Observables suitable for restricting the fidelity to multipartite maximally entangled states

We present a class of observables which are suitable for determining the fidelity of a state to the multipartite Greenberger-Horne-Zeilinger (GHZ) state. Given an expectation value of an observable belonging to the class, we give a simple formula that gives a lower bound and an upper bound for the fidelity. Applying the formula to the GHZ-state preparation experiment by Pan et al. [Nature (London) 403, 515 (2000)], we show that the observed state lies outside of the class of biseparable mixed three-qubit states. We also show that for this class of operators, adopting the principle of minimum variance [Phys. Rev. A 60, 4338 (1999)] in the state estimation always results in the state with the minimum fidelity.