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\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N > M \ge 2$$\end{document}. 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.