Genuine (k, m)-threshold controlled teleportation and its security

We propose genuine ($k$, $m$)-threshold controlling schemes for controlled teleportation via multi-particle entangled states, where the teleportation of a quantum state from a sender (Alice) to a receiver (Bob) is under the control of $m$ supervisors such that $k$ ($k\leq m$) or more of these supervisors can help Bob recover the transferred state. By construction, anyone of our quantum channels is a genuine multipartite entangled state of which any two parts are inseparable. Their properties are compared and contrasted with those of the well-known Greenberger-Horne-Zeilinger, W, and linear cluster states, and also several other genuine multipartite entangled states recently introduced in literature. We show that our schemes are secure against both Bob's dishonesty and supervisors' treacheries. For the latter case, the game theory is utilized to prove that supervisors' cheats can be well prevented. In addition to their practical importance, our schemes are also useful in seeking and exploring genuine multipartite entangled states and opening another perspective for the applications of the game theory in quantum information science.