Entanglement swapping theory and beyond

Abstract

We focus on the general theory of entanglement swapping, including entanglement swapping of pure and mixed states. We also study the theory of entangled swapping chains, which can be regarded as an application of entangled swapping. For maximally entangled states, we consider the entanglement swapping of 2-level maximally entangled states without limiting each subsystem to be in the same basis. We further 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 the case of two bipartite general pure states to multi-state cases. Besides, 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 joint measurements on multiple particles in each state, we use mathematical induction to prove the results of entanglement swapping chains of maximally entangled states and that of general pure states. Moreover, we study the entanglement swapping and entanglement swapping chains of mixed states, including X states and mixed maximally entangled states. Finally, we provide a new proof for our previous work [2022, Physica A, 585, 126400].