Complementarity of genuine multipartite Bell nonlocality

We introduce a feature of no-signaling (Bell) nonlocal theories: namely, when a system of multiple parties manifests genuine nonlocal correlation, then there cannot be arbitrarily high nonlocal correlation among any subset of the parties. We call this feature complementarity of genuine multipartite nonlocality. We use Svetlichny's criterion for genuine multipartite nonlocality and nonlocal games to derive the complementarity relations under no-signaling constraints. We find that the complementarity relations are tightened for the much stricter quantum constraints. We compare this notion with the well-known notion of monogamy of nonlocality. As a consequence, we obtain tighter nontrivial monogamy relations that take into account genuine multipartite nonlocality. Furthermore, we provide numerical evidence showcasing this feature using a bipartite measure and several other well-known tripartite measures of nonlocality.

Figure 1

Plot of maximum achievable winning probability in a particular $S_n^j$ game, ${max}_{\mathcal{NS}}\left\{p\left(S_n^j\right)\right\}$, against the observed winning probability in another $S_n^i$ game, $p\left(S_n^i\right)$ (when $S_n^i$ and $S_n^j$ are not trivially complementary). Solid blue line, maximization over no-signaling probability distributions; dashed orange curve, quantum probabilities.

Figure 2

Plot of maximum achievable winning probability in a particular $S_k^j$ game, $max\left\{p\left(S_k^j\right)\right\}$, against the observed winning probability in an $S_n^i$ game, $p\left(S_n^i\right)$ [when $S_k^j\equiv S_n^i|({\mathcal{A}}_{n-k}=a,{\mathcal{B}}_{n-k}=b)$ for some $a,b\in \{0,1\}]$. Blue curve, maximization over no-signaling probability distributions; orange curve, quantum probabilities.

Figure 3

The proof sketch for Theorem 2. It utilizes Property 3, Property 4, and the relation presented in Theorem 1.

Figure 4

Extremal form of (a) monogamy of nonlocality versus (b) complementarity of genuine nonlocality in a bipartite and tripartite scenario.

Figure 5

Plot of maximum achievable value of the sum of winning probabilities of an $S_2$ game played between $A_1,A_2$ and $A_1,A_3$, $max\{p_{A_1{A}_2}\left(S_2\right)+p_{A_1{A}_3}\left(S_2\right)\}$, against the observed winning probability in an $S_3$ game, $p\left(S_3\right)$. Solid blue line, maximization over no-signaling probability distributions; dashed orange curve, quantum probabilities.

Figure 6

The curves of maximum achievable success probability in an $S_2$ game played between $A_1$ and $A_2$, $max\left\{p_{A_1,A_2}\left(S_2\right)\right\}$, against the winning probability in an $S_2$ game played between $A_1$ and $A_3$, $p_{A_1,A_3}\left(S_2\right)$, given different observed value of the winning probability in an $S_3$ game, $p_{A_1,A_2,A_3}\left(S_3\right)$: (a) no signaling and (b) quantum constraints. The first curve from the top corresponds to the topmost legend entry, the second curve to the second legend entry, and so on.

Figure 7

Plot of maximum achievable winning probability in a $S_2$ game, $max\left\{p\left(S_2\right)\right\}$ against some tripartite measures of nonlocality: (a) the observed winning probability in the $M_3$ game, $p\left(M_3\right)$; (b) the observed winning probability in the $N_3$ game, $p\left(N_3\right)$; (c) the observed value of Mermin's facet expression, $M{F}_3$; (d) The observed value of $I_3$; and (e) The observed winning probability in the GYNI game, $p\left(GYN{I}_3\right)$. For maximization over no-signaling probability distributions, the solid blue line, and for quantum probabilities, the dashed orange curve.