This paper investigates a recently discovered geometric representation of two-qubit states, the Quantum Steering Ellipsoid, by exploring quantum state measurements on mixtures of polarization-entangled photons produced using spontaneous parametric down conversion (SPDC). Measurements of the second photon of a mixed biphoton system that are conditioned on observations of the first photon demonstrate all possible Bloch Vectors that the second photon can be collapsed to by way of ”steering,” which is only possible due to the nonlocal nature of entangled systems. This paper examines methods to experimentally verify the Quantum Steering Ellipsoid as a geometric classification of quantum states.
This paper investigates a recently discovered geometric representation of two-qubit states, the Quantum Steering Ellipsoid, by exploring quantum state measurements on mixtures of polarization-entangled photons produced using spontaneous parametric down conversion (SPDC). Measurements of the second photon of a mixed biphoton system that are conditioned on observations of the first photon demonstrate all possible Bloch Vectors that the second photon can be collapsed to by way of ”steering,” which is only possible due to the nonlocal nature of entangled systems. This paper examines methods to experimentally verify the Quantum Steering Ellipsoid as a geometric classification of quantum states.
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 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 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.
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.
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 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.