Experimentally feasible security check for $n$-qubit quantum secret sharing

In this article we present a general security strategy for quantum secret sharing (QSS) protocols based on the scheme presented by Hillery, Bužek, and Berthiaume (HBB) [Phys. Rev. A 59, 1829 (1999)]. We focus on a generalization of the HBB protocol to $n$ communication parties thus including $n$-partite Greenberger-Horne-Zeilinger states. We show that the multipartite version of the HBB scheme is insecure in certain settings and impractical when going to large $n$. To provide security for such QSS schemes in general we use the framework presented by some of the authors [M. Huber, F. Mintert, A. Gabriel, B. C. Hiesmayr, Phys. Rev. Lett. 104, 210501 (2010)] to detect certain genuine $n$-partite entanglement between the communication parties. In particular, we present a simple inequality which tests the security.