Efficient recyclable generation protocol for high-dimensional spatial-path GHZ states

Multipartite high-dimensional entanglement offers a larger space for storing and processing quantum information and is the crucial resource in future high-capacity and high-security quantum networks. The high-efficiency generation of multipartite high-dimensional entanglement is of central importance for its application. In the paper, we propose a recyclable generation protocol for the four-photon three-dimensional spatial-path Greenberger–Horne–Zeilinger (GHZ) state with linear optical elements and practical “on-off” photon detectors. Our protocol is feasible under current experimental conditions, and the generated three-dimensional GHZ state can be preserved for applications. When the generation protocol fails, the output state may evolve into the auxiliary state for the next generation round. In this way, our protocol can effectively save precious EPR resources. With the increase of repeating number, our protocol will have a prominent advantage in saving precious entanglement resources. Our protocol can provide effective guidance for the experimental preparation of the three-dimensional spatial-path GHZ state, and has important application in future multipartite high-dimensional quantum networks.