  Crucial problems of the quantum Internet are the derivation of stability
properties of quantum repeaters and theory of entanglement rate maximization in
an entangled network structure. The stability property of a quantum repeater
entails that all incoming density matrices can be swapped with a target density
matrix. The strong stability of a quantum repeater implies stable entanglement
swapping with the boundness of stored density matrices in the quantum memory
and the boundness of delays. Here, a theoretical framework of noise-scaled
stability analysis and entanglement rate maximization is conceived for the
quantum Internet. We define the term of entanglement swapping set that models
the status of quantum memory of a quantum repeater with the stored density
matrices. We determine the optimal entanglement swapping method that maximizes
the entanglement rate of the quantum repeaters at the different entanglement
swapping sets as function of the noise of the local memory and local
operations. We prove the stability properties for non-complete entanglement
swapping sets, complete entanglement swapping sets and perfect entanglement
swapping sets. We prove the entanglement rates for the different entanglement
swapping sets and noise levels. The results can be applied to the experimental
quantum Internet.
