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Reliable experimental manipulation of quantum steering direction

Shuo Wang, Ya Xiao, and Yong-Jian Gu

Open Access Open Access

Abstract

Noise-adding methods have been widely used to manipulate the direction of quantum steering, but all related experimental schemes only worked under the assumption that Gaussian measurements were performed and ideal target states were accurately prepared. Here, we prove, and then experimentally observe, that a class of two-qubit states can be flexibly changed among two-way steerable, one-way steerable and no-way steerable, by adding either phase damping noise or depolarization noise. The steering direction is determined by measuring steering radius and critical radius, each of which represents a necessary and sufficient steering criterion valid for general projective measurements and actually prepared states. Our work provides a more efficient and rigorous way to manipulate the direction of quantum steering, and can also be employed to manipulate other types of quantum correlations.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Quantum steering describes the ability of an untrusted parties, Alice, to nonlocally “steer" the state of the other trusted party, Bob, through local measurements [1]. It was originally introduced by Schrödinger [2,3] in response to the Einstein-Podolsky-Rosen paradox [4], and later strictly redefined by Wiseman et al. in terms of local hidden variable and local hidden state model [5,6]. Unlike Bell nonlocality [7] and entanglement [8], quantum steering exhibits unique asymmetry, which means the steerability from Alice to Bob is not equal to that from Bob to Alice. Especially there are also one-way steering situations where Alice can steer Bob’s state, but Bob cannot steer Alice’s state, or vice versa. A recent work found that the asymmetric property of quantum steering also exists in multipartite systems. According to the monogamy principle, two parties cannot independently demonstrate steering of a third party [9]. However, they found that the shareability of steering in reduced subsystems allows the state of one party to be steered by two or more parties [10]. This unique asymmetry, indicating the roles of parties in the steering scenario are not interchangeable, can be applied to asymmetric quantum information tasks, such as one-sided device-independent quantum key distribution [11–13].

Great efforts have been made to classify the quantum steering according to the steerable direction. In the beginning, most related works focused on verifying the existence of quantum steering by violating different inequalities. However, these inequality-based steering criteria are only sufficient but not necessary, which can not be used to verify the failure of the steering task, thus can not be used to rigorously determine the steering direction [14–18]. Since 2010, different classes of one-way steerable states were constructed for continuous variable systems [19,20] and discrete variable systems [21–24]. Experimentally, the existence of one-way steering was first demonstrated in a continuous variable system, and was restricted to the case of Gaussian measurements [25]. Then, several experiments were carried out in discrete variable systems, and the measurements were extended to projective measurements [26,27] or positive-operator-valued measurements [28]. However, these existing experiments can not conclusively demonstrate the existence of one-way steering, because there are some implicit assumptions, that is, the ideal target state can be accurately prepared, or the number of measurement settings is finite [29]. But the actual states in the experiment will inevitably deviate from the targeted states. And as the number of measurement settings increases (decreases), more (less) states can be demonstrated steerable. Thus, the current one-way steerable states will become actually two-way steerable or two-way unsteerable [30,31]. Recently, a conclusive one-way steering without ideal target state assumption and finite measurement settings assumption has been respectively observed in a qubit-qutrit system [30] and a genuine two-qubit system [31].

To make full use of quantum steering, it is necessary not only to determine the direction of quantum steering conclusively, but also to manipulate them flexibly. However, the manipulation of quantum steering direction has been mainly studied theoretically [32–38]. There are only a few related experimental works. In 2017, Qin et al. found that the direction of quantum steering can be manipulated by adding noise or losss [39]. However, it is restricted to the case of Gaussian states and Gaussian measurement. In 2019, Pramanik et al. have shown experimentally that steerable states become unsteerable with the strength of decoherence increases. However, they verified the unsteerability using a sufficient but not necessary criterion, which can not accurately determine the presence or absence of one-way quantum steering [40,41]. Two years later, Yang et al. demonstrated that the one-way quantum steering could be stably controlled in a cavity magnonic system [42]. To date, how to reliably experimentally manipulate the direction of quantum steering in discrete systems is still unknown.

In this work, we achieve the first experimental manipulation of quantum steering direction by adding phase damping noise and depolarizing noise to one side of a two-qubit system. The steering direction is rigorously determined by the steering radius criterion and the critical radius criterion, which is valid for general projective measurement and actually prepared states. We further provide the effective ranges of decoherence degree corresponding to two-way steering, one-way steering and no-way steering, respectively. Our results offer a direct reference for practical applications of quantum steering.

