Hardy is (almost) everywhere: nonlocality without inequalities for almost all entangled multipartite states

TL;DR

This paper demonstrates that almost all entangled multipartite qubit states exhibit Hardy-type nonlocality proofs without inequalities, providing a constructive method and an efficient algorithm to identify local observables that reveal this nonlocality.

Contribution

The authors establish that nearly all multipartite entangled states admit Hardy-type nonlocality proofs and provide a constructive, algorithmic approach to find the necessary local observables.

Findings

All entangled states except certain tensor products admit Hardy-type proofs.

A simple algorithm can produce the witnessing observables for nonlocality.

Only n+2 local observables are needed to demonstrate nonlocality for n-qubit states.

Abstract

We show that all $n$-qubit entangled states, with the exception of tensor products of single-qubit and bipartite maximally-entangled states, admit Hardy-type proofs of non-locality without inequalities or probabilities. More precisely, we show that for all such states, there are local, one-qubit observables such that the resulting probability tables are logically contextual in the sense of Abramsky and Brandenburger, this being the general form of the Hardy-type property. Moreover, our proof is constructive: given a state, we show how to produce the witnessing local observables. In fact, we give an algorithm to do this. Although the algorithm is reasonably straightforward, its proof of correctness is non-trivial. A further striking feature is that we show that $n+2$ local observables suffice to witness the logical contextuality of any $n$-qubit state: two each for two for the parties, and one each for the remaining $n-2$ parties.