Consequences and applications of the completeness of Hardy's nonlocality

Logical nonlocality is completely characterized by Hardy's “paradox” in $(2,2,l)$ and $(2,k,2)$ scenarios. We consider a variety of consequences and applications of this fact. (i) Polynomial algorithms may be given for deciding logical nonlocality in these scenarios. (ii) Bell states are the only entangled two-qubit states which are not logically nonlocal under projective measurements. (iii) It is possible to witness Hardy nonlocality with certainty in a simple tripartite quantum system. (iv) Noncommutativity of observables is necessary and sufficient for enabling logical nonlocality.

Figure 1

$n=3$ Hardy paradox. The blue entry (upper-left subcube) corresponds to a possible joint outcome, and the red entries (other subcubes) to impossible ones. The blank entries are not relevant, and may take either of the values.

Figure 2

(a) GHZ model. We represent only the red, impossible outcomes; all other entries are possible. (b) Hardy's paradox within the GHZ model; the blue outcome is possible.