Entanglement entropy in the Hypercube networks

TL;DR

This paper analyzes entanglement entropy in hypercube networks modeled as quantum harmonic oscillators, providing analytical and numerical results for specific partitions and conjecturing bounds for general cases.

Contribution

It introduces a method to compute entanglement entropy in hypercube networks using stratification and Schur complement, including analytical and numerical results for specific cases.

Findings

Analytical formulas for entanglement entropy between network parts.

Numerical results for H(3,2) and H(4,2) networks.

Conjectured bounds for entanglement entropy in general hypercube networks.

Abstract

We investigate the hypercube networks that their nodes are considered as quantum harmonic oscillators. The entanglement of the ground state can be used to quantify the amount of information each part of a network shares with the rest of the system via quantum fluctuations. Therefore, the Schmidt numbers and entanglement entropy between two special parts of Hypercube network, can be calculated. To this aim, first we use the stratification method to rewrite the adjacency matrix of the network in the stratification basis which is the matrix representation of the angular momentum. Then the entanglement entropy and Schmidt number for special partitions are calculated an- alytically by using the generalized Schur complement method. Also, we calculate the entanglement entropy between two arbitrary equal subsets (two equal subsets have the same number of vertices) in H(3, 2) and H(4, 2) numerically, and we give the minimum and maximum values of entanglement entropy in these two Hypercube network. Then we can conjecture the minimum and maximum values of entanglement entropy for equal subsets in H(d, 2).