Sharing nonlocality in a network using the quantum violation of chain network inequality

Based on the quantum violation of suitable $n$-local inequality in a star network for arbitrary $m$ inputs, we demonstrate the sharing of nonlocality in the network. Such a network features an arbitrary $n$ number of independent sources, $n$ edge parties, and a central party. Each party receives arbitrary $m$ inputs. We consider two different types of sharing of nonlocality in the network. i) The symmetric case - when the sharing of nonlocality is considered across all edge parties. ii) The asymmetric case - when the sharing of nonlocality is considered across only one edge party. For simplicity, we first consider the bilocal scenario $(n=2)$ with three inputs $m=3$ and demonstrate that while in the symmetric case at most two sequential observers can share nonlocality, in the asymmetric case at most four sequential observers can share nonlocality. We extend the study to $n$-local scenario by assuming each party receives three inputs and show that in the symmetric case the result remains the same for any $n$, but in the asymmetrical case, an unbounded number of sequential observers can share nonlocality across one edge for a sufficiently large value of $n$. We further extend our result for arbitrary $m$ input in $n$-local scenario. We demonstrate that for $m\geq 4$, in the symmetric case at most one sequential observer can share nonlocality irrespective of the value of $n$. For the asymmetric case, we analytically show that there exists $n(k)$ for which an arbitrary $k$ number of sequential observers can share the nonlocality across one edge. The optimal quantum violation of $m$-input $n$-local inequality is derived through an elegant SOS approach without specifying the dimension of the quantum system.