Maximal qubit violation of n-locality inequalities in a star-shaped quantum network

Bellʼs theorem was a cornerstone for our understanding of quantum theory and the establishment ofBell non-locality played a crucial role in the development of quantum information. Recently, itsextension to complex networks has been attracting growing attention, but a deep characterization ofquantum behavior is still missing for this novel context. In this work we analyze quantum correlationsarising in the bilocality scenario, that is a tripartite quantum network where the correlations betweenthe parties are mediated by two independent sources of states. First, we prove that non-bilocalcorrelations witnessed through a Bell-state measurement in the central node of the network form asubset of those obtainable by means of a local projective measurement. This leads us to derive themaximal violation of the bilocality inequality that can be achieved by arbitrary two-qubit quantumstates and arbitrary local projective measurements. We then analyze in details the relation between theviolation of the bilocality inequality and the CHSH inequality. Finally, we show how our method canbe extended to then-locality scenario consisting ofntwo-qubit quantum states distributed amongn1+nodes of a star-shaped network