The relation between Hardy's non-locality and violation of Bell inequality

TL;DR

This paper establishes a quantitative relationship between Hardy's non-locality and Bell inequality violations, showing how different physical principles constrain non-local correlations and deriving bounds consistent with known quantum limits.

Contribution

It analytically relates Hardy's non-locality to Bell operator violations and derives bounds under information causality and no-signaling principles.

Findings

Hardy's non-locality is sufficient for Bell inequality violation.

Upper bounds of Hardy's non-locality match quantum and no-signaling limits.

Derived bounds align with Tsirelson's and algebraic maximum bounds.

Abstract

We give a analytic quantitative relation between Hardy's non-locality and Bell operator. We find that Hardy's non-locality is a sufficient condition for violation of Bell inequality, the upper bound of Hardy's non-locality allowed by information causality just correspond to Tsirelson bound of Bell inequality, and the upper bound of Hardy's non-locality allowed by the principle of no-signaling just correspond to the algebraic maximum of Bell operator. Then we study the Cabello's argument of Hardy's non-locality (a generalization of Hardy's argument) and find a similar relation between it and violation of Bell inequality. Finally, we give a simple derivation of the bound of Hardy's non-locality under the constraint of information causality with the aid of above derived relation between Hardy's non-locality and Bell operator, this bound is the main result of a recent work of Ahanj \emph{et al.} [Phys. Rev. A {\bf81}, 032103(2010)].