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Estimating quantum steering and Bell nonlocality through quantum entanglement in two-photon systems

Huan Yang, Fa Zhao, Xiao‐Gang Fan, Zhiyong Ding, Dong Wang, Xue‐Ke Song, Hao Yuan, Changjin Zhang, Liu Ye

Published Jul 27, 2021• 14 citations
Publisher

Abstract

Quantum entanglement, quantum steering and Bell nonlocality, as significant quantum resources in the field of quantum information science, can achieve variously valuable quantum information tasks. Among of them, quantum entanglement and Bell nonlocality are the weakest and strongest nonlocal correlations, respectively. One can capture the quantum steering and Bell nonlocality via violating steering inequality and Bell inequality, respectively. In general, the detections of quantum steering and Bell nonlocality are strictly harder than entanglement detection. Here, based on steering inequality test and quantum state tomography, we attain various nonlocal correlations and experimentally demonstrate that the estimations of quantum steering and Bell nonlocality can be realized according to the quantum entanglement of the prepared two-photon test states. The estimated efficiency of quantum steering is stronger than the one of Bell nonlocality in this scenario, i.e., more steerable two-photon test states can be verified through quantum entanglement. In addition, quantum steering and Bell nonlocality are bounded by the corresponding upper and lower bounds, and these bounds cannot be punctured by all prepared two-photon states in experiment. These results are conducive to understand the relations among these nonlocal correlations.

