1. Introduction

Schrödinger initially introduced the concept of quantum steering in 1935 [1], which was used to describe Einstein-Podolsky-Rosen paradox [2]. The progress of research on the steering was halting, until 2007, Wiseman et al. formally and rigorously made a definition of the steering [3]. It has attracted wide attention in many fields [4]. If Alice and Bob share a quantum state $\rho _{\text {AB}}$, she chooses different bases to make a measurement on her system. With each measurement made by Alice, Bob receives a condition state. If this conditional state cannot be represented by a local hidden state (LHS) model, then this state $\rho _{\text {AB}}$ is steerable from Alice to Bob.

Quantum states which have steerability form a strict subset of entangled states, and form a strict superset of Bell nonlocal states [5–7]. The steering exhibits an inherent asymmetric feature [8–12], where there exist some entangled states which possess steerability from Alice to Bob, while they are unsteerable from Bob to Alice [13,14]. The steering plays a key role in quantum information tasks such as one-sided randomness generation [15], one-sided quantum key distribution [16–18], secure quantum teleportation [19,20], etc.

The detection of steerability can be achieved by violation of various steering inequalities. There are various steering inequalities, for instance, the linear steering inequalities [21–23], the steering inequalities based on entropic uncertainty relations [24–26], the geometric Bell-like inequality [27], and so on. In recent years, the sufficient criteria of the steering have been developed rapidly, and many practical significant results were certified experimentally. Some sufficient criteria based on entanglement have been constructed [28–31]. Bartkiewicz et al. [32] experimentally carried out the violation of Clauser-Horne-Shimony-Holt inequality in the entanglement-swapping device. According to the LHS model, Skrzypczyk et al. [33] put forward the definition of the steerable weight to quantify the steering. By using the steerable weight, Zeng [34] proposed the one-way steerable states. Lai et al. [35] presented a steering criterion based on multiple sets of measurements. Saunders et al. [36] experimentally observed the steerability of Bell local states by the steering inequalities with $N$ ($N=3,4,6,10$) sets of measurements. Motivated by the steering radius, Sun et al. [37] and Xiao et al. [38] put forward the sufficient conditions of the steering with two and multiple sets of measurements in different directions, respectively. Their results show that more steerable states can be detected as the number of measurements increases.

Generally speaking, the greater the number $N$ of measurements, the more difficult the corresponding steering inequality achieves experimentally. Clearly, when the number $N$ approaches infinity, the corresponding steering inequality can detect the most steerable states. However, it is impossible to apply infinite sets of measurements in experiments, which poses a great challenge to find many steerable states as possible experimentally. In order to detect more steerable states in two-photon systems, we theoretically derive an optimized steering criterion based on infinity measurements, which is applicable to any two-qubit states. And our steering criterion only relies on the spin correlation matrix, which eliminates the need to use the quantum state tomography in experiment. Thus, our steering criterion provides a simple solution to detect steerability without the need to perform infinity measurements. In experiment, we prepare Werner-like states in two-photon systems and measure their spin correlation matrices. We detect the steerability via using the three steering criteria: our steering criterion, the three-measurement steering criterion [22], and the geometric Bell-like inequality [27]. The experimental results show that our steering criterion can detect more steerable states than other two criteria.

2. Preliminary

In general, a two-qubit state $\rho$ can be represented by a positive definite and Hermitian operator over ${\mathcal {H}_{\text {A}}} \otimes {\mathcal {H}_{\text {B}}}$, where $\mathcal {H}_{\text {A}}$ and $\mathcal {H}_{\text {B}}$ are the Hilbert spaces with 2-dimension. If the Pauli matrices $\left \{ {\sigma _i} \right \}_{i = 0}^3$ (with $\sigma _0 = {\mathds{1}}$, $\sigma _1 = \sigma _x$, $\sigma _2 = \sigma _y$ and $\sigma _3 = \sigma _z$) are used as the basis of the Hilbert space $\mathcal {H}_{\text {A}}$ ($\mathcal {H}_{\text {B}}$), this state $\rho$ is expressed as follows

The geometric Bell-like inequality.—Żukowski et al. [27] put forward the geometric Bell-like inequality for the state $\rho$, which is given by