2. Manipulation scheme

Here, we propose two schemes to manipulate the direction of quantum steering by adding a phase damping channel (PDC) or a depolarization channel (DC) to one side of a two-qubit state $\rho _{AB}$. As shown in Fig. 1, in manipulation scheme I, the noise channel is only added to Alice’s side. Based on the operator-sum representation of the quantum channel, when PDC is employed, the state $\varepsilon (\rho _{AB})$ shared between Alice and Bob can be expressed as [43]

$$\varepsilon(\rho_{AB}) =(1-\dfrac{D}{2})\rho_{AB}+\dfrac{D}{2} (\sigma_{z}\otimes \mathbb{I})\rho_{AB}(\sigma_{z}\otimes I),$$
while when DC is employed, the state $\varepsilon (\rho _{AB})$ can be rewritten as [43]
$$\varepsilon(\rho_{AB})=(1-D)\rho_{AB}+\dfrac{D}{3}((\sigma_{x}\otimes \mathbb{I})\rho_{AB}(\sigma_{x}\otimes \mathbb{I})+(\sigma_{y}\otimes \mathbb{I})\rho_{AB}(\sigma_{y}\otimes \mathbb{I})+(\sigma_{z}\otimes \mathbb{I})\rho_{AB}(\sigma_{z}\otimes \mathbb{I}).$$

 figure: Fig. 1.

Fig. 1. Schemes for manipulating the direction of quantum steering. A two-qubit quantum state $\rho _{AB}$ is distributed to Alice and Bob, with a noise channel, on the way to Alice or Bob. The steerablity of the state shared between Alice and Bob $\varepsilon (\rho _{AB})$ is quantified by steering radius or critical radius. Two different types of noise channels, phase damping channel (PDC) and depolarizing channel (DC), are considered here. The steering direction between Alice and Bob can be manipulated by changing the noise added to Alice’s side (scheme I) or Bob’s side (scheme II).

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$\lbrace \sigma _{x},\sigma _{y},\sigma _{z} \rbrace$ are Pauli operators, $\mathbb {I}$ is identity matrix and $D$ is the decoherence degree. $D\in [0,1]$ for PDC and $D\in [0,0.75]$ for DC. In manipulation scheme II, PDC or DC is only added to Bob’s side. Similarly, the state shared between Alice and Bob can be obtained from Eq. (1) or Eq. (2) by exchanging the corresponding Pauli operators and identity matrices, i.e.,

$$\begin{array}{c}\varepsilon (\rho_{AB})=(1-\dfrac{D}{2})\rho_{AB}+\dfrac{D}{2} (\mathbb{I}\otimes \sigma_{z})\rho_{AB}(\mathbb{I}\otimes\sigma_{z}),\\ \varepsilon (\rho_{AB})=(1-D)\rho_{AB}+\dfrac{D}{3}((\mathbb{I}\otimes\sigma_{x})\rho_{AB}(\mathbb{I}\otimes\sigma_{x})+(\mathbb{I}\otimes\sigma_{y})\rho_{AB}(\mathbb{I}\otimes\sigma_{y})+(\mathbb{I}\otimes \sigma_{z})\rho_{AB}(\mathbb{I}\otimes\sigma_{z})). \end{array}$$

To accurately characterize the manipulation of steering direction, we adopt two necessary and sufficient steering criteria to verify whether the steering task is successful. The first one is based on the steering radius, which is valid for finite number of measurement settings. Considering the task that Alice wants to convince Bob that she can steer his state, in the case of $N$-measurement settings, the steering radius from Alice to Bob can be defined as [26,27]

$$R_{AB}=\max_{\lbrace \vec{n}_{1},\vec{n}_{2},\ldots\vec{n}_{N}\rbrace} \lbrace \min _{\lbrace p_{i}\rho_{i}\rbrace} \lbrace {\rm max} \lbrace \vert \vec{L}_{i} \vert \rbrace \rbrace \rbrace.$$
$\lbrace \vec {n}_{1},\vec {n}_{2},\ldots \vec {n}_{N}\rbrace$ is the direction assemblage of Alice’s $N$-measurement settings, $\lbrace p_{i}\rho _{i}\rbrace$ is the local hidden state ensemble of Bob, $\vert \vec {L}_{i} \vert$ is the length of the Bloch vector of the corresponding local hidden state $\rho _{i}$. If $R_{AB}>1$, there is no local hidden state ensemble to construct the conditional states obtained on Bob’s side, which indicates that the steering task from Alice to Bob is successful. Otherwise, the steering task fails if $R_{AB}\leq 1$. The steerability from Bob to Alice can be quantified by the steering radius $R_{BA}$, which can be defined in a similar way. The second one is based on the critical radius, which eliminates both the finite number of measurement settings constraint and the ideal target state constraint, thus can be used to conclusively determine the direction of quantum steering. The critical radius for the case in which Alice steers Bob can be defined as [44,45]
$$C_{AB}=\max_{\eta}\lbrace \eta>0:\rho^{\eta}_{AB} \ \text{is unsteerable} \rbrace,$$
where $\rho ^{\eta }_{AB}=\eta \varepsilon (\rho _{AB})+(1-\eta )\mathbb {I}_{A}/2\otimes \rho _{B}$, $\rho _B= {\rm Tr}_{A}[\varepsilon (\rho _{AB})]$ denotes the reduced state of Bob. The upper and lower bounds for the critical radius can be rigorously obtained by replacing Bob’s or Alice’s Bloch sphere inside and outside with a polyhedron, whose size is cubically scaled to the number of vertices of the polyhedron [44,45]. To accurately verify the existence of steering, we set its upper bound as the current critical radius. The steering task from Alice to Bob is successful if $C_{AB}<1$. Otherwise, the task fails if $C_{A B}\geq 1$. The steerability from Bob to Alice can be quantified by the critical radius $C_{BA}$, which can be defined in a similar way.

