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Physica Scripta

Bounding entanglement and Bell nonlocality for arbitrary two-qubit state via quantum steering ellipsoid

Ling-Ling Xing, Huan Yang

Published Mar 17, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ae5354

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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.
Abstract Quantum steering ellipsoid (QSE) can faithfully characterize arbitrary bipartite state, and provide a new method to visually investigate and capture various quantum nonlocalities. Note that the dissipation of quantum nonlocalities can be accelerated by the coupling between the system and the decoherence channel, which induces great challenges in detecting quantum nonlocalities. Herein, considering the scenario in which the subsystem of a two-qubit system is coupled with a decoherence channel, the quantum discord (QD) and concurrence are visually characterized and detected by employing the QSE. The results reveal that the QSE y and z semiaxes are responsible for the traits of the QD and concurrence under the bit flip channel. The x and y semiaxes dominate and visualize the QD and concurrence under the phase damping channel. Of particular note is that one can realize the detections of the QD and concurrence via the shape of the QSE. To be clearer, the disappearance of the QD can be ascertained according to a needle-shaped QSE or a vanishing QSE in the Bloch sphere. In contrast, one can witness the concurrence if the total semiaxis length is greater than one. Beyond this, our results visually reveal that quantum entanglement is a stronger quantum nonlocality than the QD from the perspective of geometry, and thus deepens the understanding of the quantum nonlocality.
The quantum steering ellipsoid of a two-qubit state is the set of Bloch vectors that Bob can collapse Alice's qubit to, considering all possible measurements on his qubit. We provide an elementary construction of the ellipsoid for arbitrary states, calculate its volume, and explain how this geometric representation can be made faithful. The representation provides a range of new results, and uncovers new features, such as the existence of "incomplete steering" in separable states. We show that entanglement can be analyzed in terms of three geometric features of the ellipsoid and prove that a state is separable if and only if it obeys a "nested tetrahedron" condition.
Graphical representations are invaluable for visualising physical systems and processes. In quantum information theory, the Bloch vector representation of a single qubit is ubiquitous, but visualising higher-dimensional quantum systems is far less straightforward. The quantum steering ellipsoid provides a method for geometrically representing the state of two qubits, the most fundamental system for studying quantum correlations. This thesis constitutes a significant development of the steering ellipsoid formalism. As well as offering new insight into the study of two-qubit states, we extend this powerful geometric approach to explore scenarios beyond two qubits. We find necessary and sufficient conditions for when an ellipsoid inside the Bloch ball describes a valid (i.e. positive semidefinite) two-qubit state. Combined with the notion of ellipsoid chirality, this enables a geometric characterisation of entanglement. We find a family of "maximally obese" two-qubit states whose ellipsoids have maximal volume. These states have optimal correlation properties within the set of all two-qubit states with a single maximally mixed marginal. We study a three-qubit scenario and discover that ellipsoid volume obeys an elegant monogamy of steering relationship. From this we can derive the Coffman-Kundu-Wootters (CKW) inequality for concurrence monogamy, providing an intuitive geometric derivation of this classic result. Remarkably, we find that steering ellipsoids offer a fresh perspective on questions beyond quantum state space. Entanglement witnesses are also very naturally represented and classified using the formalism. This gives a physical interpretation to any ellipsoid inside the Bloch ball as a block positive two-qubit operator, which we may then classify further. We can also use steering ellipsoids to derive some highly nontrivial results in classical Euclidean geometry, extending Euler's inequality for the circumradius and inradius of a triangle.
We prove, modulo a conjecture on quantum steering ellipsoids being true, the existence of the phenomenon of locally inaccessible hidden quantum correlations, that is, the existence of two-particle states whose hidden quantum correlations cannot be revealed by local filters implemented exclusively on one side of the experiment but that can still be revealed when both parties cooperate in applying judiciously chosen local filters. The quantum correlations considered here are the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality for Bell nonlocality and the violation of the F3{F}_{3}F3​ inequality for Einstein-Podolsky-Rosen steering. Specifically, we provide a necessary criterion for guaranteeing the presence of such phenomena for arbitrary two-qubit states. This criterion in turn relies on the conjecture that the maximal violations of CHSH inequality and F3{F}_{3}F3​ inequality are both upper bounded by functions that depend on the magnitude of the quantum steering ellipsoid center. This latter conjecture, although currently lacking an analytical proof, is supported by numerical results. We use this necessary criterion to explicitly show examples of two-qubit states with locally inaccessible hidden quantum correlations and furthermore two-qubit states with locally inaccessible maximal hidden quantum correlations.
Employing the volume of quantum steering ellipsoids (QSEs) as a measure on the fifteen-dimensional convex set of two-qubit states, we estimate the ratio of the integral of the measure over the separable states to its integral over all (separable and entangled) states to be 0.0288. This can be contrasted with the considerably larger separability ratios (probabilities) of 833=233⋅11≈0.242424\frac{8}{33} = \frac{2^3}{3 \cdot 11} \approx 0.242424338​=3⋅1123​≈0.242424 and 25341=5211⋅31≈0.0733138\frac{25}{341}=\frac{5^2}{11 \cdot 31} \approx 0.073313834125​=11⋅3152​≈0.0733138 that various forms of evidence point to with the use of the prominent Hilbert-Schmidt and Bures measures, respectively. The questions of whether the ratio in the QSE setting can be more precisely obtained or even exactly computed, as well as whether a metric can be constructed, the volume element of which yields the measure, remain to be addressed. We also investigate related issues pertaining to absolute separability. Further, we examine the behavior of the separability probability--constant in the Hilbert-Schmidt case and decreasing in the Bures--as a function of the Bloch vector norm in the QSE instance. It appears to increase approaching the pure state boundary.
The quantum steering ellipsoid visualizes the set of all qubit states that can be steered by measuring on another correlated qubit in the Bloch picture. Together with local reduced states, it provides a faithful geometric characterization of the underlying two-qubit state so that almost all nonclassical state features can be reflected in its geometric properties. Consequently, the various types of quantum ellipsoids with different geometric properties form an ellipsoid zoo, which, in this paper, is experimentally verified via measurements on many polarization-path photonic states. By generating two-qubit states with high fidelity, the corresponding ellipsoids are constructed to certify the presence of entanglement, one-way Einstein-Podolsky-Rosen steering, discord, and steering incompleteness. It is also experimentally verified that the steering ellipsoid can be reconstructed from using the 12 vertices of the icosahedron as measurement directions. Our results aid progress in applying the quantum steering ellipsoid to reveal nonclassical features of the multiqubit system.