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Physical Review A

Experimental detection of quantum steerability based on the critical radius in an all-optical system

Xiao‐Gang Fan, Fa Zhao, Huan Yang, Zhiyong Ding, Dong Wang, Liu Ye

Published Jan 20, 2023• 3 citations
Publisher

Abstract

Quantum steering is the ability that one system affects another one without delay by performing local measurements. During the past few years, the research on quantum steering has been verified in a number of experiments by the choices of measurements. But this research cannot detect all steerable states. Recently, the definition of critical radius was proposed by Nguyen et al. [Phys. Rev. Lett. 122, 240401 (2019)]. The critical radius can detect all steerable states for two-qubit states. Therefore the critical radius can be regarded as the best method for detecting the steerability of two-qubit states. However, relevant experimental research is still lacking. In this work, for comparison, we experimentally investigate three steering criteria based on the critical radius, the geometric Bell-like inequality, and the three-setting measurements steering inequality and find that the critical radius can detect the most steerable states. Furthermore, we demonstrate that the critical radius satisfies the scaling property. The results show that we only need to calculate the critical radii of some quantum states to obtain the critical radii of more scaling quantum states. Thus this property can provide the convenience for detecting the steerability of many quantum states.

DOI
10.1103/physreva.107.012419

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Quantum steering is the ability that one system affects another one without delay by performing local measurements. During the past few years, the research on quantum steering has been verified in a number of experiments by the choices of measurements. But this research cannot detect all steerable states. Recently, the definition of critical radius was proposed by Nguyen et al. [Phys. Rev. Lett. 122, 240401 (2019)]. The critical radius can detect all steerable states for two-qubit states. Therefore the critical radius can be regarded as the best method for detecting the steerability of two-qubit states. However, relevant experimental research is still lacking. In this work, for comparison, we experimentally investigate three steering criteria based on the critical radius, the geometric Bell-like inequality, and the three-setting measurements steering inequality and find that the critical radius can detect the most steerable states. Furthermore, we demonstrate that the critical radius satisfies the scaling property. The results show that we only need to calculate the critical radii of some quantum states to obtain the critical radii of more scaling quantum states. Thus this property can provide the convenience for detecting the steerability of many quantum states.
ABSTRACT Quantum steering refers to the phenomenon where one quantum system can instantaneously influence another via local measurements. The critical radius, proposed by Nguyen et al. [Phys. Rev. Lett. 122, 240401 (2019)], is a sufficient‐and‐necessary steering criterion (SNSC). However, its broad applicability has been restricted by the absence of analytical solutions for general two‐qubit states. To address this, we develop the first comprehensive analytical framework for the critical radius, unifying theory and practice. We first derive an exact SNSC for all two‐qubit T‐states under infinite measurements and verify its equivalence to the critical radius. Leveraging the critical radius's symmetry and concavity, we generalize the framework to arbitrary two‐qubit states, aiming to transform the upper and lower bounds of the critical radius into an equal pattern via optimized local operations, thereby deriving an accurate formula for the critical radius. Critically, we discover the first exact analytical SNSCs for two distinct classes of asymmetric states, precisely mapping the boundary between steerable and unsteerable regions. These breakthroughs eliminate reliance on the numerical semidefinite programming methods, provide an operationally efficient method for discovering the hidden steerability.
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.
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 system can be observed through the average coherence of 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 it cannot violate the complementary settings criteria. Three measurement settings inequality, which establish a complementarity relation between these two coherence steering criteria, always holds in experiment. Besides, we experimentally certify that the strengths of coherence steering criteria dependent on the choice of coherence measure. In comparison with two setting coherence steering criteria based on l1 norm of coherence and relative entropy of coherence, our experimental results show that the steering criterion based on skew information of coherence is more stronger in detecting the steerability of quantum states. Thus, our experimental demonstrations can deepen the understanding of the relation between the quantum steering and quantum coherence.
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.
Abstract Steering, a quantum property stronger than entanglement but weaker than non-locality in the quantum correlation hierarchy, is a key resource for one-sided device-independent quantum key distribution applications, in which only one of the communicating parties is trusted. A fine-grained steering inequality was introduced in (2014 Phys. Rev. A 90 050305), enabling for the first time the detection of steering in all steerable two-qubit Werner states using only two measurement settings. Here, we numerically and experimentally investigate this inequality for generalized Werner states and successfully detect steerability in a wide range of two-photon polarization-entangled Bell local states generated by a parametric down-conversion source.