Assessing the entanglement of three coupled harmonic oscillators

TL;DR

This work addresses the analytical challenge of entanglement in three coupled harmonic oscillators by introducing a geometrical diagonalization that constrains Euler angles and reduces the entanglement description to a single mixing angle $\theta$. Using the phase-space Wigner-function formalism, the authors derive analytical expressions for the purities $\mathbb P_x,\mathbb P_y,\mathbb P_z$ and the associated linear entropies for the bipartitions $(x|yz)$, $(y|xz)$, and $(xy|z)$, with the purities depending only on the quantum numbers $n,m,l$ and on $\theta$ through $\mu_{\theta}$. The paper reveals symmetry relations such as $S_{Ly}[(n,m,l),\theta]=S_{Lz}[(n,m,l),-\theta]$ and a monogamy-like triangle inequality via $M_k=[S_{Li}+S_{Lj}-S_{Lk}]$, illustrating how excitations and the mixing angle control entanglement distribution. Numerical results show that excitations robustly enhance entanglement and that $\theta$ acts as a tunable control parameter, with distinct maxima and symmetry across partitions; a two-oscillator limit confirms consistency with known results. Overall, the geometrical diagonalization provides a compact, actionable framework for steering continuous-variable entanglement in multi-oscillator networks, with potential implications for quantum communication and sensing applications.

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 $θ$ governs entanglement intensity, ranging from separability to maximal correlation. Moreover, we reveal the symmetry relations, notably $S_{Ly}[(n,m,l),θ]=S_{Lz}[(n,m,l),-θ]$ 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.

Figures (12)

• Figure 1: Schematic of three coupled oscillators; each two oscillators are coupled via "position-position" interaction $\hat{i}\hat{j}$ with a coupling strength of $J_{ij}$, for all $i \neq j$ and $i, j \in \{x, y, z\}$.
• Figure 2: Entanglement $S_{Lx}$ versus $\theta$ for the states $(n,0,0)$, $(0,m,0)$ and $(0,0,l)$ from left to right, respectively.
• Figure 3: The evolution of the linear entropy $S_{Lx}$ versus the mixing angle $\theta$, for various quantum states $(n,m,l)$.
• Figure 4: Evolution of the linear entropy $S_{Ly}$ as a function of the mixing angle $\theta$ for the states $(n,0,0)$, $(0,m,0)$, and $(0,0,l)$, shown from left to right, respectively.
• Figure 5: (color online) Evolution of the quantum entanglement $S_{Ly}$ as a function of the mixing angle $\theta$ for the states $(n,1,1)$, $(1,m,1)$, and $(1,1,l)$, shown from left to right, respectively.