Assessing the entanglement of three coupled harmonic oscillators

TL;DR

This paper introduces a geometrical approach to analyze quantum entanglement in three coupled harmonic oscillators, deriving analytical expressions and revealing how excitations and mixing angles influence entanglement.

Contribution

It presents a novel geometrical diagonalization method that simplifies entanglement analysis and provides analytical formulas for linear entropy and purity in three-body oscillators.

Findings

Excitations enhance correlation redistribution.

Mixing angle $ heta$ controls entanglement strength.

Symmetry relations in entanglement measures are identified.

Abstract

Quantum entanglement serves as a key phenomenon in understanding correlations in many-body systems, but analytical results remain scarce for coupled three-body oscillators. In this work, we address this gap by introducing a geometrical diagonalization approach that constrains Euler angles, thereby reducing the degrees of freedom in the entanglement analysis. It consists of deriving analytical expressions for linear entropy and purity under the bipartitions $(x|yz)$, $(y|xz)$, and $(xy|z)$ using the Wigner function framework. Our results indicate that excitations in any oscillator basically enhance the redistribution of correlations across the system. The mixing angle $\theta$ governs entanglement intensity, ranging from separability to maximal correlation. Moreover, we reveal the symmetry relations, notably $S_{Ly}[(n,m,l),\theta]=S_{Lz}[(n,m,l),-\theta]$ and an intrinsic symmetry within $(x|yz)$. Hence, we clarify how excitation levels and mixing angles create and enhance entanglement in the three coupled harmonic oscillators.