Nonlinear Bell Inequalities Tailored for Quantum Networks

In a quantum network, distant observers sharing physical resources emitted by independent sources can establish strong correlations, which defy any classical explanation in terms of local variables. We discuss the characterization of nonlocal correlations in such a situation, when compared to those that can be generated in networks distributing independent local variables. We present an iterative procedure for constructing Bell inequalities tailored for networks: starting from a given network, and a corresponding Bell inequality, our technique provides new Bell inequalities for a more complex network, involving one additional source and one additional observer. We illustrate the relevance of our method on a variety of networks, demonstrating significant quantum violations, which could not have been detected using standard Bell inequalities.

Figure 1

We consider networks consisting of distant observers ${\mathcal{A}}^j$ sharing physical resources emitted from independent sources ${\mathcal{S}}_k$, and discuss nonlocality in such networks. In our approach, starting from a network $\mathcal{N}$ (in black) with $N$ sources and $M$ parties, we define a new network ${\mathcal{N}}^{\prime }$ by adding a new independent source ${\mathcal{S}}_{N+1}$ connected to a single party ${\mathcal{A}}^M$ of $\mathcal{N}$ and to a new party ${\mathcal{A}}^{M+1}$ (in blue). We show how Bell inequalities for so-called ${\mathcal{N}}^{\prime }$-local correlations can be derived starting from Bell inequalities for $\mathcal{N}$-local correlations.

Figure 2

In the main text, we discuss a variety of networks: (a) the scenario of bilocality, (b) a general chain network featuring $N$ sources and $M=N+1$ observers, (c) a three-branch star network, (d) a general star-shaped network with $N$ branches, and (e) a network featuring a different topology, illustrating the versatility of our method.