Decomposition of symmetric multipartite observable

Production and verification of multipartite quantum state are an essential step in quantum information processing. In this work, we propose an efficient method to decompose symmetric multipartite observables, which are invariant under permutations between parties, with only $(N+1)(N+2)/2$ local measurement settings, where $N$ is the number of qubits. We apply the decomposition technique to evaluate the fidelity between an unknown prepared state and any target permutation invariant state. In addition, for some typical permutation invariant states, such as the Dicke state with a constant number of excitations, $m$, we derive a tight linear bound on the number of local measurement settings, $m(2m+3)N+1$. Meanwhile, for the $GHZ$ state, the $W$ state, and the Dicke state, we prove a linear lower bound, $\Theta(N)$. Hence, for these particular states, our decomposition technique is optimal.