1. Introduction

Since the famous EPR paradox was proposed by Einstein, Podolsky, and Rosen in 1935 [1], quantum entanglement as one of the most significant traits and nonlocal correlations of quantum mechanics has received extensive attentions [2]. For a two-qubit entangled state shared by Alice and Bob, one cannot construct local hidden state-local hidden state (LHS-LHS) model to characterize the joint probability distributions of arbitrary measurement (implement on both Alice and Bob) outcomes [2]. From the perspective of mathematics, an entangled state corresponds to the case that a composite quantum system cannot be described by the ensembles of product states. The entanglement of quantum states can be quantified by different ways, such as concurrence [3,4], entanglement of formation [4,5], entanglement of relative entropy [6], and so on. Quantum entanglement forms the base of the practical realizations for quantum computers [7], quantum errorcorrection [8], cryptographic communications [9,10], and quantum teleportation [11].

Bell nonlocality [12,13] and quantum steering [1,14–16] are other valuable traits and nonlocal correlations of quantum mechanics, and their detections are nontrivial for achieving quantum information tasks in practice. Bell nonlocality is based on the correlation of observations of separated objects, and is a stronger nonlocal correlation than quantum entanglement. Joint probability distributions of arbitrary measurement outcomes that do not satisfy local hidden variable-local hidden variable (LHV-LHV) model are deemed as Bell nonlocal [12,13]. The identification of Bell nonlocality can be achieved through the violation of Bell inequality, such as Clauser-Horne-Shimony-Holt (CHSH) inequality [17]. Different quantum information tasks, i.e., the device-independent quantum key distribution [18], communication complexity [19], and genuine random number generation [20], can be realized by virtue of Bell nonlocality.

By comparison, in 2007, quantum steering as a newer concept was formally depicted under the role of quantum information [21]. Thereafter, quantum steering has been widely investigated and discussed in the last fifteen years [16]. If and only if the statistics of measurement outcomes for a two-qubit state cannot be explained by local hidden variable-local hidden state (LHV-LHS) model, then the two-qubit state exists quantum steering [21]. As a consequence, quantum steering is a nonlocal correlation between Bell nonlocality and quantum entanglement [21,22]. Similar to the detection of Bell nonlocality, the quantum steering can be captured in terms of different steering inequalities, for instance, linear steering inequality [23,24], steering inequality based on uncertainty relations [25–28], geometric Bell-like inequalities [29], and Mermin steering inequality [30]. Also, the asymmetry [21,31] of quantum steering makes it play a crucial part in one-sided device-independent quantum key distribution [32–34].

Of particular note, the relations between different nonlocal correlations are essential roles in both theory and experiment, and the relevant investigation is always one of the promising contents in the field of quantum information science. On the one hand, it can help us deeply understand these nonlocal correlations. On the other hand, it is conducive to directly estimate one quantity of them through the other one due to that one quantity is bounded by the other one. Verstraete et al. explored the relationship between Bell nonlocality and quantum entanglement for bipartite state, and derived the range of Bell nonlocality for a certain value of quantum entanglement [35]. For a value of quantum entanglement, what extent it is reasonable to identify a large Bell violation can be displayed by using their results [35]. By employing Monte Carlo simulations, Bartkiewicz et al. compared quantum entanglement with Bell nonlocality for bipartite system [36], and attained the states that can achieve the upper and lower bounds of quantum entanglement for a given Bell violation. The relations between quantum steering and Bell nonlocality were investigated by Quan et al. [37]. Considering a general two-qubit state, Su et al. systematically derived the quantitative relation between nonlocality and entanglement, and proved the necessary and sufficient condition for achieving the upper bound [38]. Fan et al. examined the constraint relation between quantum entanglement and quantum steering of two-qubit states, and attained the steerability’s upper and lower boundaries depended on the concurrence of system [39]. In general, the detections of quantum steering and Bell nonlocality are strictly harder than entanglement detection. Fortunately, these corelations and results mentioned above provide a convenient and practical avenue to directly estimate quantum steering and Bell nonlocality via quantum entanglement in theory. Nevertheless, the experimental investigation concerning it is still lacking.

Enlighten by this, we experimentally prepare two-photon test states and investigate the relationships among quantum entanglement, quantum steering, and Bell nonlocality in this paper. It is found that the agreements between experimental results and theoretical results are satisfactory. According to the quantity of quantum entanglement for two-photon test states, one can experimentally realize the estimations of quantum steering and Bell nonlocality without implementing the tests of steering inequality and CHSH inequality. Also, the estimated efficiency of quantum steering for two-photon test states is stronger than the one of Bell nonlocality in this strategy, and thus one can certify more steerable states in terms of quantum entanglement.

