  Self-testing refers to a device-independent way to uniquely identify the
state and the measurement for uncharacterized quantum devices. The only
information required comprises the number of measurements, the number of
outputs of each measurement, and the statistics of each measurement. 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 W
state and GHZ state. We first propose and analytically prove a self-testing
criterion for the special symmetric state with equal coefficients of the
canonical basis, 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
semi-definite programming (SDP) in high precision.
