How to Share a Quantum Secret
Richard Cleve, Daniel Gottesman, Hoi‐Kwong Lo · Physical Review Letters · 1999
We investigate the concept of quantum secret sharing. In a $(k,n)$ threshold scheme, a secret quantum state is divided into $n$ shares such that any $k$ of those shares can be used to reconstruct the secret, but any set of $k\ensuremath{-}1$ or fewer shares contains absolutely no information about the secret. We show that the only constraint on the existence of threshold schemes comes from the quantum ``no-cloning theorem,'' which requires that $n<2k$, and we give efficient constructions of all threshold schemes. We also show that, for $k\ensuremath{\le}n<2k\ensuremath{-}1$, then any $(k,n)$ threshold scheme must distribute information that is globally in a mixed state.