To verify the feasibility of our manipulation schemes, we consider a family of two-qubit asymmetric states $\rho _{AB}$, which has the following form

$$\rho_{AB} =p \vert \psi(\theta)\rangle\langle\psi(\theta)\vert+(1-p)\mathbb{I}/2\otimes\rho_{B}^{\theta},$$
where $\vert \psi (\theta )\rangle =\cos \theta \vert 11\rangle +\sin \theta \vert 00\rangle$, $\rho _{B}^{\theta } ={\rm Tr}_{A} [\vert \psi (\theta )\rangle \langle \psi (\theta )\vert ]$, $p \in [0,1]$ and $\theta \in [0,\pi /4]$. The state is steerable if $p>1/2$; however, for $\cos ^{2}(2\theta )\geqslant \dfrac {2p-1}{(2-p)p^{3}}$, it becomes unsteerable from Bob to Alice [24]. The corresponding state $\varepsilon (\rho _{AB})$ shared between Alice and Bob can be obtained from Eqs. (1)–(3), see Appendix A for more details. Since the critical radii $C_{AB}$ and $C_{BA}$ are difficult to be determined analytically, we use the three-measurement settings steering radii $R_{AB}$ and $R_{BA}$ to show how PDC and DC affect the steering direction between Alice and Bob. It is easy to demonstrate that the optimal measurement settings of Alice (Bob) are $\lbrace \vec {x},\vec {y},\vec {z} \rbrace$, which are parallel to the three main axes of her (his) steering ellipsoid [37,46]. The explicit formulas of $R_{AB}$ and $R_{BA}$ are detailed in the Appendix B. The results of our manipulation schemes are shown in Fig. 2. We find that, both in manipulation scheme I and manipulation scheme II, PDC has the same effect on the direction of quantum steering. Figures 2(a) and (d) respectively show the dependence of the steering radii $R_{AB}$ and $R_{BA}$ on the decoherence degree $D$ and state parameters $\lbrace p, \theta \rbrace$. Clearly, the steerable region from Alice to Bob is larger than that from Bob to Alice, which means, a tunable $D$ allows the steerable direction of the shared state $\varepsilon (\rho _{AB})$ to be changed among two-way, one-way and no-way. However, DC behaves differently in manipulation scheme I and manipulation scheme II. As shown in Figs. 2(b) and (c), with the increases of $D$, $R_{AB}$ decreases faster in manipulation scheme I than in manipulation scheme II. However, as shown in Figs. 2(e) and (f), $R_{BA}$ behaves different from $R_{AB}$. Strikingly, in manipulation scheme II, there is a region that allows one-way steering from Bob to Alice, although there is no such region for the initial state in the form of Eq. (6).

 figure: Fig. 2.

Fig. 2. The effect of noise channel on the direction of quantum steering. The steering radii (a) $R_{AB}$ and (d) $R_{BA}$ as functions of decoherence degree $D$ and state parameters $\lbrace p, \theta \rbrace$ when PDC is either added to Alice’s side or to Bob’s side. The steering radii (b) $R_{AB}$ and (e) $R_{BA}$ as functions of decoherence degree $D$ and state parameters $\lbrace p, \theta \rbrace$ when DC is only added to Alice’s side. The steering radii (c) $R_{AB}$ and (f) $R_{BA}$ as functions of decoherence degree $D$ and state parameters $\lbrace p, \theta \rbrace$ when DC is only added to Alice’s side.

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3. Experimental setup and results

