Bounding the Persistency of the Nonlocality of W States

The nonlocal properties of the W states are investigated under particle loss. By removing all but two particles from an N-qubit W state, the resulting two-qubit state is still entangled. Hence, the W state has high persistency of entanglement. We ask an analogous question regarding the persistency of nonlocality [see N. Brunner and T. Vertesi, Phys. Rev. A 86, 042113 (2012)]. Namely, we inquire what is the minimal number of particles that must be removed from the W state so that the resulting state becomes local. We bound this value in function of N qubits by considering Bell nonlocality tests with two alternative settings per site. In particular, we find that this value is between 2N/5 and N/2 for large N. We also develop a framework to establish bounds for more than two settings per site.