An algorithm for finding Bell-type inequalities

Yu-Chun Wu, Piotr Badziąg, Marek Żukowski

We construct a simple algorithm to generate any Clauser-Horne-Shimony-Holt- (CHSH-) type Bell inequality involving a party with two local binary measurements from two CHSH-type inequalities without this party. The algorithm readily generalizes to situations where the additional observer uses three measurement settings. There, each inequality involving the additional party is constructed from three inequalities with this party excluded. With this generalization at hand, we construct and analyze a class of symmetric inequalities for four observers and three experimental settings per observer.

Lucién Hardy

Yu-Chun Wu, Marek Żukowski

A technique, which we call homogenization, is applied to transform Clauser and Horne- (CH)-type Bell inequalities that contain lower-order correlations into CH, Shimony, and Holt- (CHSH)-type Bell inequalities, which are defined for highest-order correlation functions. A homogenization leads to inequalities involving more settings, that is, a choice of one more observable is possible for each party. We show that this technique preserves the tightness of Bell inequalities: A homogenization of a tight CH-type Bell inequality is still a tight CHSH-type Bell inequality. As an example, we obtain $3\ifmmode\times\else\texttimes\fi{}3\ifmmode\times\else\texttimes\fi{}3$ CHSH-type Bell inequalities by the homogenization of $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ CH-type Bell inequalities derived by Sliwa [Phys. Lett. A 317, 165 (2003)].

Enrico G. Beltrametti, Maciej Mączyński

Mohammad Heshmatpour

Ming‐Xing Luo

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.