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arXiv:2007.10808 · Annalen der Physik

Constraint Relation Between Steerability and Concurrence for Two‐Qubit States

Xiao‐Gang Fan, Huan Yang, Fei Ming, Dong Wang, Liu Ye

Published Jun 16, 2021• 20 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1002/andp.202100098

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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.
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 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.
Abstract Quantum entanglement and steering are both fundamental concepts for describing quantum correlations, with steerable states forming a strict subset of entangled states. In general two-qubit systems, the relation between entanglement and steering can be characterized by an inequality, referred to as the constraint relation. In this work, we investigate a two-fermion spin system to explore the influence of static magnetic fields on quantum entanglement and steering. Our results demonstrate that entanglement and steering are equivalent in this system. The finding provides new perspectives for understanding quantum correlations in two-fermion spin systems. Furthermore, we examine the dynamic behavior of entanglement, showing that the maximum entanglement of this system affected by both external and internal factors.
EPR steering is a form of bipartite quantum correlation that is intermediate between entanglement and Bell nonlocality. It is detected via the violation of steering inequalities, which provide sufficient conditions for the steerability of quantum states. Furthermore, there is a sufficient criterion for guaranteeing that a two-qubit state is unsteerable. But there are some two qubit states that can not determine the steerability of the steering inequality and the sufficient criteria. Here we introduce a way of quantifying this phenomenon and use it to study the steerability of two qubit states. This article recommend a platonic solid measurement schemes. And then discuss the problem, namely expanding the scope of steerable, which guarantees that the state is steerable.
The steerability of a quantum state can be detected by steering inequalities. The linear steering inequalities show that more steerable states can be discovered with the increase of measurements. In order to detect more steerable states in two-photon systems, we first theoretically derive an optimized steering criterion based on infinity measurements for an arbitrary two-qubit state. The steering criterion is only determined by the spin correlation matrix of the state, and do not require infinity measurements. Then, we prepared the Werner-like states in two-photon systems, and measure their spin correlation matrices. Finally, we apply three steering criteria, which include our steering criterion, the three-measurement steering criterion and the geometric Bell-like inequality, to distinguish the steerability of these states. The results show that our steering criterion can detect the most steerable states under the same experimental conditions. Thus, our work provides a valuable reference for detecting the steerability of quantum states.
Quantum nonlocality is an essential resource in quantum information and is commonly classified into three distinct forms: quantum entanglement, Einstein-Podolsky-Rosen (EPR) steering, and Bell's nonlocality. Gisin's theorem shows that pure-state entanglement implies Bell nonlocality, and it motivates the question of how entanglement and EPR steering are related beyond pure states. Here we prove that every entangled rank-2 two-qubit state is EPR steerable, with rank-1 states included as a special case. The proof combines a local-unitary parametrization of rank-2 two-qubit states, a necessary and sufficient separability condition, and a state-dependent nonlinear steering inequality. This gives an analytical characterization of EPR steerability for rank-2 two-qubit entangled states. Thus, rank-2 two-qubit entangled states provide certifiable EPR-steering resources in quantum information.