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arXiv:2601.01292

Assessing the entanglement of three coupled harmonic oscillators

Ayoub Ghaba, Radouan Hab Arrih, Elhoussine Atmani, Abderrahim El Allati, Abdallah Slaoui

quant-phSubmitted Jan 3, 2026• Announced Jan 6, 2026• 0 citations
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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.

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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)(x|yz)(x∣yz), (y∣xz)(y|xz)(y∣xz), and (xy∣z)(xy|z)(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 θθθ governs entanglement intensity, ranging from separability to maximal correlation. Moreover, we reveal the symmetry relations, notably SLy[(n,m,l),θ]=SLz[(n,m,l),−θ]S_{Ly}[(n,m,l),θ]=S_{Lz}[(n,m,l),-θ]SLy​[(n,m,l),θ]=SLz​[(n,m,l),−θ] and an intrinsic symmetry within (x∣yz)(x|yz)(x∣yz). Hence, we clarify how excitation levels and mixing angles create and enhance entanglement in the three coupled harmonic oscillators.
Quantum steering is one of the most intriguing phenomena in quantum mechanics and is essential for understanding correlations in multi-body systems. Despite its importance, analytical results for coupled three-body oscillators remain scarce. In this work, we investigate this phenomenon through a geometrical diagonalization approach, which reduces the degrees of freedom associated with the system's steering properties. Specifically, we derive analytical expressions for quantum steering in all possible directions using the Wigner function framework, as it provides a complete description of the system's quantum state. Our results indicate that excitations significantly enhance quantum steering across the system; this stands in contrast to the ground state (0,0,0)(0,0,0)(0,0,0), which exhibits no steerable correlations. Furthermore, both the directionality and topology of these correlations are governed by the spatial distribution of the excitations rather than their magnitude. We also observe symmetric steering behavior between oscillators xxx, yyy, and zzz under equivalent excitation conditions, which can be formalized as Sx→z(n,m,l)(θ)=Sx→y(n,m,l)(−θ),Sz→x(n,m,l)(θ)=Sy→x(n,m,l)(−θ)S^{(n,m,l)}_{x\to z}(θ)=S^{(n,m,l)}_{x\to y}(-θ),\quad S^{(n,m,l)}_{z\to x}(θ)=S^{(n,m,l)}_{y\to x}(-θ)Sx→z(n,m,l)​(θ)=Sx→y(n,m,l)​(−θ),Sz→x(n,m,l)​(θ)=Sy→x(n,m,l)​(−θ), and Sy→z(n,m,l)(θ)=Sz→y(n,m,l)(−θ)S^{(n,m,l)}_{y\to z}(θ)=S^{(n,m,l)}_{z\to y}(-θ)Sy→z(n,m,l)​(θ)=Sz→y(n,m,l)​(−θ). Therefore, we elucidate how excitation levels and mixing angles generate and enhance steering in three coupled harmonic oscillators.
We develop an approach to study the entanglement in two coupled harmonic oscillators. We start by introducing an unitary transformation to end up with the solutions of the energy spectrum. These are used to construct the corresponding coherent states through the standard way. To evaluate the degree of the entanglement between the obtained states, we calculate the purity function in terms of the coherent and number states, separately. The result is yielded to two parameters dependance of the purity, which can be controlled easily. Interesting results are derived by fixing the mixing angle of such transformation as π/2. We compare our results with already published work and point out the relevance of these findings to a systematic formulation of the entanglement effect in two coupled harmonic oscillators.
We investigate the quantum entanglement dynamics of two coupled harmonic oscillators with a time-dependent interaction. Using the Lewis-Riesenfeld invariant method, we derive the exact analytical wave functions without any perturbative or adiabatic approximations and combine this with a phase-space analysis using the Wigner function to provide a complete description of the system's quantum state evolution. We obtain general expression form for the purity and the linear entropy SL=1−PS_L=1-\mathcal{P}SL​=1−P for arbitrary excitation numbers (n,m)(n,m)(n,m), allows a systematic study of entanglement for a large class of quantum states. We show that the entanglement dynamics is very sensitive to the interplay between the detuning parameters θθθ and ϑ2\vartheta_2ϑ2​, the frequency parameter β0β_0β0​ and the coupling strength εεε: the increase of detuning takes the system from slow irregular oscillations to fast and regular periodic behavior, and the stronger coupling systematically enhances both the amplitude and the average value of the linear entropy. Most importantly, for the resonance case ω1=ω2=1ω_1=ω_2=1ω1​=ω2​=1 and strong couplings ε≈0.99ε\approx 0.99ε≈0.99, the system shows robust undamped synchronized periodic oscillations of the linear entropy for all quantum states considered, indicating preserved quantum coherence without saturation. Our findings demonstrate that linear entropy is a sensitive and practical entanglement witness, and we establish explicit analytical relations between the coupling parameters of the system and its entanglement properties, which are directly relevant to quantum information processing and the control of quantum correlations in continuous-variable systems.
