Genuine secret-sharing states

Quantum secret sharing allows each player to have classical information for secret sharing in quantum mechanical ways. In this work, we construct a class of quantum states on which players can quantumly perform secret-sharing secure against dishonest players as well as eavesdropper. We here call them the genuine secret-sharing states. In addition, we show that if N players share an N-party genuine secret-sharing state, then arbitrary M players out of the total players can share an M-party genuine secret-sharing state by means of local operations and classical communication on the state, where N>M >= 2. We also define the distillable rate with respect to the genuine secret-sharing state and explain the connection between the distillable rate and the relative entropy of entanglement.