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Journal of Physics B Atomic Molecular and Optical Physics

Generalized quantum steering ellipsoids for a qubit-field system

K. P. Athulya, Anil Shaji

Published Jun 18, 2020• 1 citation
Publisher

Abstract

Abstract Quantum steering in a system consisting of a qubit coupled to a single-mode field are explored when classical-like measurements implemented by heterodyne detection schemes that collapse the state of the field on to a coherent state are considered with the aim of constructing the generalized quantum steering ellipsoid for the qubit. The steering ellipsoid of the qubit enables visualization of the set of states on to which it can be steered using such measurements. In some cases, the steering set does not form an ellipsoid since only heterodyne detection is considered. This is particularly the case when the initial qubit-field state is mixed. The structure of the steering set is explored in detail for this case with representative examples. Evolution of the steering ellipsoid corresponding to joint evolution of the qubit and field under the Jaynes–Cummins Hamiltonian is also studied.

DOI
10.1088/1361-6455/ab9e19

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Abstract Quantum steering in a system consisting of a qubit coupled to a single-mode field are explored when classical-like measurements implemented by heterodyne detection schemes that collapse the state of the field on to a coherent state are considered with the aim of constructing the generalized quantum steering ellipsoid for the qubit. The steering ellipsoid of the qubit enables visualization of the set of states on to which it can be steered using such measurements. In some cases, the steering set does not form an ellipsoid since only heterodyne detection is considered. This is particularly the case when the initial qubit-field state is mixed. The structure of the steering set is explored in detail for this case with representative examples. Evolution of the steering ellipsoid corresponding to joint evolution of the qubit and field under the Jaynes–Cummins Hamiltonian is also studied.
Quantum steering in a system consisting of a qubit coupled to a single-mode field is explored when classical-like measurements implemented by heterodyne detection schemes that collapse the state of the field on to a coherent state is considered. The quantum steering ellipsoid of the qubit is constructed to visualize the set of states on to which it can be steered using such measurements. In some cases, the steering set does not form an ellipsoid since only heterodyne detection is considered. Evolution of the steering ellipsoid corresponding to joint evolution of the qubit and field under the Jaynes-Cummins hamiltonian is also studied.
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 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.
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 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.