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Optics Express

Experimental verification of the relationship between first-order coherence and linear steerability

Zhiyong Ding, Pan-Feng Zhou, Liu Ji-xue, Cheng-Cheng Liu, Ming Zhao, Huan Yang, Xiao‐Gang Fan, Juan He, Ye Liu

Published Nov 15, 2021• 3 citations
Publisher

Abstract

Coherence and steerability are two essential characteristics of quantum systems. For a two-qubit state, the first-order coherence and the maximal violation of linear steering inequality are used to operationally measure the degree of coherence and steerability, respectively. Recently, a complementary relation between first-order coherence and linear steerability has been proposed. In this paper, we report an experimental verification of the complementary relation by preparing biphoton polarization entangled states in an all-optical setup. We propose an operable method for experimental measurement of the first-order coherence and linear steerability and calculate the purity of the initial states by reconstructing the density matrices of them. The experimental results coincide with the theoretical predictions very well, which provides a valuable reference for the application of optical quantum technology.

DOI
10.1364/oe.445991

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Coherence and steerability are two essential characteristics of quantum systems. For a two-qubit state, the first-order coherence and the maximal violation of linear steering inequality are used to operationally measure the degree of coherence and steerability, respectively. Recently, a complementary relation between first-order coherence and linear steerability has been proposed. In this paper, we report an experimental verification of the complementary relation by preparing biphoton polarization entangled states in an all-optical setup. We propose an operable method for experimental measurement of the first-order coherence and linear steerability and calculate the purity of the initial states by reconstructing the density matrices of them. The experimental results coincide with the theoretical predictions very well, which provides a valuable reference for the application of optical quantum technology.
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.
Coherence and nonlocal correlations are two fundamental features of quantum mechanics and the relations between them have received much attention recently. While the relations of coherence to Bell nonlocality and to entanglement have been established, an interesting question left open is how to quantitatively characterize the relation of coherence to steering, which is one kind of nonlocal correlation lying between Bell nonlocality and entanglement. In this paper, we address this question by considering the first-order coherence and the three-setting linear steering inequality for a two-qubit system. Interestingly, we find that the first-order coherence for both subsystems has a complementary relation with the maximum violation of the three-setting linear steering inequality. A weighted sum of their squares keeps invariant under unitary operations, although each of them changes in general. This suggests that the maximum violation of the three-setting linear steering inequality can be taken as a measure of hidden coherence unveiled by unitary operations.
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.
We report an all-optical experiment to investigate the uncertainty relations based on linear entropy. First we prepare two kinds of biphoton states initially, i.e., Bell-like states and Werner states. Then we perform projective measurements on one subsystem over a complete or incomplete set of mutually unbiased bases; the corresponding measurement probabilities can be directly determined by the photon coincidence counting. Together with the initial state and the postmeasurement states of the system available by quantum state tomography, linear-entropy-based uncertainty relations can be fully verified. It is shown that the experimental results agree very well with the theoretical predictions. Also, it is revealed that entanglement of the initial biphoton states is anticorrelated with the total of uncertainties. Our experimental results could not only deepen the understanding of uncertainty relations, but also facilitate potential applications to quantum technologies.
The bipartite state is said to be steerable iff it does not have single system description, i.e., the bipartite state can not be explained by local hidden state model [Phys. Rev. Lett. {\bf 98}, 140402 (2007)]. Several steering inequalities have been derived using different local uncertainty relations to verify the control on the state of one subsystem by other. Here, we derive complementarity kind of relations between coherences measured on mutually unbiased bases using various coherence measures like the l1l_1l1​-norm, relative entropy and skew information. Using these relations, we formulate steering inequalities to check whether one subsystem has control to the coherence of the other entangled subsystem. According to our results, all pure entangled states are maximally steerable but for mixed entangled states, coherence steerability needs stronger non-local correlation than the existing state steering inequalities. We also show that our steering criteria is monogamous.