Entanglement swapping theory and beyond

Abstract: We focus on the theoretical research of entanglement swapping and entanglement swapping chains of pure states and mixed states. We show the general theory of entanglement swapping, through which the final states after entanglement swapping can be obtained. We consider the entanglement swapping of 2-level maximally entangled states without limiting each subsystem to be in the same basis. We also consider the entanglement swapping between d-level maximally entangled states,which is realized by performing a joint measurement on the particles that contain the first particle of the selected entangled states and the particles without involving the first particle in other entangled states. For the entanglement swapping of general pure state, we generalize case of two bipartite general pure states to multi-state cases. Moreover, we propose the entanglement swapping between bipartite general pure states and maximally entangled states. For entanglement swapping chains, we propose the entanglement swapping chains for maximally entangled states, which is realized by performing measurements on multiple particles. We prove the results of entanglement swapping chains of general pure states and maximally entangled states. In addition, we study the entanglement swapping and entanglement swapping chains of mixed states, including $X$ states and mixed maximally entangled states. What is more, we provide a new proof for our previous work [2022, Physica A, 585, 126400]. Finally, we propose a new algorithm for deriving entanglement swapping results, and verify the correctness of the algorithm by comparing it with the entanglement swapping of two Bell states.