Entanglement Generalization in Coupled Harmonic Oscillators

TL;DR

This paper presents a general method for analyzing entanglement in coupled harmonic oscillators interacting with a thermal bath, enabling the determination of quantum correlations between any two sites.

Contribution

It introduces an exact approach using simultaneous diagonalization to study entanglement in coupled harmonic oscillators, generalizing previous methods.

Findings

Derived a condition for entanglement existence between two sites.

Developed a method to construct the covariance matrix for the system.

Applied the PPT-criterion to identify quantum correlations.

Abstract

A general and in principle exact approach for the continuous variable entanglement in a system of coupled harmonic oscillators in contact with a thermal bath is formulated. This allows a generalization to describe entanglement's existence between two sites in any system of this kind. It is employed a method of simultaneous diagonalization of two quadratic forms to obtain the uncoupled quantum Hamiltonian. Making use of the transformations that uncouple the system, the covariance matrix for two positions is constructed, and through the positive partial transpose criterion (PPT-criterion), the condition that determines the existence or not of quantum correlation between the sites is obtained.