How to Share a Quantum Secret

TLDR

In a ((k,n)) threshold scheme, a secret quantum state is divided into n shares such that any k of those shares can reconstruct the secret, but any set of k‑1 or fewer shares contains absolutely no information about the secret. We investigate the concept of quantum secret sharing and explore its similarities and differences with quantum error‑correcting codes. We analyze the relationship between quantum secret sharing schemes and quantum error‑correcting codes, highlighting their structural distinctions. We show that the only constraint on threshold schemes is the quantum no‑cloning theorem (n < 2k), provide an efficient construction for all such cases, and note that schemes with k ≤ n < 2k − 1 must distribute information that is globally in a mixed state, contrasting with most quantum codes that encode pure states as pure states.

Abstract

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 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, in all such cases, we give an efficient construction of a ((k,n)) threshold scheme. We also explore similarities and differences between quantum secret sharing schemes and quantum error-correcting codes. One remarkable difference is that, while most existing quantum codes encode pure states as pure states, quantum secret sharing schemes must use mixed states in some cases. For example, if k <= n < 2k-1 then any ((k,n)) threshold scheme must distribute information that is globally in a mixed state.