Foundations of Physics

, Volume 42, Issue 5, pp 709–719 | Cite as

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

  • Shane MansfieldEmail author
  • Tobias Fritz


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.


Non-locality Bell inequality Hardy paradox Possibilistic 



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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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of OxfordOxfordUK
  2. 2.ICFO—Institut de Ciènces FotòniquesCastelldefelsSpain

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