Figure 3(a) shows our experimental setup. A 405 nm continuous-wave diode laser is used to pump a 10 mm long periodically poled potassium titanyl phosphate (PPKTP) crystal inside a polarization Sagnac interferometer clockwise and counter-clockwise to generate a class of two-qubit polarization-entangled states ${\rm cos}\theta \vert HH\rangle +{\rm sin}\theta \vert VV\rangle$ [47]. $\vert H \rangle$ and $\vert V \rangle$ denote the horizontal and vertical polarization, respectively. A dichroic mirror (DM) and two interference filters (IFs) are used to filter the pump beam. These down-conversion photons are separated by a polarization beam splitter (PBS), one of which is sent to Bob, while the other is collected by a single-mode fiber, and then sent to Alice through an unbalanced interferometer (UI). A pair of half-wave plates (HWPs, H2 and H3) is used to correct the unitary operations applied by the fiber. The beam splitter (BS) in the UI separates photons into two paths. In the upper path, the state remains unchanged, while in the lower path, the composition of two sufficiently long birefringent crystals (PC1, PC2) and an HWP (H4) set at 22.5$^{\circ }$ can completely destroy the coherence between $\vert H \rangle$ and $\vert V \rangle$. By combining these two paths into one, arbitrary two-qubit states in the form of Eq. (7) can be prepared. The parameters $p$ and $\theta$ can be flexibly controlled by the removable shutters (RSs) and the HWP (H0), respectively. Two semi-circular HWPs (H5 and H6) are employed to implement the steering direction manipulation schemes. As shown in Figs. 3(b) and (c), by changing the axes and action time of these HWPs, we can simulate the PDC and DC as desired [48]. The measurement setup consists of a quarter wave plate (QWP), an HWP and a PBS on both sides, allowing us to perform all desired measurements. Thus, we can check whether the state is two-way steerable, one-way steerable, or no-way steerable by analyzing the steering radii and critical radii.

 figure: Fig. 3.

Fig. 3. (a). Experimental setup. A pair of photons in a state $\vert \psi (\theta )\rangle$ is generated via the spontaneous parametric down-conversion process by pumping a type-II cut PPKTP crystal located in a Sagnac interferometer. A half-wave plate (HWP, H0) is used to control the parameter $\theta$. These down-conversion photons are filtered by interference filters (IFs). One of the photons is send to Bob. The other one passes through an unbalanced interferometer (UI), and then is sent to Alice. In the upper path, the state remains unchanged. In the lower path, a half-wave plate set at 22.5$^{\circ }$ (H4) and two sufficiently long calcite crystals (PC1 and PC2) with the length of the last one being two times larger than the first one to completely destroy the coherence between $\vert H\rangle$ and $\vert V\rangle$. By combining these two paths into one, arbitrary two-qubit states in the form of Eq. (7) can be prepared. Two removable shutters (RSs) are used to control the parameter $p$. Two semi-circular HWPs (H5 and H6) are used to implement the steering direction manipulation schemes. Quarter-wave plates (QWP), HWPs, and polarization beam splitters (PBSs) on both sides of Alice and Bob are used to perform the desired measurements. The photons are detected by single-photon detectors, and the signals are sent for coincidence. (b). Implementation of phase damping channel. $T$ is the repetition time. $\tau$ represents the activation time of $\sigma _{z}$; $\tau =0$ corresponds to noiseless channel ($I$), and $\tau >0$ corresponds to phase damping channel. (c). Implementation of depolarizing channel. $T$ is the repetition time, $\tau _{1}$, $\tau _{2}$ and $\tau _{3}$ represent the time intervals of $\sigma _{x}$, $\sigma _{y}$ and $\sigma _{z}$ activation, respectively. $\tau =\tau _{1}+\tau _{2}+\tau _{3}$ is the total Pauli operator time: $\tau =0$ corresponds to noiseless channel ($I$), and $0<\tau _{1} =\tau _{2} =\tau _{3}<T/3$ corresponds to depolarizing channel.

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Here, we use a specific two-way steerable state $\rho _{AB}$ with $p=0.95$ and $\theta =0.32$ as the initial state to investigate how the direction of quantum steering can be manipulated by adding a noise channel. As mentioned in the previous section, the effect of PDC on the steerability of state $\rho _{AB}$ is the same for both manipulation schemes. Experimentally, we take the manipulation scheme I as an example to investigate the performance of PDC. As shown in Fig. 3(b), the PDC can be simulated by a consist of two semi-circular HWPs (H5 and H6). For some appropriate axes, H5 and H6 can act as $\sigma _{z}$ or $I$. We increase the decoherence degree $D=\tau /T$ from $0$ to $1$ with a step of $0.1$ by fixing the activation time $\tau$ of the Pauli operator $\sigma _{z}$ and changing the repetition time $T$. Figure 4(a) shows the dependence of steering radii $R_{AB}$ (blue dots) and $R_{BA}$ (red squares) on the decoherence degree $D$ in the case of three-measurement settings. Obviously, with the increases of $D$, $R_{BA}$ decrease faster than $R_{AB}$, the two-way steering can be turned to either one-way steering or no-way steering. To eliminate the restriction of finite number of measurement settings, we further adopt critical radius to quantify the steerability. And we adopt an outer polyhedron with 162 vertices for approximating the Bloch sphere of Alice’s (Bob’s) reconsrtucted states to calculate the corresponding critical radii $C_{AB}$ ( $C_{BA}$). Figures 4(c) and (d) respectively depict the experimental reduced states of Alice and Bob (blue dots), which are close to their corresponding target states (red dots) with an average fidelity of $0.998 \pm 0.0014$. The dependence of critical radii $C_{AB}$ (blue dots) and $C_{BA}$ (red squares) on the decoherence degree $D$ is shown in Fig. 4(b). Clearly, more steerability can be discovered. The one-way steering in the range of $D\in [0.60,0.73]$ in Fig. 4(a) becomes two-way steering in Fig. 4(b). And the no-way steering in the range of $D\in [0.72,0.85]$ in Fig. 4(a) becomes one-way steering in Fig. 4(b). Obviously, with the help of the critical radius criterion, one can more robustly and rigorously determine the direction of quantum steering. Thus, we will adopt the critical radius criterion in the following discussion. We increase the decoherence degree $D=\tau /T$ from $0$ to $1$ with a step of $0.1$ by fixing the activation time $\tau$ of the Pauli operator $\sigma _{z}$ and changing the repetition time $T$. Figure 4(a) shows the dependence of steering radii $R_{AB}$ (blue dots) and $R_{BA}$ (red squares) on the decoherence degree $D$ in the case of three-measurement settings. Obviously, with the increases of $D$, $R_{BA}$ decrease faster than $R_{AB}$, the two-way steering can be turned to either one-way steering or no-way steering.

