Maximal qubit violation of n-local inequalities in a quantum network

Source-independent quantum networks are considered as a natural generalization to the Bell scenario where we investigate the nonlocal properties of quantum states distributed and measured in a network. Considering the simplest network of entanglement swapping, recently Gisin et al. Phys. Rev. A 96, 020304 (R) (2017) and Andreoli New. J. Phys. 19, 113020 (2017) independently provided a systematic characterization of the set of quantum states leading to violation of the so-called bilocality inequality. In this work, we consider the complexities in quantum networks with an arbitrary number of parties distributed in chain-shaped and star-shaped networks. We derive the maximal violation of the ``n-local'' inequality that can be achieved by arbitrary two-qubit states for such chain- and star-shaped networks. This would further provide a deeper understanding of quantum correlations in complex structures.