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Computationally Efficient Nonlinear Bell Inequalities for General 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 these multipartite correlations, which are generated by multiple sources. The main problem is that no computationally efficient method is available for constructing useful Bell inequalities for any quantum network. 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 which allows constructing a polynomial-time algorithm. For any bipartite entangled pure states and Greenberger-Horne-Zeilinger (GHZ) states as quantum resources, we prove the generic non-multilocality of quantum networks with multiple independent observers using the new Bell inequalities. The violations are maximal with respect to the presented Tsirelson’s bound for the maximally entangled Einstein-Podolsky-Rosen (EPR) states and GHZ states. Moreover, these violations hold for Werner states or some general noisy states. Our results suggest that the new Bell inequalities can be used to characterize experimental quantum networks.