 figure: Fig. 4.

Fig. 4. The effect of PDC on the direction of quantum steering. (a). The steering radii $R_{AB}$ (blue dots) and $R_{BA}$ (red squares) as a function of decoherence degree $D$. (b). The critical radii $C_{AB}$ (blue dots) and $C_{BA}$ (red squares) as a function of decoherence degree $D$. Theoretical predictions in (a) and (b) are represented as solid curves with the corresponding colors. The dashed black lines denote boundaries between steerable and unsteerable. (c). The distributions of Alice’s experimental reduced states (blue dots) and theoretical reduced states (red dots) with different decoherence degrees. (d). The distributions of Bob’s experimental reduced states (blue dots) and theoretical reduced states (red dots) with different decoherence degrees. Theoretically, the reduced states of Alice and Bob are independent of the decoherence degree. Error bars are estimated by the Poissonian statistics of two-photon coincidences, which are too small to be visible.

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To eliminate the restriction of finite number of measurement settings, we further adopt critical radius to quantify the steerability. And we adopt an outer polyhedron with 162 vertices for approximating the Bloch sphere of Alice’s (Bob’s) reconsrtucted states to calculate the corresponding critical radii $C_{AB}$ ( $C_{BA}$). Figure 4(c) and (d) respectively depict the experimental reduced states of Alice and Bob (blue dots), which are close to their corresponding target states (red dots) with an average fidelity of $0.998 \pm 0.0014$. The dependence of critical radii $C_{AB}$ (blue dots) and $C_{BA}$ (red squares) on the decoherence degree $D$ is shown in Fig. 4(b). Clearly, more steerability can be discovered. The one-way steering in the range of $D\in [0.60,0.73]$ in Fig. 4(a) becomes two-way steering in Fig. 4(b). And the no-way steering in the range of $D\in [0.72,0.85]$ in Fig. 4(a) becomes one-way steering in Fig. 4(b). Obviously, with the help of the critical radius criterion, one can more robustly and rigorously determine the direction of quantum steering. Thus, we will adopt the critical radius criterion in the following discussion. We increase the decoherence degree $D=\tau /T$ from $0$ to $1$ with a step of $0.1$ by fixing the activation time $\tau$ of the Pauli operator $\sigma _{z}$ and changing the repetition time $T$. Fig. 4(a) shows the dependence of steering radii $R_{AB}$ (blue dots) and $R_{BA}$ (red squares) on the decoherence degree $D$ in the case of three-measurement settings. Obviously, with the increases of $D$, $R_{BA}$ decrease faster than $R_{AB}$, the two-way steering can be turned to either one-way steering or no-way steering. To eliminate the restriction of finite number of measurement settings, we further adopt critical radius to quantify the steerability. And we adopt an outer polyhedron with 162 vertices for approximating the Bloch sphere of Alice’s (Bob’s) reconsrtucted states to calculate the corresponding critical radii $C_{AB}$ ( $C_{BA}$). Figures 4(c) and (d) respectively depict the experimental reduced states of Alice and Bob (blue dots), which are close to their corresponding target states (red dots) with an average fidelity of $0.998 \pm 0.0014$. The dependence of critical radii $C_{AB}$ (blue dots) and $C_{BA}$ (red squares) on the decoherence degree $D$ is shown in Fig. 4(b). Clearly, more steerability can be discovered. The one-way steering in the range of $D\in [0.60,0.73]$ in Fig. 4(a) becomes two-way steering in Fig. 4(b). And the no-way steering in the range of $D\in [0.72,0.85]$ in Fig. 4(a) becomes one-way steering in Fig. 4(b). Obviously, with the help of the critical radius criterion, one can more robustly and rigorously determine the direction of quantum steering. Thus, we will adopt the critical radius criterion in the following discussion.