DOI
10.1364/oe.430964

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Quantum entanglement, quantum steering and Bell nonlocality, as significant quantum resources in the field of quantum information science, can achieve variously valuable quantum information tasks. Among of them, quantum entanglement and Bell nonlocality are the weakest and strongest nonlocal correlations, respectively. One can capture the quantum steering and Bell nonlocality via violating steering inequality and Bell inequality, respectively. In general, the detections of quantum steering and Bell nonlocality are strictly harder than entanglement detection. Here, based on steering inequality test and quantum state tomography, we attain various nonlocal correlations and experimentally demonstrate that the estimations of quantum steering and Bell nonlocality can be realized according to the quantum entanglement of the prepared two-photon test states. The estimated efficiency of quantum steering is stronger than the one of Bell nonlocality in this scenario, i.e., more steerable two-photon test states can be verified through quantum entanglement. In addition, quantum steering and Bell nonlocality are bounded by the corresponding upper and lower bounds, and these bounds cannot be punctured by all prepared two-photon states in experiment. These results are conducive to understand the relations among these nonlocal correlations.
Recently quantum nonlocality has been classified into three distinct types: quantum entanglement, Einstein-Podolsky-Rosen steering, and Bell's nonlocality. Among which, Bell's nonlocality is the strongest type. Bell's nonlocality for quantum states is usually detected by violation of some Bell's inequalities, such as Clause-Horne-Shimony-Holt inequality for two qubits. Steering is a manifestation of nonlocality intermediate between entanglement and Bell's nonlocality. This peculiar feature has led to a curious quantum phenomenon, the one-way Einstein-Podolsky-Rosen steering. The one-way steering was an important open question presented in 2007, and positively answered in 2014 by Bowles et al., who presented a simple class of one-way steerable states in a two-qubit system with at least thirteen projective measurements. The inspiring result for the first time theoretically confirms quantum nonlocality can be fundamentally asymmetric. Here, we propose another curious quantum phenomenon: Bell nonlocal states can be constructed from some steerable states. This novel finding not only offers a distinctive way to study Bell's nonlocality without Bell's inequality but with steering inequality, but also may avoid locality loophole in Bell's tests and make Bell's nonlocality easier for demonstration. Furthermore, a nine-setting steering inequality has also been presented for developing more efficient one-way steering and detecting some Bell nonlocal states.
Quantum entanglement is a key resource for foundational quantum information and communication tasks, such as teleportation, entanglement swapping and quantum key distribution [1]. Quantum steering is a quantum correlation that is distinct from other nonclassical correlations such as Bell nonlocality [2] and entanglement. Steering describes the effect of measuring one system, let's say Alice, and affecting the measurement results on the other system, let's say Bob. Moving through the hierarchy of nonlocal classes, from Bell nonlocality towards nonsepa-rability, requires an increasing number of parties and apparatus that need to be trusted but gives access to protocols which are more robust to noise for projective measurements [3,4]. In a quantum steering task [5] the correlations of Alice and Bob's measurement outcomes are tested against a classical probability bound which can be only violated if the state shared between the parties is quantum steerable.
The Einstein-Podolsky-Rosen (EPR) steering is an intermediate quantum nonlocality between entanglement and Bell nonlocality, which plays an important role in quantum information processing tasks. In the past few years, the investigations concerning the EPR steering have been demonstrated in a series of experiments. However, these studies rely on the relevant steering inequalities and the choices of measurement settings. Here, we experimentally verify the EPR steering via entanglement detection without using any steering inequality and measurement setting. By constructing two new states from a two-qubit state, we observe the EPR steering by detecting the entanglement of these new states. The results show that the entanglement of the newly constructed states can be regarded as an unusual kind of steering witness for these states. Compared to the results of Xiao et al. [Phys. Rev. Lett. 118, 140404 (2017)], we find that the ability to detect the EPR steering in our scenario is stronger than that of two-setting projective measurements, which can observe more steerable states. Hence, our demonstrations can deepen the understanding of the connection between the EPR steering and entanglement.
The ability that one system immediately affects another one by using local measurements is regarded as quantum steering, which can be detected by various steering criteria. Recently, Mondal et al. [Phys. Rev. A 98, 052330 (2018)] derived the complementarity relations of coherence steering criteria, and revealed that the quantum steering of a system can be observed through the average coherence of a subsystem. Here, we experimentally verify the complementarity relations between quantum steering criteria by employing two-photon Bell-like states and three Pauli operators. The results demonstrate that if prepared quantum states can violate two setting coherence steering criteria and turn out to be steerable states, then they cannot violate the complementary settings criteria. Three measurement settings inequalities, which establish a set of complementarity relations between these two coherence steering criteria, are always obeyed by all prepared quantum states in experiment. In addition, we experimentally certify that the strengths of coherence steering criteria depend on the choice of coherence measure. In comparison with two setting coherence steering criteria based on l1{l}_{1}l1​ norm of coherence and relative entropy of coherence, our experimental results show that the steering criterion based on skew information of coherence is the strongest in detecting the steerability of two-photon Bell-like states. Thus, our experimental demonstration can deepen the understanding of the relation between the quantum steering and quantum coherence.
Quantum steering is one form of quantum correlations interpolating between entanglement and Bell nonlocality, which in some cases can be detected by various steering inequalities. Recently, a remarkable and useful steerability criterion via geometric Bell-like inequalities was established [M. Zukowski, A. Dutta, and Z. Yin, Phys. Rev. A 91, 032107 (2015)]. We report an experimental investigation of this steering criterion and verify the geometric Bell-like steering inequality experimentally by using of the Werner states. The results demonstrate that the geometric Bell-like steering inequality is a convenient tool to detect quantum steering both theoretically and practically.
Quantum inseparabilities can be classified into three inequivalent forms: entanglement, Einstein-Podolsky-Rosen (EPR) steering, and Bell's nonlocality. Bell-nonlocal states form a strict subset of EPR steerable states which also form a strict subset of entangled states. Recently, EPR steerable states have been shown to be fundamental resources for one-sided device-independent quantum information processing tasks and, hence, identification of EPR steerable states becomes important from foundational as well as informational theoretic perspectives. In the present paper, we propose a criterion to detect whether a given two-qubit state is EPR steerable. From an arbitrary given two-qubit state, another two-qubit state is constructed in such a way that the given state is EPR steerable if the new constructed state is entangled. Hence, EPR steerability of an arbitrary two-qubit state can be detected by detecting entanglement of the newly constructed state. Apart from providing a distinctive way to detect EPR steering without using any steering inequality, the present paper also provides an alternative way to avoid locality loopholes in EPR steering tests and to reduce the ``complexity cost'' present in experimentally detecting EPR steering. We also generalize our criteria to detect EPR steering of higher dimensional quantum states. Finally, we illustrate our result by using our proposed technique to detect EPR steerability of various families of mixed states.