Verifying hierarchical network nonlocality in general quantum networks

Recent, the notions of fully and l-level quantum network nonlocality (FQNN and l-QNN) have been proposed to investigate quantum network nonlocality who cannot be generated when at least l sources in a quantum network distribute classical physical systems. Such notions are regarded to be beyond the standard quantum network nonlocality (QNN), which can only guarantee that there is one non-classical source in this quantum network. Up to now, the topic of FQNN and l-QNN has been established for star and chain networks in [A. Pozas-Kerstjens et al. Phys. Rev. Lett. 128, 010403 (2022)] and [M. X. Luo et al. Arxiv. 2306.15717 quant-ph Jan. 2024]. In the paper, we focus on the case of more complex-structure general networks. We first establish the Bell-type inequalities for verifying l-QNN in acyclic and cyclic networks respectively. Furthermore, some cases of general networks are also discussed. For instance, the well-known Butterfly networks.