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

TL;DR

This paper demonstrates the universality of Hardy's non-locality paradox across various Bell scenarios, linking it to possibilistic non-locality and exploring its limitations and implications for computational complexity.

Contribution

It establishes that Hardy's paradox is necessary and sufficient for possibilistic non-locality in certain scenarios and introduces a new paradox in the (2,3,3) case.

Findings

Hardy's paradox is universal in (2,2,l) and (2,k,2) scenarios.

A new proof without inequalities is found for the (2,3,3) scenario.

Results impact understanding of the complexity of recognizing non-local correlations.

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 occurence 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.