva, M., “Repeated measurements in stochastic mechanics,” Phys. Rev. D 34, 3732 (1986).

    Bohm,D., “A suggested interpretation of the quantum theory in terms of hidden variables. I,” Phys. Rev. 85, 166 (1952), “A suggested interpretation of the quantum theory in terms of hidden variables. II,” ibid. 85, 180 (1952).

    Borac, S., “On the algebra generated by two projections,” J. Math. Phys. 36, 863 (1995)JMAPAQ000036000002000863000001.

    Busch, P., Schonbek, T. P., and Schroeck, J. F. E., “Quantum observables: Compatibility versus commutativity and maximal information,” J. Math. Phys. 28, 2866 (1987)JMAPAQ000028000012002866000001.

    Wenger, J., Hafezi, M., Grosshans, G., Tualle-Brouri, R., and Grangier, P., “Maximal violation of Bell inequalities using continuous-variable measurements,” Phys. Rev. A 67, 012105 (2003).

    Werner, R. F., “A generalization of stochastic mechanics and its relation to quantum mechanics,” Phys. Rev. D 34, 463 (1986).


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