The $N$-setting-measurement steering inequality.—The steering inequality with $N$ sets of measurements on each side can be given by [4]

3. Optimized steering criterion based on infinity measurements

When the number $N$ of measurements becomes bigger, the steering inequality in Eq. (3) can detect more steerable states. Inspired by this, we investigate the steering inequality when the number $N$ of measurements approaches infinity. The left-hand and right-hand sides in Eq. (3) might be infinity if $N \to \infty$. In order to avoid this divergent behaviour, we construct the infinity-setting-measurement steering inequality as follows

First of all, we consider for what condition does $A_k$ satisfy, the expectation ${{\text {Tr}}\left [ {\rho ({A_k} \otimes {B_k}}) \right ]}$ reaches maximum. Based on this formula ${{\text {Tr}}\left [ {\rho ({\sigma _i} \otimes {\sigma _j}}) \right ]}={T_{ij}}$ for $i,j \in \{ 1,2,3\}$, the expectation can be given by

And then, we determine the computable expression of the maximum violation ${F_{\max }}$. When the number $N$ is a finite value, the maximum value of the left-hand side in Eq. (3) can be expressed as

Equation. (8) shows that we can choose $N$ different measurement settings $\left \{ {{\boldsymbol{r}_k}} \right \}_{k = 1}^N$ among all measurement settings to obtain the maximum value. However, when $N \to \infty$, we need to consider all measurement settings rather than selectively. Thus the maximum violation can be defined as

In order to further simplify Eq. (9), we assume that these unit vectors ${\left \{ {{\boldsymbol{r}_k}} \right \}_{k = 1}^N}$ are evenly distributed within the Bloch sphere (the starting points of these vectors are at the center of the Bloch sphere). For example, when $N=3$, the vectors ${\left \{ {{\boldsymbol{r}_k}} \right \}_{k = 1}^3}$ can be a set of vectors that are orthogonal to each other, which has exactly the same concept as the three-measurement steering inequality [22]; when $N=4$, the ending points of these vectors ${\left \{ {{\boldsymbol{r}_k}} \right \}_{k = 1}^4}$ are located at the four vertices of a regular tetrahedron. As the number $N$ grows, the more densely the points are stacked on the surface of the Bloch sphere. When $N \to \infty$, the ending points of these vectors ${\left \{ {{\boldsymbol{r}_k}} \right \}_{k = 1}^\infty }$ will fill the entire sphere, which leads to the area of each ending point to be $4\pi /N$. Thus, the face element $dS$ of the Bloch sphere can be expressed as

Combining Eqs. (9) and (10), the maximum violation $F_{\max }$ can be calculated by an integral

Finally, we determine the value of the constant $C$ in Eq. (5). Here, we consider the operator $O_N$, which is given by

To obtain the value of $C$, we provide an optimal distribution

Considering Eqs. (11) and (16), we can obtain the conclusions: (i) the maximum violation $F_{\max }$ is completely determined by the spin correlation matrix $T$; (ii) a two-qubit state is steerable if $F_{\max }>1/2$.

Since our steering criterion is the result of extending the steering inequality with $N$ measurements to the steering inequality with infinite measurements, it can detect more steerable states than the result as expressed in Eq. (3). For Bell diagonal states, we can obtain $\iint {\sqrt {\left \langle \boldsymbol{r} \right |{T^2}\left | \boldsymbol{r} \right \rangle } dS }>2\pi$, which satisfies the sufficient and necessary steering criterion [39–41]. Thus, our steering criterion can detect all steerable states for Bell diagonal states.

Applying Eq. (7) to the geometric Bell-like inequality, the left hand side in Eq. (2) can be expressed as

Similarly, the maximum value of the left hand side in Eq. (4) can be given by

4. Experimental detection of steerability

In this section, we report an experimental detection of the steerability by using our steering criterion in two-photon systems. Here, we choose to prepare the Werner-like states, which are given by

