Demonstrating One-Way Einstein-Podolsky-Rosen Steering in Two Qubits
Authors: Kai Sun1, Xiang-Jun Ye1, Jin-Shi Xu1, Jing-Ling Chen2, Chuan-Feng Li1, Guang-Can Guo1
1Key Laboratory of Quantum Information, CAS Centre for Excellence and Synergetic Innovation Centre in Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei, China.
2Chern Institute of Mathematics, Nankai University, Tianjin, China.
Asymmetric Einstein-Podolsky-Rosen (EPR) steering is an important “open question” proposed when EPR steering is reformulated in 2007 . Suppose Alice and Bob share a pair of two-qubit states; it is easy to imagine that if Alice entangles with Bob, then Bob must also entangle with Alice. Such a symmetric feature holds for both entanglement and Bell nonlocality . However, the situation is dramatically changed when one turns to a novel kind of quantum nonlocality, the EPR steering, which stands between entanglement and Bell nonlocality. It may happen that for some asymmetric bipartite quantum states, Alice can steer Bob but Bob cannot steer Alice. This distinguished feature would be useful for the one-way quantum tasks. The first experimental verification of one-way EPR steering was performed by using two entangled continuous variable systems in 2012 . However, the experiments demonstrating one-way EPR steering [3,4] are restricted to Gaussian measurements, and for more general measurements, like projective measurements, there is no experiment realizing the asymmetric feature of EPR steering, even though the theoretical analysis has been proposed .
Figure 1: Illustration of one-way EPR steering. In one direction (red), EPR steering is realized and this direction is safe for quantum information. In the other direction (blue), steering task fails and this direction is not safe.
Recently, we for the first time quantified the steerability and demonstrated one-way EPR steering in the simplest entangled system (two qubits) using two-setting projective measurements . The asymmetric two-qubit states in the form of ρAB = η |Ψ(θ)⟩⟨Ψ(θ)| + (1-η) |Φ(θ)⟩⟨Φ(θ)|, where 0 ≤ η≤ 1, |Ψ(θ)⟩ = cos θ |0A 0B⟩ + sinθ |1A 1B⟩, |Φ(θ)⟩ = cosθ |1A 0B⟩ + sinθ |0A 1B⟩, are prepared in this experiment (see Figure 2(a) ). Based on the steering robustness , an intuitive criterion R called as “steering radius” is defined to quantify the steerability (see Figure 2 (c) ). The different values of R on two sides clearly illustrate the asymmetric feature of EPR steering. Furthermore, the one-way steering is demonstrated when R > 1 on one side and R < 1 on the other side (see Figure 2 (b)).
Figure 2: (click on the figure to view with higher resolution) Experimental results for asymmetric EPR steering. (a) The distribution of the experimental states. The right column shows the entangled states we prepared, and the left column is a magnification of the corresponding region in the right column. The states located in the yellow (grey) regions are predicted to realize one-way (two-way) steering theoretically in the case of two-setting measurements. The blue points and red squares represent the states realizing one-way and two-way EPR steering, respectively. The black triangles represent the states for which EPR steering task fails for both observers. (b) The values of R for the states are labeled in the left column in (a). The red squares represent the situation where Alice steers Bob's system, and the blue points represent the case where Bob steers Alice's system. (c) Geometric illustration of the strategy for local hidden states (black points) to construct the four normalized conditional states (red points) obtained from the maximally entangled state.
For the failing EPR steering process, the local hidden state model, which provides a direct and convinced contradiction between the nonlocal EPR steering and classical physics, is prepared experimentally to reconstruct the conditional states obtained in the steering process (see Figure 3).
Figure 3. (click on the figure to view with higher resolution) The experimental results of the normalized conditional states and local hidden states shown in the Bloch sphere. The theoretical and experimental results of the normalized conditional states are marked by the black and red points (hollow), respectively. The blue and green points represent the results of the four local hidden states in theory and experiment, respectively. The normalized conditional states constructed by the local hidden states are shown by the brown points. Spheres (a) and (c) are for the case in which Alice steers Bob's system, whereas (b) and (d) show the case in which Bob steers Alice's system. The parameters of the shared state in (a) and (b) are θ = 0.442 and η = 0.658; the parameters of the shared state in (c) and (d) are θ = 0.429 and η = 0.819. The spheres (a), (b) and (d) show that the local hidden state models exist, and the steering tasks fail. The sphere (c) Shows that no local hidden state model exists for the steering process with the constructed hidden states located beyond the Bloch sphere and R = 1.076.
The quantification of EPR steering provides an intuitional and fundamental way to understand the EPR steering and the asymmetric nonlocality. The demonstrated asymmetric EPR steering is significant within quantum foundations and quantum information, and shows the applications in the tasks of one-way quantum key distribution  and the quantum sub-channel discrimination , even within the frame of two-setting measurements.
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