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. Vértesi, 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.

Figure 1

A schematic view of the Bell polytope (region L) and the space attainable with quantum systems (region Q). The correlation point $P_1$ corresponding to the $W_N$ state is outside the Bell polytope, whereas the point $P_0$ corresponding to the product state falls inside the Bell polytope. $p_{\mathrm{crit}}$ is fully determined by points $P_0,P_1$ and the facets of the polytope.

Figure 2

For $\rho (n,p)$ states of $n$ qubits (7), the graph shows lower bounds on the threshold probability $p_{\mathrm{crit}}$ obtained using our numerical method up to $m=6$ measurement settings.

Figure 3

For $\rho (n,p)$ states of $n$ qubits (7), the graph shows lower bounds on the threshold probability $p_{\mathrm{crit}}$ obtained using our numerical method for $m=2$ measurement settings. These bounds coincide with the bounds coming from our class of Bell inequalities for two measurement settings up to $n=14$ and we conjecture to be the same for any even $n$ particles. We show results for our Bell inequalities with optimized and nonoptimized settings (in the latter case, the two observables are the Pauli operators $Z$ and $X$). The lower curve reproduces the results of Ref. [28] in case of the WWWZB inequalities [43].

Figure 4

For $W$ states of $N$ qubits, the graph shows upper bounds (marked by a triangle pointing down) and lower bounds (marked by triangles pointing up) on the persistency of nonlocality $P_{NL}^{m=2}\left(W_N\right)$.