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arXiv:2601.01307

Einstein-Podolsky-Rosen Steering in Three Coupled Harmonic Oscillators

Ayoub Ghaba, Radouan Hab Arrih, Elhoussine Atmani, Abdallah Slaoui

quant-phmath-phSubmitted Jan 4, 2026• Announced Jan 6, 2026• 0 citations
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Abstract

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 $(0,0,0)$, 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 $x$, $y$, and $z$ under equivalent excitation conditions, which can be formalized as $S^{(n,m,l)}_{x\to z}(θ)=S^{(n,m,l)}_{x\to y}(-θ),\quad S^{(n,m,l)}_{z\to x}(θ)=S^{(n,m,l)}_{y\to x}(-θ)$, and $S^{(n,m,l)}_{y\to z}(θ)=S^{(n,m,l)}_{z\to y}(-θ)$. Therefore, we elucidate how excitation levels and mixing angles generate and enhance steering in three coupled harmonic oscillators.

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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.
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.
Using the Wigner function in phase space, we study quantum steering and entanglement between two coupled harmonic oscillators. We derive expressions for purity and quantum steering in both directions and identify several important selection rules. Our results extend the work reported in {\color{blue} [Phys. Rev. E 97, 042203 (2018)]} focused on the weak coupling regime, revealing significant deviations in the ultra-strong coupling regime. In particular, Makarov's prediction of a separable ground state contrasts with our exact calculations, highlighting the limitations of his approach under strong coupling conditions. We show that quantum steering between excited oscillators is completely absent even in the ultra-strong coupling regime. Similarly, resonant oscillators have no steering, and ground states cannot steer any receiver state. We find that quantum steering becomes notably more pronounced as the system approaches resonance and within specific ranges of ultra-strong coupling. This behavior is marked by a clear asymmetry, where steering is present in only one direction, highlighting the delicate balance of interaction strengths that govern the emergence of quantum correlations. These results advance our understanding of how excitation levels and coupling strengths influence quantum steering and entanglement in coupled harmonic oscillators.
Einstein-Podolsky-Rosen (EPR) steering is a form of quantum correlation that exhibits a fundamental asymmetry in the properties of quantum systems. Given two observers, Alice and Bob, there are known to exist bipartite entangled states which are one-way steerable in the sense that Alice can steer Bob's state, but Bob cannot steer Alice's state. Here, we generalize this phenomenon to three parties and find a cyclic property of tripartite EPR steering. In particular, we identify a three-qubit state whose reduced bipartite states are one-way steerable for arbitrary projective measurements. Moreover, the three-qubit state has a cyclic steering property in the sense that by arranging the system in a triangular configuration the neighboring parties can steer each other's states only in the same (e.g., clockwise) direction. That is, Alice can steer Bob's state, Bob can steer Charlie's state, and Charlie can steer Alice's state, but not the other way around.
Einstein-Podolsky-Rosen (EPR) steering is a form of bipartite quantum correlation that is intermediate between entanglement and Bell nonlocality. It allows for entanglement certification when the measurements performed by one of the parties are not characterised (or are untrusted) and has applications in quantum key distribution. Despite its foundational and applied importance, EPR steering lacks a quantitative assessment. Here we propose a way of quantifying this phenomenon and use it to study the steerability of several quantum states. In particular we show that every pure entangled state is maximally steerable, the projector onto the anti-symmetric subspace is maximally steerable for all dimensions, we provide a new example of one-way steering, and give strong support that states with positive-partial-transposition are not steerable.
We consider two systems A and B that share Einstein–Podolsky–Rosen (EPR) steering correlations and study how these correlations will decay when each of the systems is independently coupled to a reservoir. EPR steering is a directional form of entanglement, and the measure of steering can change depending on whether system A is steered by B, or vice versa. First, we examine the decay of the steering correlations of the two-mode squeezed state. We find that if system B is coupled to a reservoir, then the decoherence of the steering of A by B is particularly marked, to the extent that there is a sudden death of steering after a finite time. We find a different directional effect if the reservoirs are thermally excited. Second, we study the decoherence of the steering of a Schrödinger cat state, modeled as the entangled state of a spin and harmonic oscillator, when the macroscopic system (the cat) is coupled to a reservoir.
The Einstein-Podolsky-Rosen (EPR) paradox is one of the milestones in quantum foundations, arising from the lack of local realistic description of quantum mechanics. The EPR paradox has stimulated an important concept of "quantum nonlocality", which manifests itself by three different types: quantum entanglement, quantum steering, and Bell nonlocality. Although Bell nonlocality is more often used to show the "quantum nonlocality", the original EPR paradox is essentially a steering paradox. In this work, we formulate the original EPR steering paradox into a contradiction equality,thus making it amenable to an experimental verification. We perform an experimental test of the steering paradox in a two-qubit scenario. Furthermore, by starting from the steering paradox, we generate a generalized linear steering inequality and transform this inequality into a mathematically equivalent form, which is more friendly for experimental implementation, i.e., one may only measure the observables in xxx-, yyy-, or zzz-axis of the Bloch sphere, rather than other arbitrary directions. We also perform experiments to demonstrate this scheme. Within the experimental errors, the experimental results coincide with the theoretical predictions. Our results deepen the understanding of quantum foundations and provide an efficient way to detect the steerability of quantum states.