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Quantum Information Processing

Quantum steering borders in three-qubit systems

Joanna K. Kalaga, W. Leoński

Published Jun 2, 2017• 32 citations
Publisher

Abstract

We discuss a family of states describing three-qubit systems in a context of quantum steering phenomena. We show that symmetric steering cannot appear between two qubits—only asymmetric steering can appear in such systems. The main aim of this paper is to discuss the possible relations between the entanglement measures and steering parameter for two-mode mixed state corresponding to the qubit–qubit subsystem. We have derived the conditions determining boundary values of the negativity parametrized by concurrence. We show that two-qubit mixed state cannot be steerable when the negativity of such state is smaller than, or equal to, its boundary value. Finally, we have found ranges of the values of the mixedness measure, parametrized by concurrence and negativity for steerable and unsteerable two-qubit mixed states.

DOI
10.1007/s11128-017-1627-6

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We discuss a family of states describing three-qubit systems in a context of quantum steering phenomena. We show that symmetric steering cannot appear between two qubits—only asymmetric steering can appear in such systems. The main aim of this paper is to discuss the possible relations between the entanglement measures and steering parameter for two-mode mixed state corresponding to the qubit–qubit subsystem. We have derived the conditions determining boundary values of the negativity parametrized by concurrence. We show that two-qubit mixed state cannot be steerable when the negativity of such state is smaller than, or equal to, its boundary value. Finally, we have found ranges of the values of the mixedness measure, parametrized by concurrence and negativity for steerable and unsteerable two-qubit mixed states.
Considering the system of three interacting qubits, we analyze four families of states from the point of view of bipartite correlations appearing in two-qubit subsystems of a three-qubit model, such as Einstein-Podolsky-Rosen steering, entanglement, and coherence. We reveal mutual relations among the steering parameter, concurrence, and three measures of coherence (degree of coherence, first-, and second-order correlation functions). Analyzing in parallel the steerable and unsteerable states, we derive analytical formulas giving the maximal and minimal values of coherence measures as concurrence varies.
Abstract Entanglement and steering are used to describe quantum correlation. And steerable states form a strict subset of entangled states. A natural question arises concerning how much territory steerability occupies entanglement for a general two‐qubit entangled state. In this work, the constraint relation between steerability and concurrence is investigated by using two kinds of evolutionary states and any two‐qubit states (or lots of randomly generated states). By combining the theoretical and numerical proofs, the upper and lower boundaries of steerability are obtained. And the lower boundary can be used as a sufficient criterion for steering detection. Furthermore, a special kind of mixed state transformed by performing an arbitrary unitary transformation on Werner‐like state is considered, and a sufficient steering criterion described by concurrence and purity is proposed.
We put forward complementary relations of entanglement, coherence, steering inequality violation, and Bell nonlocality for arbitrary three-qubit states. We show that two families of genuinely entangled three-qubit pure states with a single parameter exist, and they exhibit maximum coherence and steering inequality violation for a fixed amount of negativity, respectively. It is found that the negativity is exactly equal to the geometric mean of bipartite concurrences for the three-qubit pure states, although the negativity is always less than or equal to the latter for three-qubit mixed states. Moreover, the complementary relation between negativity and first-order coherence for tripartite entanglement states is established. Furthermore, we investigate the close relation between the negativity and the maximum steering inequality violation. In addition, the complementary relation between negativity and the maximum Bell-inequality violation for arbitrary three-qubit states is obtained. The results provide reliable evidence of fundamental connections among entanglement, coherence, steering inequality violation, and Bell nonlocality.
Einstein-Podolsky-Rosen (EPR) steering is the ability that an observer persuades a distant observer to share entanglement by making local measurements. Determining a quantum state is steerable or unsteerable remains an open problem. Here, we derive a new steering inequality with infinite measurements corresponding to an arbitrary two-qubit T state, from consideration of EPR steering inequalities with N projective measurement settings for each side. In fact, the steering inequality is also a sufficient criterion for guaranteering that the T state is unsteerable. Hence, the steering inequality can be viewed as a necessary and sufficient criterion to distinguish whether the T state is steerable or unsteerable. In order to reveal the fact that the set composed of steerable states is the strict subset of the set made up of entangled states, we prove theoretically that all separable T states can not violate the steering inequality. Moreover, we put forward a method to estimate the maximum violation from concurrence for arbitrary two-qubit T states, which indicates that the T state is steerable if its concurrence exceeds 1/4.
We investigate the steerability of two-qubit Bell-diagonal states under projective measurements by the steering party. In the simplest nontrivial scenario of two projective measurements, we solve this problem completely by virtue of the connection between the steering problem and the joint-measurement problem. A necessary and sufficient criterion is derived together with a simple geometrical interpretation. Our study shows that a Bell-diagonal state is steerable by two projective measurements iff it violates the Clauser-Horne-Shimony-Holt (CHSH) inequality, in sharp contrast with the strict hierarchy expected between steering and Bell nonlocality. We also introduce a steering measure and clarify its connections with concurrence and the volume of the steering ellipsoid. In particular, we determine the maximal concurrence and ellipsoid volume of Bell-diagonal states that are not steerable by two projective measurements. Finally, we explore the steerability of Bell-diagonal states under three projective measurements. A simple sufficient criterion is derived, which can detect the steerability of many states that are not steerable by two projective measurements. Our study offers valuable insight on steering of Bell-diagonal states as well as the connections between entanglement, steering, and Bell nonlocality.