To generalize the schemes of Hu et al. (Quantum Inf. Process. 21, 3 2022) from two-level and three-level quantum systems to high-level quantum system, a definitive three-party quantum state sharing (QSS) protocol in an amplitude damping environment is proposed to distribute an arbitrary unknown single-qudit quantum state. To achieve this goal, we initially propose the detailed steps for sharing a pure entangled three-qudit state as a quantum resource in an amplitude damping environment via entanglement compensation. Then, based on the pure entangled three-qudit state shared among trader and two agents, the secret quantum information of the trader, i.e., an unknown single-qudit quantum state is divided in such a manner that it can be probabilistically reconstructed by one recipient through the introduction of an auxiliary qubit and the application of appropriate unitary operations by the recipient, provided that two other participants cooperate. Subsequently, the above three-party sharing scheme is generalized to the case of multiparty sharing. The findings indicate that the success rate of the recipient in recovering the secret quantum information is solely determined by the smaller of the absolute values of the coefficients that define the quantum channel. In addition, regardless of the noise intensity in the amplitude damping channel, the fidelity of quantum state transmitted in our schemes can always reach 100%, which is impossible for existing QSS schemes in high-dimensional quantum system based on noisy environments.
