Self-testing measurements in networks from star shaped to tree shaped based on genuine network nonlocality

Network self-testing enables the unique identification of unknown measurements in a network using only observed measurement statistics. In this work, we first establish the self-testing statements for measurements in a three-branch star-shaped network (3BSN) scenario based on genuine network nonlocality. In this network, the central node performs a single joint measurement, while each branch node implements three binary-outcome projective measurements. We propose a set of statistical constraints for one pair of measurements (the first and second) at each branch node, and, by exploiting the maximal violation of the Mermin inequality for another pair (the second and third), we demonstrate the self-testing of the measurements in this 3BSN scenario. Subsequently, by performing a joint measurement on branch nodes from two discussed 3BSNs, we construct a three-layer two-forked tree-shaped network. We then establish the self-testing protocol of the joint measurement in this newly formed network scenario. In conjunction with the self-testing statements for each individual 3BSN scenario, this establishment enables the self-testing of all measurements in the tree-shaped network scenario. Our self-testing approach does not rely on any network Bell inequalities, thus providing a different perspective for device-independent quantum information processing tasks in networks.