2. Preliminaries

Concurrence is one of the significant ways to quantify quantum entanglement [3,4]. The concurrence of a bipartite pure state $\left | \psi \right \rangle$ is given by $C\left ( {\left | \psi \right \rangle } \right ) = | {\langle {\psi }|{{\tilde \psi }}\rangle }|$ with $| {\tilde \psi } \rangle \textrm { = }({\sigma _y} \otimes {\sigma _y})\left | {{\psi ^*}} \right \rangle$ [4], where ${\sigma _y}$ is Pauli matrice and $\left | {{\psi ^*}} \right \rangle$ is the complex conjugate of $\left | \psi \right \rangle$. Considering all possible decompositions of pure states for any bipartite state ${\rho _{AB}}$, namely, ${\rho _{AB}} = \sum \nolimits _n {{q_n}\left | {{\psi _n}} \right \rangle } \left \langle {{\psi _n}} \right |$, the concurrence of ${\rho _{AB}}$ is $C({\rho _{AB}}) = \mathop {\min }_{{q_{n,}}\left | {{\psi _n}} \right \rangle } \sum \nolimits _n {{q_n}} C(\left | {{\psi _n}} \right \rangle )$ [5]. $C({\rho _{AB}})$ can be reduced to $C({\rho _{AB}}) = \{ 0,{\lambda _1} - {\lambda _2} - {\lambda _3} - {\lambda _4}\}$ [4] for calculating conveniently. The eigenvalues of the Hermitian matrix $\sqrt {\sqrt {{\rho _{AB}}} {{\tilde \rho }_{AB}}\sqrt {{\rho _{AB}}} }$ are represented by $\{ {\lambda _1},{\lambda _2},{\lambda _3},{\lambda _4}\}$ with the order of ${\lambda _1} \ge {\lambda _2} \ge {\lambda _3} \ge {\lambda _4}$, here ${\tilde \rho _{AB}}\textrm { = }({\sigma _y} \otimes {\sigma _y})\rho _{AB}^*({\sigma _y} \otimes {\sigma _y})$. In addition, the Bell nonlocality of ${\rho _{AB}}$ can be detected by using CHSH inequality which can be written as

Here, ${\boldsymbol {a}}$, ${\boldsymbol {a'}}$, ${\boldsymbol {b}}$ and ${\boldsymbol {b'}}$ represent the unit vectors, and ${\boldsymbol {\sigma }} = \left ( {{\sigma _x},{\sigma _y},{\sigma _z}} \right )$ is vector of Pauli matrix. The maximum expected value of the Bell operator [40] is

The results imply that one can verify the Bell nonlocality when the lower bound in Eq. (4) is not equal to zero, and the estimation of Bell nonlocality for two-qubit states can be achieved via concurrence.

Quantum steering of two-qubit states can be captured in terms of steering inequalities, such as Cavalcanti-Jones-Wiseman-Reid (CJWR) inequality [23,24]. Considering three measurements, the CJWR inequality is described by

The eigenvalues of $\sqrt {{T^T}({\rho _{AB}})T({\rho _{AB}})}$ are indicated by ${\{ }{t_1}({\rho _{AB}}),{t_2}({\rho _{AB}}),{t_3}({\rho _{AB}}) {\} }$ with decreasing order. The steerability of ${\rho _{AB}}$ can be characterized by [39]

This quantity is restricted by concurrence and purity [39], namely,

As a consequence, the quantum steering of ${\rho _{AB}}$ can be estimated by employing the lower bound in Eq. (8) which depends on the concurrence and purity.

3. Experiment and results

The qubits are encoded by polarized photons (the central wavelength is 810 nm) in experiment. The states $\left | 0 \right \rangle$ and $\left | 1 \right \rangle$ are encoded by horizontally polarized states $\left | H \right \rangle$ and vertically polarized states $\left | V \right \rangle$, respectively. In this paper, the two-photon test states are chosen by

 

Fig. 1. Experimental setup. The two-photon Bell-like states $\left | {{\phi _{AB}}} \right \rangle$ are prepared in Fig. 1(a). Figure 1(b) is used to achieve the experimental test states ${\rho _{AB}}(\textit {p},\theta )$. Figure 1(c) is used to obtain $S({\rho _{AB}}(\textit {p},\theta ))$ and carry out tomography for quantum states. Abbreviations: PBS, polarizing beam splitter; HWP, half-wave plate; BBO, type-I $\beta$ -barium borate; BS, beam splitter; ATT, attenuator; QWP, quarter-wave plate; IF: interference filter; SPD: single photon detector.

