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.
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.
Abstract An arbitrary bipartite state can be visually characterized through a quantum steering ellipsoid (QSE), and different nonclassical correlations can be visually revealed via the geometric traits of the QSE. Here, we characterize the QSE of bipartite Bell diagonal states subject to the influences of the bit flip (BF), the depolarizing (DP), and the phase damping (PD) channels, and ascertain the influences of various noise channels on the QSE. By employing nonlinear asymmetric quantum steering criterion (NAC), linear asymmetric quantum steering criterion (LAC), quantum steering criterion from the entropic uncertainty relation (SCEUR), and the shape of the QSE, we reveal and witness the quantum steering and the nonlocal advantage of quantum coherence (NAQC) of the system under these noise channels. It turns out that noise channels can significantly affect the shape of the QSE, one can effectively capture quantum steering and the NAQC via the QSE. Among the NAC, LAC, and SCEUR, the LAC (the NAC) is the most promising avenue for detecting the quantum steering under the DP and PD channels (under the the BF channels). It is particularly important to note that the shape of the QSE can visually display the abilities of these criteria in witnessing quantum steering. Moreover, based on the l1 norm of coherence, the NAQC can be achieved under the BF, DP, and PD channels if the sum of three semiaxis lengths of the QSE can surpass $\sqrt 6 $.
A quantum steering ellipsoid (QSE) is a visual characterization for bipartite qubit systems, and it is also a novel avenue for describing and detecting quantum correlations. Herein, by using a QSE, we visualize and witness the first-order coherence (FOC), Bell nonlocality (BN) and purity under non-inertial frames. Also, the collective influences of the depolarizing channel and the non-coherence-generating channel (NCGC) on the FOC, BN and purity are investigated in the QSE formalism. The results reveal that the distance from the center of the QSE to the center of the Bloch sphere visualizes the FOC of a bipartite system, the lengths of the QSE semiaxis visualize the BN, and the QSE’s shape and position dominate the purity of the system. One can capture the FOC, BN and purity via the shape and position of the QSE in the non-inertial frame. The depolarizing channel (the NCGC) gives rise to the shrinking and degradation (the periodical oscillation) of the QSE. One can use these traits to visually characterize and detect the FOC, BN and purity under the influence of external noise. Of particular note is that the condition for the QSE to achieve the center of the Bloch sphere cannot be influenced by the depolarizing channel and the NCGC. The characterization shows that the conditions for the disappearance of the FOC are invariant under the additional influences of the depolarizing channel and NCGC.
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.
The effect of a local trace-preserving single-qubit channel on a two-qubit state is investigated in the frame of quantum steering ellipsoids (QSEs). The phenomenon of locally increased quantum discord is visualized in this picture. We strictly prove that a B-side two-qubit discordant state can be locally prepared from a classical state by a trace-preserving channel on qubit B if and only if its QSE of B is a nonradial line segment. For states with higher-dimensional QSEs, the phenomenon of locally increased quantum correlation generally happens when the shape of the QSE is like a baguette. Based on this observation, we find a class of entangled states whose quantum discord can be increased by the local amplitude-damping channel. Further, we find that the local quantum channel does not increase the size of the QSEs of either qubit A or qubit B, for the needle-shaped QSE states, or for the Bell diagonal states with higher-dimensional QSEs.
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 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.