Entanglement entropy and Schmidt numbers in quantum networks of coupled quantum oscillator

TL;DR

This paper studies entanglement entropy and Schmidt numbers in quantum networks of coupled harmonic oscillators, deriving conditions for equal entropy in certain graph partitions and analyzing the effects of graph conductance.

Contribution

It introduces a theorem using the generalized Schur complement to identify graphs with equal entanglement entropy across partitions and explores entropy behavior in large systems and couplings.

Findings

Graphs with complete connections between subsets have the same entropy.

Entanglement entropy increases with coupling strength and system size.

Conductance correlates with entanglement entropy in various graphs.

Abstract

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 parts 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 separates into four subsets. The nodes of the first and last subsets are those which are not connected to the nodes of other part. In theorem I, by using generalized Schur complement method in these four subsets, we prove that all graphs which their connections between all 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 large coupling and large size of system, is investigated in these graphs. One of important quantities about partitioning, is conductance of graph. The con- ductance of graph is considered in some various graphs. In these graphs we compare the conductance of graph and the entanglement entropy.