Hardy’s proof of nonlocality in the presence of noise

We extend the validity of Hardy’s nonlocality without inequalities proof to cover the case of special one-parameter classes of nonpure statistical operators. These mixed states are obtained by mixing the Hardy states with a completely chaotic noise or with a colored noise and they represent a realistic description of imperfect preparation processes of (pure) Hardy states in nonlocality experiments. Within such a framework we are able to exhibit a precise range of values of the parameter measuring the noise affecting the nonoptimal preparation of an arbitrary Hardy state, for which it is still possible to put into evidence genuine nonlocal effects. Equivalently, our work exhibits particular classes of bipartite mixed states whose constituents do not admit any local and deterministic hidden variable model reproducing the quantum mechanical predictions.

Figure 1

Values of $1-p$ versus $p_1$, for $d_1{d}_2=6$ and $p_2=1∕\sqrt{3}$ (left figure) and for $d_1{d}_2=12$ and $p_2=1∕\sqrt{2}$, with the constraint that $p_1^2+p_2^2⩽1$.