Verifying hierarchical network nonlocality in general quantum networks

Recently, a class of innovative notions on quantum network nonlocality (QNN), called full quantum network nonlocality (FQNN), have been proposed in Phys. Rev. Lett. 128 010403 (2022). As the generalization of full network nonlocality (FNN), l -level quantum network nonlocality ( l -QNN) was defined in arxiv. 2306.15717 quant-ph (2024). FQNN is a NN that can be generated only from a network with all sources being non-classical. This is beyond the existing standard network nonlocality, which may be generated from a network with only a non-classical source. One of the challenging tasks is to establish corresponding Bell-like inequalities to demonstrate the FQNN or l -QNN. Up to now, the inequality criteria for FQNN and l -QNN have only been established for star and chain networks. In this paper, we devote ourselves to establishing Bell-like inequalities for networks with more complex structures. Note that star and chain networks are special kinds of tree-shaped networks. We first establish the Bell-like inequalities for verifying l -QNN in k -forked tree-shaped networks. Such results generalize the existing inequalities for star and chain networks. Furthermore, we find the Bell-like inequality criteria for l -QNN for general acyclic and cyclic networks. Finally, we discuss the demonstration of l -QNN in the well-known butterfly networks.