Quantum Secret Sharing Schemes and Reversibility of Quantum Operations
  Quantum secret sharing schemes encrypting a quantum state into a multipartite entangled state are treated. The lower bound on the dimension of each share given by Gottesman [Phys. Rev. A \textbf{61}, 042311 (2000)] is revisited based on a relation between the reversibility of quantum operations and the Holevo information. We also propose a threshold ramp quantum secret sharing scheme and evaluate its coding efficiency. 
Tomohiro Ogawa, Akira Sasaki, Mitsugu Iwamoto, Hirosuke Yamamoto