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Hardy’s Non-locality Paradox and Possibilistic Conditions for Non-locality

Abstract

Hardy’s non-locality paradox is a proof without inequalities showing that certain non-local correlations violate local realism. It is ‘possibilistic’ in the sense that one only distinguishes between possible outcomes (positive probability) and impossible outcomes (zero probability). Here we show that Hardy’s paradox is quite universal: in any (2,2,l) or (2,k,2) Bell scenario, the occurrence of Hardy’s paradox is a necessary and sufficient condition for possibilistic non-locality. In particular, it subsumes all ladder paradoxes. This universality of Hardy’s paradox is not true more generally: we find a new ‘proof without inequalities’ in the (2,3,3) scenario that can witness non-locality even for correlations that do not display the Hardy paradox. We discuss the ramifications of our results for the computational complexity of recognising possibilistic non-locality.

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References

  1. Bell, J.S.: Physics 1, 195 (1964)

    Google Scholar 

  2. Bell, J.S.: Speakable and Unspeakable in Quantum Mechanics. Cambridge University Press, Cambridge (1987). Collected papers on quantum philosophy

    Google Scholar 

  3. Heywood, P., Redhead, M.: Found. Phys. 13, 481 (1983). doi:10.1007/BF00729511

    MathSciNet  ADS  Article  Google Scholar 

  4. Greenberger, D.M., Horne, M.A., Zeilinger, A.: In: Kafatos, M. (ed.) Bell’s Theorem, Quantum Theory, and Conceptions of the Universe, pp. 69–72. Kluwer Academic, Norwell (1989)

    Google Scholar 

  5. Greenberger, D.M., Horne, M.A., Shimony, A., Zeilinger, A.: Am. J. Phys. 58, 1131 (1990). http://link.aip.org/link/?ajp/58/1131

    MathSciNet  ADS  Article  Google Scholar 

  6. Hardy, L.: Phys. Rev. Lett. 71(11), 1665 (1993). http://link.aps.org/doi/10.1103/PhysRevLett.71.1665

    MathSciNet  ADS  MATH  Article  Google Scholar 

  7. Boschi, D., Branca, S., De Martini, F., Hardy, L.: Phys. Rev. Lett. 79(15), 2755 (1997). http://link.aps.org/doi/10.1103/PhysRevLett.79.2755

    MathSciNet  ADS  MATH  Article  Google Scholar 

  8. Choudhary, S.K., Ghosh, S., Kar, G., Kunkri, S., Rahaman, R., Roy, A.: Quantum Inf. Comput. 10(9–10) (2010)

    Google Scholar 

  9. Cereceda, J.L.: Phys. Lett. A 327(5–6), 433 (2004)

    MathSciNet  ADS  MATH  Article  Google Scholar 

  10. Abramsky, S.: arXiv ePrints (2010). http://arxiv.org/abs/1007.2754

  11. Brandenburger, A., Yanofsky, N.: J. Phys. A, Math. Theor. 41, 425302 (2008). http://iopscience.iop.org/1751-8121/41/42/425302

    MathSciNet  ADS  Article  Google Scholar 

  12. Barbieri, M., Cinelli, C., De Martini, F., Mataloni, P.: Eur. Phys. J. D, At. Mol. Opt. Plasma Phys. 32(2), 261 (2005). http://www.springerlink.com/index/k5y7k27ttjga27un.pdf

    Google Scholar 

  13. Fritz, T.: arXiv ePrints (2010). http://arxiv.org/abs/1006.2497

  14. Popescu, S., Rohrlich, D.: Found. Phys. 24(3), 379 (1994). http://www.springerlink.com/index/J842V3324U512NX0.pdf

    MathSciNet  ADS  Article  Google Scholar 

  15. Zavodny, J.: In preparation

  16. Pitowsky, I.: Math. Program. 50(1), 395 (1991). http://www.springerlink.com/index/njt1m90jtgrw6607.pdf

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

SM would like to thank Samson Abramsky and Rui Soares Barbosa for valuable discussions, and acknowledges financial support from the National University of Ireland Travelling Studentship programme.

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Correspondence to Shane Mansfield.

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Mansfield, S., Fritz, T. Hardy’s Non-locality Paradox and Possibilistic Conditions for Non-locality. Found Phys 42, 709–719 (2012). https://doi.org/10.1007/s10701-012-9640-1

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  • DOI: https://doi.org/10.1007/s10701-012-9640-1

Keywords

  • Non-locality
  • Bell inequality
  • Hardy paradox
  • Possibilistic