Codes for Quantum Secret Sharing with a Helper

Abstract

Helper quantum secret sharing is a form of secret sharing defined by its unique access structure. One special fixed party, called the helper, can work together with any other party to fully decode the secret. A blind helper is one who can provide this assistance while not holding any local information about the encoded secret. In this work, we analyze the general structure of QSS helper codes and present new code constructions. We fully characterize the structure of blind helper stabilizer codes and show that for the encoding of a single qubit, recovery is always possible using one-way local operations and classical communication (LOCC) from the helper to the targeted party. Furthermore, we demonstrate how such codes also allow for the helper to target larger subsets of parties by one-way LOCC, enabling them to be authorized to recover the secret. Finally, when each party only holds a qubit, we identify the general form of all helper codes (including non-stabilizer codes) and find that LOCC recovery is only possible in special cases.