Entanglement entropy in the quantum networks of a coupled quantum harmonic oscillator

We investigate the entanglement of the ground state in the quantum networks that their nodes are considered as quantum harmonic oscillators. To this aim, the Schmidt numbers and entanglement entropy between two arbitrary partitions of a network are calculated. In partitioning an arbitrary graph into two parts there are some nodes in each part which are not connected to the nodes of the other part. So, these nodes of each part can be in distinct subsets. Therefore, the graph is separated into four subsets. The nodes of the first and last subsets are those which are not connected to the nodes of the other part. In theorem 1, by using the generalized Schur complement method in these four subsets, we prove that all the graphs whose connections between the two alternative subsets are complete, have the same entropy. A large number of graphs satisfy this theorem. Then the entanglement entropy in the limit of the large coupling and large size of the system is investigated in these graphs. Also, the asymptotic behaviors of the Schmidt numbers and entanglement entropy in the limit of infinite coupling are shown. One important quantity about partitioning is the conductance of the graph. The conductance of the graph is considered in various graphs. In these graphs we compare the conductance of the graph and the entanglement entropy.