How to Share a Quantum Secret

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 $k-1\mathrm{}$ 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 $n<\mathrm{}2k$, and we give efficient constructions of all threshold schemes. We also show that, for $k\le n<\mathrm{}2k-1\mathrm{}$, then any $(k,n)$ threshold scheme must distribute information that is globally in a mixed state.