Genuine Quantum Nonlocality in the Triangle Network

Quantum networks allow in principle for completely novel forms of quantum correlations. In particular, quantum nonlocality can be demonstrated here without the need of having various input settings, but only by considering the joint statistics of fixed local measurement outputs. However, previous examples of this intriguing phenomenon all appear to stem directly from the usual form of quantum nonlocality, namely via the violation of a standard Bell inequality. Here we present novel examples of “quantum nonlocality without inputs,” which we believe represent a new form of quantum nonlocality, genuine to networks. Our simplest examples, for the triangle network, involve both entangled states and joint entangled measurements. A generalization to any odd-cycle network is also presented.

Figure 1

The triangle network features three observers (green circles), connected by three independent bipartite sources. Here, the sources distribute the quantum states ${\psi }_{\alpha },{\psi }_{\beta }$, and ${\psi }_{\gamma }$.

Figure 2

Step 1 shows that any trilocal model compatible with the quantum distribution $P_Q(a,b,c)$ must have a specific structure. Specifically, for each source, the classical variable set can be divided into two equal weight [i.e., $P(\alpha \in X_0)=\dots =1/2$] disjoint subsets containing all output information on the coarse grained distribution where $\chi =\{{\chi }_0,{\chi }_1\}$ group together two outputs. For instance, when $\alpha \in X_0$, $\beta \in Y_1$, and $\gamma \in Z_0$, the outputs must be $a=\downarrow$ for Alice, $b=\chi$ for Bob, $c=\uparrow$ for Charlie. Note that the ordering is important.