In this paper, we study the entanglement structure of a system of N quantum oscillators with distinctive coupling strengths, all linearly coupled to a common massless scalar quantum field. This study is helpful in characterizing the notion of an entanglement domain and its symmetry features, which is useful for understanding the interplay between different levels of structure in many-body quantum systems. The effect of the quantum field on the system is derived via the influence functional and the correlation functions are obtained from the solutions of the evolutionary operator of the reduced density matrix. They are then used to construct the covariance matrix, which forms the basis for our analysis of the structure of quantum entanglement in this open system. To make the physical features explicit, we consider a system of three quantum coupled oscillators placed at the vertices of an equilateral triangle with disparate pairwise couplings. We analyze the entanglement between one oscillator and the other two with equal (symmetric) and unequal (asymmetric) coupling strengths. As a physical illustration, we apply the results for these two different configurations to address some basic issues in macroscopic quantum phenomena from the quantum entanglement perspective.
Abstract Quantum steering is one of the most intriguing phenomena in quantum mechanics and is essential for understanding correlations in multipartite systems. Despite its importance, analytical results for coupled three-body oscillators remain scarce. In this work, we investigate this phenomenon using a geometrical diagonalization approach, which reduces the degrees of freedom associated with the system’s steering properties to a single effective mixing parameter, making the problem fully tractable within the Wigner function framework. We derive closed-form expressions for quantum steering in all possible directions. Our analysis shows that steering vanishes in the ground state and increases with the quantum number of the excited mode, while excitations significantly enhance quantum steering across the system. However, steering disappears for multi-excitation states involving more than one nonzero quantum number. Furthermore, both the directionality and topology of these correlations are governed by the spatial distribution of the excitations rather than their magnitude. In particular, exciting one mode produces a hub-and-spoke structure in which that mode influences the other two, whereas exciting a different mode leads to a distinct configuration in which no steering is observed between the remaining modes and the excited one. We further uncover a cyclic balance relation representing the equivalence between clockwise and counterclockwise steering flows, as well as a sign-sensitive directionality in which steering in a given direction appears only for specific signs of the effective mixing parameter. We also observe symmetric steering behavior between oscillators under equivalent excitation conditions, reflecting underlying algebraic symmetries consistent with the root structure of the associated Lie algebra. These findings provide an analytical description of quantum steering in tripartite quantum states of coupled oscillators and offer guidance for designing new protocols based on quantum steering in continuous-variable systems.
We study the dynamics and redistribution of entanglement and coherence in three time-dependent coupled harmonic oscillators. We resolve the Schrödinger equation by using time-dependent Euler rotation together with a linear quench model to obtain the state of vacuum solution. Such state can be translated to the phase space picture to determine the Wigner distribution. We show that its Gaussian matrix [Formula: see text] can be used to directly cast the covariance matrix [Formula: see text]. To quantify the mixedness and entanglement of the state, one uses respectively linear and von Neumann entropies for three cases: fully symmetric, bi-symmetric and fully nonsymmetric. Then we determine the coherence, tripartite entanglement and local uncertainties and derive their dynamics. We show that the dynamics of all quantum information quantities are driven by the Ermakov modes. Finally, we use an homodyne detection to redistribute both resources of entanglement and coherence.