Nonlocality of $W$ and Dicke states subject to losses

We discuss the nonlocality of $W$ and Dicke states subject to losses. We consider two noise models, namely, loss of excitations and loss of particles, and investigate how much loss can be tolerated such that the final state remains nonlocal. This leads to a measure of robustness of the nonlocality of Dicke states, with a clear physical interpretation. Our results suggest that the relation between nonlocality and entanglement of Dicke states is not monotonic.

Figure 1

For $W$ states of $n$ qubits, the graph shows lower bounds on the threshold probability $p_{th}\left(n\right)$ of losing an excitation for the Hardy, WWZB, and MABK inequalities. The best bound is obtained using the WWZB inequality, $p_{th}\left(n\right)\ge 31.4\%$, albeit for small values of $n$. For the Mermin and the Hardy inequalities and large values of $n$, we get $p_{th}\left(n\right)\ge 27.41\%$ and $p_{th}\left(n\right)\ge 18.89\%$, respectively.

Figure 2

Bounds on the threshold probability $p_{th}(100,k)$ of losing an excitation for Dicke states of $n=100$ qubits with $k$ excitations. Here we consider the Hardy inequality ${\mathcal{S}}_{100}$. The most robust Dicke state features only a few excitations, here $k=5$. Note that for $\frac{k}{n}$ approaching 1 the numerics become unstable.

Figure 3

We consider Dicke states of $n$ qubits with $k$ excitations and give bounds $p_{th}(n,k)$ considering the Hardy inequality. The bottommost curve, i.e., $n=100$, corresponds to Fig. 2. Clearly, robustness increases as the number of qubits $n$ increases. For large values of $n$, numerics could be performed only for small $k$.

Figure 4

A comparison of the threshold values $p_{th}(n,k)$ for the Hardy, WWZB, and MABK inequalities. Here we consider the case $n=30$ qubits. Again, the WWZB inequality is optimal, slightly outperforming the MABK inequality. Interestingly, the most robust states, for each inequality, have only a few excitations.

Figure 5

Robustness of Dicke states of $n=200$ qubits with $k$ excitations to loss of particles. We give bounds on the fraction of qubits remaining in the final state $n_f$ for the Hardy inequality.

Figure 6

Robustness of Dicke states of $n=30$ qubits with $k$ excitations to loss of particles. We give bounds on the number of qubits in the final state $n_f$ for the Hardy and the WWZB inequalities.