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Table 1. The settings of optical axis angles of HWP and QWP in Fig. 1(c) for achieving different Pauli measurements.

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In the following, we focus attention on experimentally estimating Bell nonlocality and quantum steering with the help of quantum entanglement. The experimental results and theoretical predictions are displayed in Figs. 2–4. It deserves emphasizing that some of error bars are too small to observe in Figs. 2–4. In Figs. 2 and 4(a) (Figs. 3 and 4(b)), the black solid circles and purple squares indicate the experimental upper and lower bounds, respectively. These bounds depend on the concurrence (concurrence and purity) of test states, and they are also used to estimate the Bell nonlocality (quantum steering) of ${\rho _{AB}}(\textit {p},\theta )$. The Bell nonlocality (quantum steering) of test states can be detected if the lower bounds in Figs. 2 and 4(a) (Figs. 3 and 4(b)) are not equal to zero. The orange hollow rhombuses in Figs. 2 and 4(a) (Figs. 3 and 4(b)) represent the experimental Bell nonlocality (quantum steering) for test states. Corresponding theoretical results are expressed by employing dashed or solid lines with different colours. In Figs. 2–4, the Bell nonlocality and quantum steering of ${\rho _{AB}}(\textit {p},\theta )$, whose parameter (p or $\theta$) is occupied by the light blue area, can be theoretically estimated by using concurrence.

 

Fig. 2. Experimental results and the corresponding theoretical predictions. The Bell nonlocality of the states, whose parameter p is occupied by the light blue area, can be estimated by concurrence in theory.

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Fig. 3. Experimental results and the corresponding theoretical predictions. The quantum steering of the states, whose parameter p is occupied by the light blue area, can be estimated by concurrence in theory.

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As demonstrated from Figs. 2–4, the experimental results are in good agreement with the theoretical results. The experimental Bell nonlocality or quantum steering are bounded by the upper and lower bounds, and cannot puncture these bounds. For all test states with $\theta$=${45^\textrm {o}}$, the Bell nonlocality and quantum steering can achieve the corresponding upper bounds in experiment, as revealed in Fig. 2(a) and Fig. 3(a), and the Bell nonlocality and quantum steering disappear for the test state ${\rho _{AB}}(\textit {p}\textrm { = }0.5,\theta \textrm { = }{45^\textrm {o}})$. There are ten prepared states (labeled by 1, 2, 3, 4, 5, 7, 8, 9, 10, and 11 in Fig. 2(a) and Fig. 3(a) ) that are both steerable and Bell nonlocal. Notably, one cannot use concurrence to detect the Bell nonlocality or quantum steering for all test states. Explicitly, in Fig. 2(a), the experimental lower bounds of four test states with p=0, 0.05, 0.95, and 1 (labeled by 1, 2, 10, and 11) are not zero, and one can verify the Bell nonlocality for these Bell nonlocal states through concurrence. In Fig. 3(a), using concurrence and purity, one can observe the quantum steering of six steerable states with p=0, 0.05, 0.15, 0.85, 0.95, and 1 (labeled by 1, 2, 3, 9, 10, and 11) due to that the experimental lower bounds are not zero for these states. That is to say, one can estimate the Bell nonlocality of four test states among the ten prepared Bell nonlocal states, and observe the quantum steering of six test states for the ten prepared steerable states. Hence, the efficiency in estimating quantum steering is stronger than the one in estimating Bell nonlocality for prepared two-photon test states ${\rho _{AB}}(\textit {p}, \theta \textrm { = }{45^\textrm {o}})$, i.e., more steerable states can be verified by quantum entanglement.

 

Fig. 4. Experimental results and the corresponding theoretical predictions. The Bell nonlocality and quantum steering of the states, whose parameter $\theta$ is occupied by the light blue area, can be estimated by concurrence in theory.