We further investigate the influence of DC on the direction of quantum steering. Similarly, the DC can also be simulated by H5 and H6. As shown in Fig. 3(c), for some appropriate axes, H5 and H6 can act as $\sigma _{x}$, $\sigma _{y}$ or $\sigma _{z}$. And we can fix their corresponding activation times $\tau _{1}$, $\tau _{2}$ and $\tau _{3}$, and then change the repetition time $T$ to adjust the decoherence degree $D=(\tau _{1}+\tau _{2}+\tau _{3})/T$, where $\tau _{1}=\tau _{2}=\tau _{3}$. The results for the manipulation scheme I and the manipulation scheme II are shown in Figs. 5(a) and (b), respectively. Experimentally measured results of critical radii $C_{AB}$ and $C_{BA}$ are respectively represented by blue dots and red squares, which are in good agreement with theoretical predictions. Similar to the above PDC-based manipulation scheme, both Alice and Bob can manipulate their steering directions by varying the degree of DC-induced decoherence $D$. However, the steerabilities of Alice and Bob disappear faster in these two DC-based manipulation schemes. In addition, unlike manipulation scheme I, $C_{BA}$ decreases slower than $C_{AB}$ in manipulation scheme II. We find that asymmetric steering becomes symmetric steering when $D=0.044$. The results in range of $D\in [0,0.4]$ are enlarged in Figs. 5(c) and (d). Surprisingly, even if the one-way steering from Bob Alice is absent for all states in the form Eq. (7), it can occur when the degree of decoherence on Bob’s side increases to $D\in [0.22,0.32]$. These results show that the change of quantum steering direction depends on which side the noise channel is added.

 figure: Fig. 5.

Fig. 5. The effect of DC on the direction of quantum steering. (a). The critical radii $C_{AB}$ (blue dots) and $C_{BA}$ (red squares) as a function of decoherence degree $D$ when DC is only added to Alice’s side. (b). The critical radii $C_{AB}$ (blue dots) and $C_{BA}$ (red squares) as a function of decoherence degree $D$ when DC is only added to Bob’s side. (c). The enlarged view of the part inside the yellow box in Fig. 5(a). (d). The enlarged view of the part inside the yellow box in Fig. 5(b). Theoretical predictions in (a)-(d) are represented as solid curves with the corresponding colors. The dashed black lines denote the steerable boundaries. Error bars are estimated by the Poissonian statistics of two-photon coincidences, which are too small to be visible.

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4. Conclusion

In summary, we propose two schemes for manipulating the direction of quantum steering by respectively adding a noise channel to one side of two-qubit states share between Alice and Bob. By measuring steering radius and critical radius, we find that, whether adding PDC or adding DC, a tunable degree of decoherence allows the steering direction between Alice and Bob to be shifted from a two-way steerable range to a one-way steerable range, and finally, to a range that is unsteerable in both directions. Unlike PDC-based manipulation scheme, the steering direction is changed depending on which side the DC is added to. Surprisingly, when DC is added to Bob’s side, a new one-way steering from Bob to Alice is discovered. Compared with the previous noise-adding manipulating scheme, our scheme is valid for general projective measurement, provides a more efficient way to understand the asymmetric characteristic of quantum steering, and can also be employed to manipulate other types of quantum correlations.

5. Appendix A: the corresponding state after adding noise channel

As mentioned in the maintext, we propose two schemes for manipulating the direction of quantum steering by respectively adding a noise channel to one side of two-qubit states $\rho _{AB}$, which has the following form

$$\rho_{AB}=p\vert \psi(\theta)\rangle\langle\psi(\theta)\vert+(1-p) \mathbb{I} /2\otimes\rho_{B}^{\theta},$$
where $\vert \psi (\theta )\rangle =\cos \theta \vert 11\rangle +\sin \theta \vert 00\rangle$, $\rho _{B}^{\theta } ={\rm Tr}_{A} [\vert \psi (\theta )\rangle \langle \psi (\theta )\vert ]$, $p \in [0,1]$ and $\theta \in [0,\pi /4]$.

In the manipulation scheme I, the noise channel is only added to Alice’s side. Mathematically, the noise channel corresponds to a completely positive trace-preserving map [43]. The phase damping channel (PDC) maps the initial state $\rho _{AB}$ into

$$\varepsilon (\rho_{AB})=\frac{1}{2}\left( \begin{array}{cccc} (p+1) \cos ^2\theta & 0 & 0 & (1-D) p \sin (2\theta ) \\ 0 & (1-p) \sin ^2\theta & 0 & 0 \\ 0 & 0 & (1-p) \cos ^2\theta & 0 \\ (D-1) p \sin (2\theta ) & 0 & 0 & (p+1) \sin ^2\theta \\ \end{array} \right),$$
where $D\in [0,1]$ is the decoherence degree. While the depolarizing channel (DC) maps the initial state $\rho _{AB}$ into
$$\varepsilon (\rho_{AB})=\frac{1}{6}\left( \begin{array}{cccc} (t+3) \cos ^2\theta & 0 & 0 & t \sin (2\theta )\\ 0 & (3-t) \sin ^2\theta & 0 & 0 \\ 0 & 0 & (3-t) \cos ^2\theta & 0 \\ t \sin (2\theta ) & 0 & 0 & (t+3) \sin ^2\theta \\ \end{array} \right),$$
where $t=(3-4D)p$, $D\in [0,0.75]$ is the decoherence degree.

