Witnessing the distribution of sources in quantum networks via hierarchical nonlocality

Quantum networks with multiple sources always face performance challenges due to the vulnerability of quantum systems. Thus, it is highly desirable to have the capability to continuously monitor and determine the exact number of quantum sources versus classical sources present within the network. Hierarchical network nonlocality can reveal the relationship between network nonlocality and the number of quantum sources within the network, thus becoming a way to address the aforementioned issue. However, up to now, precise hierarchical network nonlocality inequalities have only been established for star networks, and cannot be obtained for other non-star structured networks Phys. Rev. Lett.128 010403 (2022) and Phys. Rev. A110 022617 (2024). In this paper, we establish more refined criteria for hierarchical network nonlocality inequalities in arbitrary network structures. Violating such inequalities can reveal the exact number of quantum sources in the network. These results enhance the understanding of quantum source distribution in complex network topologies.