Quantifying Multipartite Nonlocality

The nonlocal correlations of multipartite entangled states can be reproduced by a classical model if sufficiently many parties join together or if sufficiently many parties broadcast their measurement inputs. The maximal number $m$ of groups and the minimal number $k$ of broadcasting parties that allow for the reproduction of a given set of correlations quantify their multipartite nonlocal content. We show how upper bounds on $m$ and lower bounds on $k$ can be computed from the violation of the Mermin-Svetlichny inequalities. While $n$-partite Greenberger-Horne-Zeilinger states violate these inequalities maximally, we find that $W$ states violate them only by a very small amount.

Figure 1

Different groupings of $n=4$ parties into $m$ groups. Within each group, every party can communicate to any other party, as indicated by the arrows. (a) If all parties join into one group ($m=1$), they can achieve any correlations. (b),(c) If they split into $m=2$ groups, they can realize some nonlocal correlations but not all. (d) If they are all separated ($m=n$), they can only reproduce local correlations.

Figure 2

Maximal values of $M_n$ and $M_n^{+}$ for partially entangled GHZ states for $3\le n\le 6$. The dots are values found by numerical optimization, and the solid lines are the conjectured violation $M_n=M_n^{+}=2^{(n-1)/2}\sin 2\theta$ (valid only above 1 for $M_n$ and $\sqrt{2}$ for $M_n^{+}$).

Figure 3

Maximal values of $M_n$ (solid line) and $M_n^{+}$ (dashed line) for $n$-partite $W$ states. The curves were obtained by a general numerical optimization for $n\le 9$ and under the hypothesis that all parties use identical measurement settings for $10\le n\le 19$. The asymptotic values for $n\to \infty$ computed as explained in the text are also shown.