In the manipulation scheme II, the noise channel is only added to Bob’s side. Since phase damping channel models the decoherence situations in which there is no loss of energy while the phases die out progressively, it is mathematically equivalent to reducing the off-diagonal elements of the density matrix of $\rho _{AB}$. Therefore, after adding a PDC, the final state $\varepsilon (\rho _{AB})$ is the same as that in manipulation scheme I, i.e. it has same form of Eq. (8). However, depolarizing noise will induce bit flip, phase flip, and a combination of bit and phase flip errors. After the action of DC, the state $\rho _{AB}$ involves to

$$\scalebox{0.9}{$\displaystyle\varepsilon (\rho_{AB})=\frac{1}{6} \left( \begin{array}{cccc} 2 D (1\!-\!p)\!+\!m \cos^2\theta & 0 & 0 & t \sin(2\theta)\\ 0 & 2 D (1\!+\!p)\!-\!n \sin^2\theta & 0 & 0 \\ 0 & 0 & 2 D (1\!+\!p)\!-\!n\cos ^2\theta & 0 \\ t \sin(2\theta) & 0 & 0 & 2 D (1\!-\!p)\!+\!m \sin^2\theta\\ \end{array} \right),$}$$
where $m=3p-4D+3$, $n=3p-4D-3$, $D$ is decoherence degree.

6. Appendix B: the detalied method for determining the steering radius

In the maintext, we employ the steering radii $R_{AB}$ and $R_{BA}$, obtained in the case of three-setting measurements, to show how manipulation scheme I and manipulation scheme II affect the direction of quantum steering. Here, we take the case of manipulation scheme I and add PDC as an example to calculate $R_{AB}$ and $R_{BA}$. As mentioned above, after adding the PDC, the state $\rho _{AB}$ evolves to Eq. (8), which can also be rewritten as

$$\varepsilon (\rho_{AB})=\frac{1}{4}\sum_{\mu ,\nu =0}^{3}\Theta _{\mu \nu }\sigma _{\mu}\otimes \sigma _{\nu},$$
where ${\sigma _\mu,_\nu }= \{\mathbb {I}, \sigma _x,\sigma _y,\sigma _z\}$, $\mu,\nu =0, 1,2,3$, and $\Theta _{\mu \nu } = tr(\rho \sigma _\mu \otimes \sigma _\nu )$ is real for all $\mu, \nu$. As a block matrix, we have $\Theta = \begin {pmatrix} 1 & \boldsymbol {b}^T \\ \boldsymbol {a} & T \end {pmatrix}$, where $\boldsymbol {a}$ and $\boldsymbol {b}$ corespond to the Bloch vectors of the reduced states $\varepsilon (\rho _A)=Tr_{B}(\varepsilon (\rho _{AB}))$ and $\varepsilon (\rho _B)=Tr_{A}(\varepsilon (\rho _{AB}))$ of $\varepsilon (\rho _{AB})$ respectively, and $T$ is a 3 $\times$ 3 matrix encoding the correlations [49]. Consider the steering task between Alice and Bob, the set of states to which Bob can steer Alice forms an ellipsoid sphere center at $\boldsymbol {c}_{A}=\boldsymbol {a}-T \boldsymbol {b}/(1-b^2)$, with orientation and semiaxes lengths $s_{i}=\sqrt {q_{i}}$ given by the eigenvectors and eigenvalues $q_{i}$ of the ellipsoid matrix [46]
$$Q_A =\frac{1}{1-b^2}\left( T-\boldsymbol{a}\boldsymbol{b} ^{T}\right) \left( \mathbb{I}+\frac{\boldsymbol{ b}\boldsymbol{b}^{T}}{ 1-b^{2}}\right) \left( T^{T}-\boldsymbol{b}\boldsymbol{a}^{T}\right).$$

