The quantum paradox has provided an intuitive way to reveal the essential difference between quantum mechanics and classical theory. In 1935, by considering a continuous-variable entangled state
Photonics Research, Volume. 9, Issue 6, 992(2021)
Steering paradox for Einstein–Podolsky–Rosen argument and its extended inequality
The Einstein–Podolsky–Rosen (EPR) paradox is one of the milestones in quantum foundations, arising from the lack of a local realistic description of quantum mechanics. The EPR paradox has stimulated an important concept of “quantum nonlocality,” which manifests itself in three types: quantum entanglement, quantum steering, and Bell’s nonlocality. Although Bell’s nonlocality is more often used to show “quantum nonlocality,” the original EPR paradox is essentially a steering paradox. In this work, we formulate the original EPR steering paradox into a contradiction equality, thus making it amenable to experimental verification. We perform an experimental test of the steering paradox in a two-qubit scenario. Furthermore, by starting from the steering paradox, we generate a generalized linear steering inequality and transform this inequality into a mathematically equivalent form, which is friendlier for experimental implementation, i.e., one may measure the observables only in the
1. INTRODUCTION
The quantum paradox has provided an intuitive way to reveal the essential difference between quantum mechanics and classical theory. In 1935, by considering a continuous-variable entangled state
Undoubtedly, the EPR paradox is a milestone in quantum foundations, as it has opened the door of “quantum nonlocality.” In 1964, Bell made a distinct response to the EPR paradox by showing that quantum entangled states may violate Bell’s inequality, which hold for any local-hidden-variable model [4]. This indicates that local-hidden-variable models cannot reproduce all quantum predictions, and the violation of Bell’s inequality by entangled states directly implies a kind of nonlocal property—Bell’s nonlocality. Since then, Bell’s nonlocality has achieved rapid and fruitful development in two directions [5]. (i) On one hand, more and more Bell’s inequalities have been introduced to detect Bell’s nonlocality in different physical systems, e.g., the Clause–Horne–Shimony–Holt (CHSH) inequality for two qubits [6], the Mermin–Ardehali–Belinskii–Klyshko (MABK) inequality for multipartite qubits [7], and the Collins–Gisin–Linden–Masser–Popescu inequality for two qudits [8]. (ii) On the other hand, some novel quantum paradoxes, or the all-versus-nothing (AVN) proofs, have been suggested to reveal Bell’s nonlocality without inequalities. Typical examples are the Greenberger–Horne–Zeilinger (GHZ) paradox [9] and the Hardy paradox [10]. Experimental verifications of Bell’s nonlocality have also been carried out; for instance, Aspect
Despite being developed from the EPR paradox, Bell’s nonlocality does not directly correspond to the EPR paradox. As pointed out in Ref. [3], inspired by the EPR argument, one can derive three types of “quantum nonlocality”: quantum entanglement, quantum steering, and Bell’s nonlocality. The original EPR paradox is actually a special case of quantum steering [14]. Although quantum steering has been experimentally demonstrated in various quantum systems [15–24], all of these experiments just indirectly illustrate the EPR paradox, in which most of them are based on statistical inequalities. Here the direct illustration of a quantum paradox means that we can find a contradiction equality for this paradox and demonstrate it (Ref. [16] is an AVN proof but not a contradiction equality). For example, (i) the GHZ paradox [9] can be formulated as a contradiction equality “
The purpose of this paper is two-fold. (i) Based on our previous results of the steering paradox “
2. EPR PARADOX AS A STEERING PARADOX “k = 1”
Following Ref. [25], let us consider an arbitrary two-qubit pure entangled state
By performing a projective measurement on her qubit along the
Here we show that a more general steering paradox “
Experimentally, we test the EPR paradox for a two-qubit system in the simplest case of
3. GENERALIZED LINEAR STEERING INEQUALITY
Just as Bell’s inequalities may be derived from the GHZ and Hardy paradoxes [26,27], this is also the case for the EPR paradox. In turn, from the steering paradox “
The GLSI has two remarkable advantages over the usual LSI [15]. (i) Based on its own form as in the inequality (5), the GLSI includes naturally the usual LSI as a special case, and thus can detect more quantum states. In particular, the GLSI can detect the steerability for all pure entangled states Eq. (1) in the whole region
To be more specific, we give an example of the three-setting GLSI from the inequality (5), where Alice’s three measuring directions are
In the experiment to test the inequalities, Alice prepares two qubits and sends one of them to Bob, who trusts his own measurements but not Alice’s. Bob asks Alice to measure at random
4. EXPERIMENTAL RESULTS
Figure 1.Experimental setup. Polarization-entangled photons pairs are generated via nonlinear crystal. An asymmetric loss interferometer along with half-wave plates (HWPs) is used to prepare two-qubit pure entangled states. The projective measurements are performed using wave plates and polarization beam splitter (PBS).
