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arXiv:2607.00346

Extreme volume monogamy via bound-state engineering

Wen-Jie Zhang, Jun-Hong An

quant-phSubmitted Jul 1, 2026• Announced Jul 2, 2026• 0 citations
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Abstract

Quantum steering ellipsoid (QSE) provides a faithful representation of a two-qubit state. When extended to tripartite systems, the steerability from a trusted party to different receivers is subject to volume monogamy relations, which only constrain the total steerability but cannot individually eliminate the steerability of an untrusted third party, leaving a potential channel for information leakage via steering. Here, we show that this residual steerability can be completely suppressed by selectively engineering bound states in local qubit-environment subsystems, without compromising the steerability between trusted parties. Specifically, when bound states are formed in the subsystems formed by the trusted parties and their environments but absent in the untrusted one, the untrusted party's QSE volume decays to zero, while the trusted party's QSE volume remains finite. Our results establish selective bound-state engineering as a mechanism for extreme volume monogamy, with potential applications in secure quantum communication with an untrusted third party.

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Physica Scripta•Mar 17, 2026•0 citations
Quantum steering ellipsoid (QSE) provides a faithful representation of a two-qubit state. When extended to tripartite systems, the steerability from a trusted party to different receivers is subject to volume monogamy relations, which only constrain the total steerability but cannot individually eliminate the steerability of an untrusted third party, leaving a potential channel for information leakage via steering. Here, we show that this residual steerability can be completely suppressed by selectively engineering bound states in local qubit-environment subsystems, without compromising the steerability between trusted parties. Specifically, when bound states are formed in the subsystems formed by the trusted parties and their environments but absent in the untrusted one, the untrusted party's QSE volume decays to zero, while the trusted party's QSE volume remains finite. Our results establish selective bound-state engineering as a mechanism for extreme volume monogamy, with potential applications in secure quantum communication with an untrusted third party.
Quantum steering ellipsoids (QSEs) provide a geometric representation, within the Bloch picture, of all possible states to which one qubit can be steered through measuring another correlated qubit. However, in realistic settings, quantum systems are inevitably coupled to their environment, resulting in decoherence and degradation of the QSE. Here, by investigating how local dissipative environments coupled to each qubit affect the quantum steering, we find that the geometry of each party's QSE is closely tied to the non-Markovian effect and the formation of a bound state in the energy spectrum of the total qubit-environment system. The bound state provides the ability and the non-Markovian effect provides the dynamical way for preserving the QSE. We systematically examine the characteristics of QSEs under three distinct scenarios, i.e., two-sided bound states, one-sided bound states, and no bound state, revealing a diverse range of steering types. Our work establishes quantum reservoir engineering as a tunable strategy for protecting and controlling quantum steering in open systems, offering a practical pathway toward robust steering-based quantum technologies.
The quantum steering ellipsoid can be used to visualize 2-qubit states, and thus provides a generalization of the Bloch picture for the single qubit. Recently, a monogamy relation for the volumes of steering ellipsoids has been derived for pure 3-qubit states and shown to be stronger than the celebrated Coffman-Kundu-Wootters inequality. We first demonstrate the close connection between this volume monogamy relation and the classification of pure 3-qubit states under stochastic local operations and classical communication. We then show that this monogamy relation does not hold for general mixed 3-qubit states and derive a weaker monogamy relation that does hold for such states. We also prove a volume monogamy relation for pure 4-qubit states (further conjectured to hold for the mixed case), and generalize our 3-qubit inequality to nnn qubits. Finally, we study the effect of noise on the quantum steering ellipsoid and find that the volume of any 2-qubit state is nonincreasing when the state is exposed to arbitrary local noise. This implies that any volume monogamy relation for a given class of multiqubit states remains valid under the addition of local noise. We investigate this quantitatively for the experimentally relevant example of isotropic noise.
Any two-qubit state can be faithfully represented by a steering ellipsoid inside the Bloch sphere, but not every ellipsoid inside the Bloch sphere corresponds to a two-qubit state. We give necessary and sufficient conditions for when the geometric data describe a physical state and investigate maximal volume ellipsoids lying on the physical-unphysical boundary. We derive monogamy relations for steering that are strictly stronger than the Coffman–Kundu–Wootters (CKW) inequality for monogamy of concurrence. The CKW result is thus found to follow from the simple perspective of steering ellipsoid geometry. Remarkably, we can also use steering ellipsoids to derive non-trivial results in classical Euclidean geometry, extending Eulerʼs inequality for the circumradius and inradius of a triangle.
The set of all qubit states that can be steered to by measurements on a correlated qubit is predicted to form an ellipsoid-called the quantum steering ellipsoid-in the Bloch ball. This ellipsoid provides a simple visual characterization of the initial two-qubit state, and various aspects of entanglement are reflected in its geometric properties. We experimentally verify these properties via measurements on many different polarization-entangled photonic qubit states. Moreover, for pure three-qubit states, the volumes of the two quantum steering ellipsoids generated by measurements on the first qubit are predicted to satisfy a tight monogamy relation, which is strictly stronger than the well-known monogamy of entanglement for concurrence. We experimentally verify these predictions, using polarization and path entanglement. We also show experimentally that this monogamy relation can be violated by a mixed entangled state, which nevertheless satisfies a weaker monogamy relation.
Quantum steering plays a fundamental role in quantum information processing regime. In particular, there are two steering-related concepts: one is the quantum steering ellipsoid, which has been used to visualize the two-qubit state, and the other is the Einstein-Podolsky-Rosen (EPR) steering that is a nonlocal correlation between entanglement and Bell nonlocality. In this work, we study the monogamy phenomenon for these two steering properties of the general n-qubit states with n > 4. First, we conjecture that there is a universal tradeoff relation in terms of the spin correlation matrices of reduced bipartite states. From this relation, we can derive the similar monogamy relation for the volume of quantum steering ellipsoids and the linear EPR-steering witness.
Abstract Quantum steering ellipsoid (QSE) provides a faithful and geometric representation for arbitrary two-qubit state, and many properties of two-qubit state can be represented by the parameters of the QSE. The volume of the QSE is one of the nontrivial and visual tools to explore quantum resources. Previous investigations are mostly restricted to special two-qubit states. Here, considering arbitrary two-qubit state, we demonstrate how entanglement and Bell nonlocality can be revealed through the QSE, and use the QSE volume to bound entanglement and Bell nonlocality for arbitrary two-qubit state. The upper and lower bounds of entanglement and Bell nonlocality are investigated through the QSE, and the two-qubit states that can achieve these bounds are obtained. Also, our results reveal that if and only if the two-qubit pure state is an entangled state, the corresponding QSE must fill the whole Bloch sphere (the QSE volume reaches the maximum). For the rank-2 separable states, the corresponding QSE must be a degenerate ellipsoid. For arbitrary two-qubit separable state, the volume of the QSE is not necessarily equal to zero. If arbitrary two-qubit state cannot violate the CHSH inequality, then the QSE volume may be greater than zero or equal to zero. These findings provide examples demonstrating the possibility of unveiling quantum resource of arbitrary two-qubit state in a geometrically visible scenario.