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arXiv:1604.00265 · Physical Review A

Nonseparability and steerability of two-qubit states from the geometry of steering outcomes

H. Chau Nguyen, Thanh Vu

Published Jul 20, 2016• 35 citations
arXivHTMLPDF

Abstract

When two qubits, $A$ and $B$, are in an appropriate state, Alice can remotely steer Bob's system $B$ into different ensembles by making different measurements on $A$. This famous phenomenon is known as quantum steering, or Einstein-Podolsky-Rosen steering. Importantly, quantum steering establishes the correspondence not only between a measurement on $A$ (made by Alice) and an ensemble of $B$ (owned by Bob) but also between each of Alice's measurement outcomes and an unnormalized conditional state of Bob's system. The unnormalized conditional states of $B$ corresponding to all possible measurement outcomes of Alice are called Alice's steering outcomes. We show that, surprisingly, the four-dimensional geometry of Alice's steering outcomes completely determines both the nonseparability of the two-qubit state and its steerability from her side. Consequently, the problem of classifying two-qubit states into nonseparable and steerable classes is equivalent to geometrically classifying certain four-dimensional skewed double cones.

DOI
10.1103/physreva.94.012114

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•Jan 1, 2013•48 citations
When two qubits, AAA and BBB, are in an appropriate state, Alice can remotely steer Bob's system BBB into different ensembles by making different measurements on AAA. This famous phenomenon is known as quantum steering, or Einstein-Podolsky-Rosen steering. Importantly, quantum steering establishes the correspondence not only between a measurement on AAA (made by Alice) and an ensemble of BBB (owned by Bob) but also between each of Alice's measurement outcomes and an unnormalized conditional state of Bob's system. The unnormalized conditional states of BBB corresponding to all possible measurement outcomes of Alice are called Alice's steering outcomes. We show that, surprisingly, the four-dimensional geometry of Alice's steering outcomes completely determines both the nonseparability of the two-qubit state and its steerability from her side. Consequently, the problem of classifying two-qubit states into nonseparable and steerable classes is equivalent to geometrically classifying certain four-dimensional skewed double cones.
Einstein-Podolsky-Rosen steering is a form of quantum nonlocality exhibiting an inherent asymmetry between the observers, Alice and Bob. A natural question is then whether there exist entangled states which are one-way steerable, that is, Alice can steer Bob's state, but it is impossible for Bob to steer the state of Alice. So far, such a phenomenon has been demonstrated for continuous variable systems, but with a strong restriction on allowed measurements, namely, considering only Gaussian measurements. Here we present a simple class of entangled two-qubit states which are one-way steerable, considering arbitrary projective measurements. This shows that the nonlocal properties of entangled states can be fundamentally asymmetrical.
Fully characterizing the steerability of a quantum state of a bipartite system has remained an open problem ever since the concept of steerability was first defined. In this paper, using our recent geometrical approach to steerability, we suggest a necessary and sufficient condition for a two-qubit state to be steerable with respect to projective measurements. To this end, we define the critical radius of local models and show that a state of two qubits is steerable with respect to projective measurements from Alice's side if and only if her critical radius of local models is less than 1. As an example, we calculate the critical radius of local models for the so-called T-states by proving the optimality of a recently suggested local hidden state model.
Einstein-Podolsky-Rosen steering is a form of quantum nonlocality intermediate between entanglement and Bell nonlocality. Although Schrödinger already mooted the idea in 1935, steering still defies a complete understanding. In analogy to "all-versus-nothing" proofs of Bell nonlocality, here we present a proof of steering without inequalities rendering the detection of correlations leading to a violation of steering inequalities unnecessary. We show that, given any two-qubit entangled state, the existence of certain projective measurement by Alice so that Bob's normalized conditional states can be regarded as two different pure states provides a criterion for Alice-to-Bob steerability. A steering inequality equivalent to the all-versus-nothing proof is also obtained. Our result clearly demonstrates that there exist many quantum states which do not violate any previously known steering inequality but are indeed steerable. Our method offers advantages over the existing methods for experimentally testing steerability, and sheds new light on the asymmetric steering problem.
Einstein-Podolsky-Rosen (EPR) steering refers to the ability of one observer to convince a distant observer that they share entanglement by making local measurements. Determining which states allow a demonstration of EPR steering remains an open problem in general. Here, we outline and demonstrate a method of analytically constructing new classes of two-qubit states which are nonsteerable by arbitrary projective measurements, from consideration of local operations performed by the steering party on states known to be nonsteerable.
Quantum steering is an important nonclassical resource for quantum information processing. However, even though lots of steering criteria exist, it is still very difficult to efficiently determine whether an arbitrary two-qubit state shared by Alice and Bob is steerable or not, because the optimal measurement directions of Alice are unknown. In this work, we provide an efficient quantum steering detection scheme for an arbitrary two-qubit state with the help of machine learning, where Alice and Bob only need to measure in a few fixed measurement directions. In order to prove the validity of this method, we first realize a high-performance quantum steering classifier with the whole information. Furthermore, a high-performance quantum steering classifier with partial information is realized, where Alice and Bob only need to measure in three fixed measurement directions. Our method outperforms the existing methods in generic cases in terms of both speed and accuracy, opening up the avenues to explore quantum steering via the machine learning approach.
Einstein-Podolsky-Rosen steering is a form of quantum nonlocality intermediate between entanglement and Bell nonlocality. Although Schro¨dinger already mooted the idea in 1935, steering still defies a complete understanding. In analogy to ‘‘all-versus-nothing’’ proofs of Bell nonlocality, here we present a proof of steering without inequalities rendering the detection of correlations leading to a violation of steering inequalities unnecessary. We show that, given any two-qubit entangled state, the existence of certain projective measurement by Alice so that Bob’s normalized conditional states can be regarded as two \ndifferent pure states provides a criterion for Alice-to-Bob steerability. A steering inequality equivalent to the all-versus-nothing proof is also obtained. Our result clearly demonstrates that there exist many quantum states which do not violate any previously known steering inequality but are indeed steerable. Our method offers advantages over the existing methods for experimentally testing steerability, and sheds new light on the asymmetric steering problem.