Computationally Efficient Nonlinear Bell Inequalities for Quantum Networks

The correlations in quantum networks have attracted strong interest with new types of violations of the locality. The standard Bell inequalities cannot characterize the multipartite correlations that are generated by multiple sources. The main problem is that no computationally efficient method is available for constructing useful Bell inequalities for general quantum networks. In this work, we show a significant improvement by presenting new, explicit Bell-type inequalities for general networks including cyclic networks. These nonlinear inequalities are related to the matching problem of an equivalent unweighted bipartite graph that allows constructing a polynomial-time algorithm. For the quantum resources consisting of bipartite entangled pure states and generalized Greenberger-Horne-Zeilinger (GHZ) states, we prove the generic nonmultilocality of quantum networks with multiple independent observers using new Bell inequalities. The violations are maximal with respect to the presented Tsirelson’s bound for Einstein-Podolsky-Rosen states and GHZ states. Moreover, these violations hold for Werner states or some general noisy states. Our results suggest that the presented Bell inequalities can be used to characterize experimental quantum networks.

Figure 1

Schematic network in terms of the GLCM. There are $m$ independent sources $S_1,S_2,\dots ,S_m$ that distribute hidden states ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_m$, respectively. Each spacelike separated party ${\mathcal{A}}_i$ receives hidden states ${\mathrm{\Lambda }}_i=\{{\lambda }_{j_1},{\lambda }_{j_2},\dots ,{\lambda }_{j_{{\ell }_i}}\}$ that are distributed by the corresponding sources ${\mathsf{S}}_i=\{S_{j_1},S_{j_2},\dots ,S_{j_{{\ell }_i}}\}$, where ${\cup }_{i=1}^n{\mathsf{S}}_i=\{S_1,S_2,\dots ,S_m\}$. In the experiment, each party ${\mathcal{A}}_i$ obtains one bit $a_i$ dependent on input bit $x_i$ and hidden states ${\mathrm{\Lambda }}_i$, $i=1,2,\dots ,n$. One vector connects a source set and one party (cubic).

Figure 2

Schematic quantum network with $k$ independent observers. Each pair of observers ${\mathcal{A}}_i$ and $\mathcal{B}$ share some entangled states, $i=1,2,\dots ,k$. $\mathcal{B}$ may contain states that are not shared with ${\mathcal{A}}_i\mathrm{s}$. ${\mathcal{A}}_i$ and ${\mathcal{A}}_j$ do not share entangled states for any $i\ne j,i,j=1,2,\dots ,k$. The graph of two black dots connected by one long line and one short line schematically represents quantum resources.

Figure 3

(a) Long-distance entanglement distribution. All parties ${\mathcal{A}}_2,\dots ,{\mathcal{A}}_{n-1}$ jointly distribute a bipartite entangled state to two parties ${\mathcal{A}}_1$ and ${\mathcal{A}}_n$ with bipartite entangled states as quantum resources. (b) Hybrid star-shaped network. ${\mathcal{B}}_i\mathrm{s}$ and ${\mathcal{C}}_i\mathrm{s}$ jointly distribute a four-partite GHZ state to all parties ${\mathcal{A}}_j\mathrm{s}$ with bipartite entangled states and generalized four-partite GHZ states. (c) Cyclic network consisting of bipartite entangled states. One edge connects two entangled particles (black dots). One cubic denotes an observer who has several particles.