A simple proof of the converse of Hardy's theorem

TL;DR

This paper presents a straightforward proof that, for two spin-1/2 particles, a unique state exhibits Hardy-type nonlocality, providing explicit probability expressions and correcting a previous misconception in the literature.

Contribution

It offers a simple proof of the uniqueness of Hardy-type nonlocality states and corrects an error in Mermin's earlier proof.

Findings

Identifies a unique Hardy-type nonlocality state for two spin-1/2 particles.

Provides explicit probability formulas for observing nonlocality.

Corrects a mistake in Mermin's previous proof.

Abstract

In this paper we provide a simple proof of the fact that for a system of two spin-1/2 particles, and for a choice of observables, there is a unique state which shows Hardy-type nonlocality. Moreover, an explicit expression for the probability that an ensemble of particle pairs prepared in such a state exhibits a Hardy-type nonlocality contradiction is given in terms of two independent parameters related to the observables involved. Incidentally, a wrong statement expressed in Mermin's proof of the converse [N.D. Mermin, Am. J. Phys. 62, 880 (1994)] is pointed out.