Deterministic hierarchical sharing of a partially known quantum operation via a non-maximally entangled state

Abstract Recent studies have demonstrated the critical role of hierarchical quantum operation sharing in enhancing security and efficiency within quantum information systems. This approach enables the structured distribution of quantum operations among multiple parties, facilitating collaborative tasks essential for various practical applications. In this paper, we explore the hierarchical sharing of a partially known single-qubit operation, and propose a four-party hierarchical sharing scheme utilizing a non-maximally entangled six-qubit state as the quantum channel. This scheme can be regarded as two sub-schemes, wherein three agents are divided into two power levels for information access: one at a higher power level and the other at a lower level. Each agent can reconstruct the target state with complete certainty, independent of the coefficients of the non-maximally entangled channel, through the cooperation of some or all other agents. Additionally, we extend this framework to accommodate m + n agents, where n agents operate at the higher power level while the remaining m agents function at the lower power level.