Optimal Quantum Violations of n-Locality Inequalities with Conditional Dependence on Inputs

Bell experiment in a network gives rise to a form of quantum nonlocality which is conceptually different from traditional multipartite Bell nonlocality. In this work, the star-network configuration involving arbitrary n independent sources and $(n+1)$ parties, including n edge parties and one central party is considered. Each of the n edge parties shares a physical system with the central party. Each edge party receives $2^{m-1}$ number of inputs, and the central party receives an arbitrary m number of inputs. The conditional dependence on the inputs of each edge party is imposed so that the local probabilities satisfy a set of constraints. A family of generalized n-locality inequalities is proposed in the arbitrary input scenario by imposing the set of constraints on inputs. The optimal quantum violation of the inequalities is derived by using an elegant sum-of-squares approach without specifying the dimension of the quantum system. Notably, the optimal quantum value is achieved only when the set of linear constraints on inputs is satisfied, which, in turn, self-tests the observables required for each edge party. It shows that while conditional dependence on inputs significantly reduces the n-local bound of the inequalities, the optimal quantum violation remains invariant. It argues that this implies a more robust test of network non-locality, which can be revealed for smaller visibility parameters of the corresponding state. Further, the network nonlocality is characterized and examine its correspondence with suitably derived standard Bell nonlocality.