Figure 2.Experimental results for pure states. (a) Experimental results concerning the steering paradox “
Second, we experimentally address the violations of the GLSI using the above pure states
Figure 3.Experimental results for mixed states. (a), (b) Steering detection for the generalized Werner state
5. CONCLUSION
In summary, we have advanced the study of the EPR paradox in two aspects. (i) We have presented a generalized steering paradox “
Recently, quantum steering has been applied to the one-sided device-independent quantum key distribution protocol to secure shared keys by measuring the quantum steering inequality [33]. Our GLSI can also be applied to this scenario to implement the one-sided device-independent quantum key distribution (one-sided DIQKD). In addition, our results may be applied to applications such as quantum random number generation [34,35] and quantum sub-channel discrimination [36,37].
APPENDIX A: GENERALIZED LINEAR STEERING INEQUALITY OBTAINED FROM THE GENERAL STEERING PARADOX “k = 1”
Actually, from the steering paradox “
The derivation procedure is as follows: in the steering scenario
The quantity on the left-hand-side of Eq.?(
Remark 1. In Ref. [
Let us rewrite the projective measurements Eq.?(
Let
Remark 2. We may rewrite the GLSI (
APPENDIX B: EPR STEERING BY USING THE THREE-SETTING GLSI
In this experimental work, we demonstrate EPR steering for the two-qubit generalized Werner state by using the GLSI. We focus on the three-setting GLSI. In the steering scenario
(i)?
Similarly, for
(ii)?
Similarly, for
Thus, in summary, the classical bound is given by [here
Let
Example 1. Let us consider the two-qubit pure state
Example 2. Let us consider the two-qubit generalized Werner state
For the state Eq.?(
Figure 4.Detecting EPR steerability of the generalized Werner state by using the usual three-setting LSI (blue line) and three-setting GLSI (red line). For a fixed parameter
Figure 5.Generalized Werner states violate the usual three-setting LSI in the blue region and three-setting generalized LSI in the red region. It can be observed that the GLSI is stronger than the usual LSI in detecting EPR steerability.
Figure 6.Detecting EPR steerability of the mixed state Eq. (
Figure 7.Mixed states Eq. (
APPENDIX C: EXPERIMENTAL DETAILS
A 404?nm laser is sent into a nonlinear BBO crystal to generate the maximally entangled state of the form
In our experiments, the verification of the mixed state is achieved by probabilistically mixing the corresponding pure states. Specifically, we measured the corresponding observables in different pure states and post-processed the data (changing the probability of these pure states and mixing them together) to obtain experimental data of different mixed states. Now we show how to construct two types of mixed states, the generalized Werner state, and an asymmetric mixed state [
Figure 8.Experimental setup and the specific angles for state preparation.
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Tianfeng Feng, Changliang Ren, Qin Feng, Maolin Luo, Xiaogang Qiang, Jing-Ling Chen, Xiaoqi Zhou, "Steering paradox for Einstein–Podolsky–Rosen argument and its extended inequality," Photonics Res. 9, 992 (2021)
Category: Quantum Optics
Received: Oct. 6, 2020
Accepted: Mar. 17, 2021
Published Online: May. 20, 2021
The Author Email: Changliang Ren (renchangliang@hunnu.edu.cn), Jing-Ling Chen (chenjl@nankai.edu.cn), Xiaoqi Zhou (zhouxq8@mail.sysu.edu.cn)