Entanglement swapping theory and beyond

Entanglement swapping is not only one of the important resources for quantum information processing, but also the key to building large-scale quantum networks. In this paper, we mainly 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 one can get the final states after entanglement swapping. We consider entanglement swapping of 2-level maximally entangled states without limiting each subsystem to be in the same basis, which has certain inspirations for studying entanglement swapping of graph states. We consider 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 entanglement swapping of general pure state, we generalize case of two bipartite general pure states to multi-state cases. Moreover, we propose 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 through mathematical induction. In addition to pure states entanglement swapping, we also study the entanglement swapping and entanglement swapping chains of mixed states, including X states and mixed maximally entangled states. We provide a new proof for our previous work [2022, Physica A, 585, 126400]. Finally, we propose a simple theoretical model that provides a new perspective for considering quantum mechanics principles.