The whole experimental setup is shown in Fig. 1, which includes three modules distinguished by rectangular boxes of different colors. The module (a) is composed of a high-power continuous laser (130 mW, 405 nm), a polarization beam splitter (PBS), a half-wave plate (HWP) with optical axis angle of $\alpha$ and two type-I $\beta$-barium borate (BBO) crystals ($6.0 \times 6.0 \times 0.5 \,{\text {mm}^3}$). The pump beam undergoes spontaneous parametric down conversion through BBO to produce the entangled photon pairs $\left | {{\phi \left ( \alpha \right ) }} \right \rangle = \cos 2\alpha \left | {HH} \right \rangle + \sin 2\alpha \left | {VV} \right \rangle$ [42], and the angle parameter $\alpha$ can be easily adjusted. The module (b) includes a HWP and three 2.6-mm yttrium orthovanadate ($\text {YVO}_4$) crystals. The module (b) is used to generate the maximally mixed state $\frac {{{\mathds {1}}}}{2} \otimes \frac {{{\mathds {1}}}}{2}$. Firstly, we set the optical axis angle $\alpha$ of the HWP in the module (a) to be $22.5^{\text {o}}$, so that we can prepare the maximally entangled state $\left | {{\phi \left ( 22.5^{\text {o}} \right ) }} \right \rangle$. And then, let the photon of the B path pass through the module (b). The HWP and the three crystals in the module (b) can change the $\left | {{\phi \left ( 22.5^{\text {o}} \right ) }} \right \rangle$ into the maximally mixed state $\frac {{{\mathds {1}}}}{2} \otimes \frac {{{\mathds {1}}}}{2}$ [43–45]. By using modules (a), (b) and the time-mixing technique [36,46–49], we can prepare the Werner-like states ${\rho _\text {W}}\left ( p, \alpha \right )$. And the parameter $p$ in the ${\rho _\text {W}}\left ( p, \alpha \right )$ can be controlled by the ratio between the measurement duration of the coincidence counts for the Bell-like state and the maximally mixed state. The module (c) includes two quarter-wave plates (QWPs), two HWPs, two 3-nm interference filter (IFs), two PBSs, four single-photon detectors (SPDs), and a logic coincidence unit. In the experiment, we choose six Pauli measurement bases $\Pi _0^x=\frac {1}{2}\left ( {{\mathds {1}} + {\sigma _x}} \right )$, $\Pi _1^x=\frac {1}{2}\left ( {{\mathds {1}} - {\sigma _x}} \right )$, $\Pi _0^y=\frac {1}{2}\left ( {{\mathds {1}} + {\sigma _y}} \right )$, $\Pi _1^y=\frac {1}{2}\left ( {{\mathds {1}} - {\sigma _y}} \right )$, $\Pi _0^z=\frac {1}{2}\left ( {{\mathds {1}} + {\sigma _z}} \right )$ and $\Pi _1^z=\frac {1}{2}\left ( {{\mathds {1}} - {\sigma _z}} \right )$, which can be realized by adjusting the angles of the optical axis of QWPs and HWPs in Fig. 1(c). After this experimental processing, we can obtain the measurement probabilities and process these probabilities to calculate the elements $T_{ij}$ of the spin correlation matrix $T$.

 

Fig. 1. Experimental scheme for detecting steerability. The experimental setup is constructed by three modules: (a) state preparation module, (b) dephase component module, and (c) state measurement module. The setup consists of some Key components: barium borate (BBO) crystal, mirror (MIR), half-wave plate (HWP), quarter-wave plate (QWP), polarizing beam splitter (PBS), interference filter (IF), and single photon detector (SPD).

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Detection of steerability.—Based on the sufficient condition of unsteering [50], the Werner-like states in Eq. (19) are unsteerable if $p \leqslant 1/2$. However, for the case $p>1/2$, whether the Werner-like states have the steerability needs further discussion. To solve this problem, we choose the parameters $p=\left \{ {0.5,0.6,0.7,0.8,0.9,1} \right \}$ and $\alpha = \left \{{{2.5^{\text {o}}},{5^{\text {o}}},{10^{\text {o}}},{15^{\text {o}}},{20^{\text {o}}},{22.5^{\text {o}}}} \right \}$, and prepare 36 Werner-like states according to the experimental device shown in Fig. 1(c). Since our steering criterion, the three-measurement steering criterion and the geometric Bell-like inequality are directly related to the spin correlation matrix for a quantum state, it is not necessary for us to reconstruct the density matrix. In order to experimentally probe the steerability of these states, we need to introduce the relation between the spin correlation matrix and the measurement probabilities for an arbitrary two-qubit state $\rho$.

