  We focus on the general theory of entanglement swapping in this paper,
through which one can get final states that the original entangled states
collapse onto after entanglement swapping. We consider the entanglement
swapping between 2-level maximally entangled states without limiting each
subsystem to be in the same basis. For the entanglement swapping between any
number of d-level maximally entangled states, we consider the case realized by
performing a joint measurement on the particles that contain the first particle
of the selected entangled states and the ones without involving the first
particle in other entangled states. We generalize the entanglement swapping
between two bipartite general pure states to multi-state cases. We propose the
entanglement swapping chains for maximally entangled states, which is realized
by performing measurements on multiple particles. We use mathematical induction
to prove the results of entanglement swapping chains of general pure states and
maximally entangled states. We propose the entanglement swapping between
bipartite general pure states and maximally entangled states. In addition to
pure states entanglement swapping, we also study the entanglement swapping and
chains of mixed states, including X states and mixed maximally entangled
states. We propose an entanglement model and give a graphical description of
entanglement and entanglement swapping. Finally, we provide a new proof for our
previous work [2022, Physica A, 585, 126400], which is more concise than the
original one.
