Quantum secret sharing and Mermin operator

Quantum secret sharing is well known as a method for players to share a classical secret for secret sharing in quantum mechanical ways. Most of the results associated with quantum secret sharing are based on pure multipartite entangled states. In reality, however, it is difficult for players to share a pure entangled state, although they can share a state close to the state. Thus, it is necessary to study how to perform the quantum secret sharing based on a general multipartite state. We here present a quantum secret sharing protocol on an $N$-qubit state close to a pure $N$-qubit Greenberger-Horne-Zeilinger state. In our protocol, $N$ players use an inequality derived from the Mermin inequality to check secure correlation of classical key bits for secret sharing. We show that if our inequality holds then every legitimate player can have key bits with positive key rate. Therefore, for sufficiently many copies of the state, the players can securely share a classical secret with high probability by means of our protocol.