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arXiv:1411.7449 · Physical Review A

Effect of local channels on quantum steering ellipsoids

Xueyuan Hu, Heng Fan

Published Feb 4, 2015• 23 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.91.022301

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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 BBB-side two-qubit discordant state can be locally prepared from a classical state by a trace-preserving channel on qubit BBB if and only if its QSE of BBB 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 AAA or qubit BBB, for the needle-shaped QSE states, or for the Bell diagonal states with higher-dimensional QSEs.
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.
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.
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{l_1}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 $.
Nonclassical correlations are a key resource to explore foundational quantum information tasks and find applications in device-independent protocols. Quantum steering was formalized in 2007 [1] and is distinct from other nonclassical correlations such as Bell nonlocality [2] and entanglement. Steering describes the effect of a local measurement on one system and affecting the measurement results on the other system. This can be visualized for two-qubit states using quantum steering ellipsoids (QSE), which is the set of Bloch vectors that Alice can collapse Bob's state to. They are theoretically well studied [3-6] and have been recently experimentally demonstrated [7].
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.
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.