For a bilateral measurement operator ${M_{\text {A}}} \otimes {M_{\text {B}}}$ on both photons A and B, the corresponding measurement probability $P\left ( {{M_{\text {A}}},{M_{\text {B}}}} \right )$ can be expressed as

In this experiment, we make use of the six Pauli measurement bases to combine into sixteen bilateral measurement operators, whose corresponding probabilities are expressed as follows

Once the sixteen probabilities are measured experimentally, we can obtain the elements of the spin correlation matrix $T$ by

Next, we start to use our steering criterion, three-measurement steering criterion and the geometric Bell-like inequality to detect the steerability of the 36 prepared states. In order to better compare the three steering criteria, we rewrite them as

Clearly, when a two-qubit state $\rho$ violates one of the conditions ${S_\infty } \leqslant 1/2$, $S_3 \leqslant 1/2$ and ${S_{\text {Q}} } \leqslant 1/2$, the state $\rho$ is steerable.

With these preparations, we focus attention on the experimental detection of steerability. In Fig. 2, by taking the parameter $p$ as the variable under six situations (a) $\alpha = 2.5^{\text {o}}$, (b) $\alpha = 5^{\text {o}}$, (c) $\alpha = 10^{\text {o}}$, (d) $\alpha = 15^{\text {o}}$, (e) $\alpha = 20^{\text {o}}$ and (f) $\alpha = 22.5^{\text {o}}$, we present the experimental results and theoretical predictions of our steering criterion, the three-measurement steering criterion and the geometric Bell-like inequality. It is worth emphasising that the error bars are too small to be observed in Fig. 2. Here, the horizontal axis represents the variable $p$, while the vertical axis denotes ${S_\infty }$, ${S_3}$ and $S_{\text {Q}}$. Blue dashed lines, green dashed lines and red dashed lines in every subfigure stand for the theoretical results of ${S_\infty }$, $S_3$ and ${S_{\text {Q}}}$ respectively. In addition, blue dots, green prisms and red triangles which are obtained by our experimental data, stand for ${S_\infty }$, $S_3$ and ${S_{\text {Q}}}$ respectively. Obviously, our experimental results match very well with the theoretical predictions. The value of the black solid lines is the constant $1/2$, which means the bound of the above steering criteria. Once one of ${S_\infty }$, $S_3$ and ${S_{\text {Q}}}$ is greater than $1/2$, the quantum state is steerable. From Fig. 2, it is demonstrated that our steering criterion can experimentally capture more steerable states than the other two criteria.

 

Fig. 2. Experimental results and theoretical predictions in the different parameter $\alpha$ : (a) $\alpha = 2.5^{\text {o}}$, (b) $\alpha = 5^{\text {o}}$, (c) $\alpha = 10^{\text {o}}$, (d) $\alpha = 15^{\text {o}}$, (e) $\alpha = 20^{\text {o}}$ and (f) $\alpha = 22.5^{\text {o}}$. Blue dashed lines, green dashed lines and red dashed lines in every subfigure stand for the theoretical results of ${S_\infty }$, ${S_3}$ and $S_{\text {Q}}$ in Eq. (23). The value of the black solid lines is the constant $1/2$, which means the bound of the above steering criteria. In addition, blue dots, green prisms and red triangles which are obtained by our experimental datum, stand for the steering criteria ${S_\infty }$, ${S_3}$ and $S_{\text {Q}}$ respectively.

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5. Conclusion

In this work, we derive an optimized steering criterion based on infinite measurements, and experimentally detect the steerability by the steering criterion. First, we construct a linear steering inequality in the case of infinite measurements, and deduce the sufficient criterion to provide an assurance with respect to an arbitrary two-qubit state being steerable. And then, we prepare Werner-type states in two-photon systems and obtain their spin correlation matrices via measuring the sixteen probabilities. Finally, we adopt our steering criterion, the three-measurement steering criterion and the geometric Bell-like inequality to detect the steerability of these prepared states. The theoretical and experimental researches show that our steering criterion can detect a wider range of steerable states and capture more steerable states. We believe that through this criterion, more steerable states can be found to be applied to future quantum information tasks.