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If we set state parameter $\theta$ to ${30^\textrm {o}}$ in experiment, the Bell nonlocality and quantum steering cannot achieve the corresponding upper bounds for most test states, as depicted in Fig. 2(b) and Fig. 3(b). There are six Bell nonlocal test states ${\rho _{AB}}(\textit {p}, \theta \textrm { = }{30^\textrm {o}})$ with p=0, 0.05, 0.15, 0.85, 0.95, and 1 (labeled by 1, 2, 3, 9, 10, and 11 in Fig. 2(b)) and eight steerable test states ${\rho _{AB}}(\textit {p}, \theta \textrm { = }{30^\textrm {o}})$ with p=0, 0.05, 0.15, 0.25, 0.75, 0.85, 0.95, and 1 (labeled by 1, 2, 3, 4, 8, 9, 10, and 11 in Fig. 3(b)). Four Bell nonlocal states (labeled by 1, 2, 10, and 11 in Fig. 2(b)) and six steerable states (labeled by 1, 2, 3, 9, 10, and 11 in Fig. 3(b)) can be captured by utilizing concurrence, respectively. The above-mentioned results demonstrate that one can effectively estimate the Bell nonlocality and quantum steering of ${\rho _{AB}}(\textit {p},\theta )$ according to the concurrence, especially for the ${\rho _{AB}}(\textit {p},\theta )$ with the conditions of $\textit {p} \to 0$ and $\textit {p} \to 1$.

At last stage, we experimentally estimate the Bell nonlocality and quantum steering of ${\rho _{AB}}(\textit {p}\textrm { = }0.05,\theta )$ with different parameters $\theta$. Among the seven prepared states ${\rho _{AB}}(\textit {p}\textrm { = }0.05,\theta )$, five test states ${\rho _{AB}}(\textit {p}\textrm { = }0.05,\theta )$ with $\theta$=${15^\textrm {o}}$, ${30^\textrm {o}}$, ${45^\textrm {o}}$, ${60^\textrm {o}}$, and ${75^\textrm {o}}$ (labeled by 2, 3, 4, 5, and 6 in Fig. 5) are both Bell nonlocal and steerable in experiment. One can reveal from Fig. 5(a) that only three prepared Bell nonlocal states (labeled by 3, 4, and 5, in Fig. 5(a)) can be verified by using the lower bound in experiment. All prepared steerable states can be experimentally captured through the corresponding lower bound, as demonstrated in Fig. 5(b). These results further demonstrate that the ability in estimating quantum steering via concurrence is stronger than the one in estimating Bell nonlocality for prepared test states ${\rho _{AB}}(\textit {p},\theta )$ in experiment.

 

Fig. 5. Experimental results and the corresponding theoretical predictions of different nonlocal correlations for prepared two-photon states ${\rho _{AB}}(\textit {p},\theta )$.

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In order to demonstrate the hierarchy of different nonlocal correlations more clearly, we depict the concurrence $C({\rho _{AB}}(p,\theta ))$, Bell nonlocality $N({\rho _{AB}}(p,\theta ))$, and quantum steering $S({\rho _{AB}}(p,\theta ))$ of prepared two-photon states ${\rho _{AB}}(\textit {p},\theta )$ in Fig. 5. It is verified that the experimental results are in good agreement with the theoretical results, and the quantum steering is an intermediate nonlocal correlation between entanglement and Bell nonlocality.

4. Conclusions

Based on steering inequality test and reconstructed density matrices of prepared two-photon states ${\rho _{AB}}(\textit {p},\theta )$, we estimate the quantum steering and Bell nonlocality of ${\rho _{AB}}(\textit {p},\theta )$ by using quantum entanglement in experiment. The results certify that the Bell nonlocality is bounded by the upper and lower bounds which are determined by the concurrence of two-photon states, and the quantum steering is constrained by the upper and lower bounds which depend on the concurrence and purity of test states. These bounds cannot be punctured by all prepared two-photon states in experiment. One can directly and effectively estimate the Bell nonlocality and quantum steering of ${\rho _{AB}}(\textit {p},\theta )$ via these corresponding lower bounds without carrying out steering inequality test and CHSH inequality test, especially for ${\rho _{AB}}(\textit {p},\theta )$ with state parameters $\textit {p} \to 0$, $\textit {p} \to 1$ and $\theta \to {45^\textrm {o}}$. In contrast to the estimation of Bell nonlocality, one can more effectively estimate quantum steering for test states ${\rho _{AB}}(\textit {p},\theta )$ by virtue of the quantum entanglement of ${\rho _{AB}}(\textit {p},\theta )$, and more steerable states can be observed in this strategy. Therefore, our experimental results demonstrate that the quantum entanglement can be used to witness quantum steering and Bell nonlocality for two-qubit states, and they can also help us to understand the relationships among quantum entanglement, quantum steering, and Bell nonlocality.