Equivalence between sharing quantum and classical secrets and error correction

Abstract

We present a general scheme for sharing quantum secrets, and an extension to sharing classical secrets, which contain all known quantum secret sharing schemes. In this framework we show the equivalence of existence of both schemes, that is, the existence of a scheme sharing a quantum secret implies the extended classical secret sharing scheme works, and vice versa. As a consequence of this we find new schemes sharing classical secrets for arbitrary access structures. We then clarify the relationship to quantum error correction and observe several restrictions thereby imposed, which for example indicates that for pure state threshold schemes the share size $q$ must scale with the number of players $n$ as $q\geq \sqrt{n}$. These results also provide a new way of searching for quantum error correcting codes.