Quantum steering as a resource for secure tripartite quantum state sharing

Quantum state sharing (QSS) is a protocol by which a (secret) quantum state may be securely split into shares, shared between multiple potentially-dishonest players, and reconstructed. Crucially, the players are each assumed to be dishonest, and so QSS requires that only a collaborating authorized subset of players can access the original secret state; any dishonest unauthorized conspiracy cannot reconstruct it. We analyze a QSS protocol involving three untrusted players and demonstrate that quantum steering is the required resource enabling the protocol to proceed securely. We analyze the level of steering required to share any single-mode Gaussian secret which enables the states to be shared with the optimal use of resources.