Self-Testing of Symmetric Three-Qubit States

Self-testing refers to a device-independent way to uniquely identify an unknown quantum device based only on the observed statistics. Earlier results on self-testing of multipartite state were restricted either to Dicke states or Graph states. In this paper, we propose self-testing schemes for a large family of symmetric three-qubit states, namely the superposition of <inline-formula> <tex-math notation="LaTeX">$W$ </tex-math></inline-formula> state and <inline-formula> <tex-math notation="LaTeX">$GHZ$ </tex-math></inline-formula> state. We first propose and analytically prove a self-testing criterion for the special symmetric state with equal coefficients of the canonical bases, by designing subsystem self-testing of partially and maximally entangled state simultaneously. Then we demonstrate for the general case, the states can be self-tested numerically by the swap method combining semidefinite programming (SDP) in high precision.