And the ellipsoid sphere $Q_B$ at Bob’s side can be obtained by transposing $\Theta$ and sending $\boldsymbol {a}\rightarrow \boldsymbol {b}$, $\boldsymbol {b}\rightarrow \boldsymbol {a}$, $T\rightarrow T^{T}$. Combining Eq. (8) and Eq. (12), we can obtain the eigenvectors of $q_{i}$, which are $\{1,0,0\},\{0,1,0\}$, $\{0,0,1\}$ respectively. In addition, in the case of three-setting measurements, the steering radius of Alice can be expressed as [26,27]

$$R_{AB}=\max_{\lbrace \vec{n}_{1},\vec{n}_{2},\vec{n}_{3}\rbrace} \lbrace \min _{\lbrace p_{i}\rho_{i}\rbrace} \lbrace {\rm max} \lbrace \vert \vec{L}_{i} \vert \rbrace \rbrace \rbrace,$$
$\lbrace \vec {n}_{1},\vec {n}_{2},\vec {n}_{3}\rbrace$ is the direction assemblage of Alice’s three-measurement settings, $\lbrace p_{i}\rho _{i}\rbrace$ is the local hidden state ensemble of Bob, $\vert \vec {L}_{i} \vert$ is the length of the Bloch vector of the corresponding local hidden state $\rho _{i}$. And Bob’s steering radius $R_{BA}$ can be defined similarly. According to the symmetry of $Q_A$, it is easy to demonstrate that the optimal measurement settings of Alice are parallel to the three semiaxes of $Q_A$, which are $\lbrace \vec {x},\vec {y},\vec {z} \rbrace$ [37]. Similarly, we can obtain the optimal measurement directions of Bob, which also are $\lbrace \vec {x},\vec {y},\vec {z} \rbrace$. With the optimal measurement settings and state Eq. (8), $R_{AB}$ can be rewritten as
$$\scalebox{0.9}{$\displaystyle R_{AB}=\sqrt{(D-1)^2 p^2 \sin ^2(2 \theta )+1-\frac{(D-1) p^2 \sin (2 \theta )}{2}*A+\frac{(D-1) \tan (2 \theta ) }{p}*B+(D-1)^2 \tan ^2(2 \theta )},$}$$
where
$$\scalebox{0.96}{$\begin{aligned} & A=\sqrt{\frac{ \left((D-1)^2 p^4 \sin ^4(4 \theta ) \csc ^2(2 \theta )-8 \left(((D-2) D+2) p^2-1\right) \sin ^2(4 \theta )+16 (D-1)^2 \sin ^2(2 \theta )\right)}{\left(p^2 \cos (4 \theta )+p^2-2\right)^2}},\\ &B=\sqrt{\frac{\sec^2 (2 \theta )((D-1)^2 p^6 \sin ^2(2 \theta )+p^2 \tan ^2(2 \theta ) \left({-}2 ((D-2) D+2) p^2+(D-1)^2 \sec ^2(2 \theta )+2\right))}{\left(p^2-\sec ^2(2 \theta )\right)^2}}. \end{aligned}$}$$

And $R_{BA}$ can be rewritten as

$$R_{BA}=\sqrt{p^2 \left(2 (D-1)^2 \sin ^2(2 \theta )+1\right)}.$$

Similarly, in the case of manipulation scheme I and add DC, the steering radii $R_{AB}$ and $R_{BA}$ of state Eq. (9) can be expressed as

$$\begin{aligned} R_{AB}&=\sqrt{\sec ^2(2 \theta )+\frac{\tan (2 \theta ) \sec (2 \theta )}{(4 D-3) p}* C+ \frac{((4 D-3) p \sin (2 \theta ))^2}{9}*E},\\ R_{BA}&=\frac{1}{3} \sqrt{-(3-4 D)^2 p^2 (\cos (4 \theta )-2)}, \end{aligned}$$
where
$$\begin{aligned}& C= \sqrt{\frac{(3-4 D)^6 p^6 \sin ^2(2 \theta )+9 (3-4 D)^2 p^2 \tan ^2(2 \theta ) \left({-}4 (3-4 D)^2 p^2+9 \sec ^2(2 \theta )+18\right)}{\left((3-4 D)^2 p^2-9 \sec ^2(2 \theta )\right)^2}},\\ & E= 1-\sqrt{\frac{(3-4 D)^6 p^6 \sin ^2(2 \theta )+9 (3-4 D)^2 p^2 \tan ^2(2 \theta ) \left({-}4 (3-4 D)^2 p^2+9 \sec ^2(2 \theta )+18\right)}{\left((3-4 D)^2 p^2-9 \sec ^2(2 \theta )\right)^2*((4 D-3) p \sin (2 \theta ))^2}}. \end{aligned}$$

And in the case of manipulation scheme II and add DC, the steering radii $R_{AB}$ and $R_{BA}$ of state Eq. (10) can be expressed as

$$\begin{aligned} R_{AB}&=\frac{1}{3}\sqrt{\frac{4 D-3}{p}((4 D-3) p^3 \sin ^2(2 \theta )+\sec ^2(2 \theta ) *(F+(4 D-3) p)- p^2 \sin (2 \theta )* G)},\\ R_{BA}&=\frac{p\sin(2 \theta ) \csc(4 \theta ) \left((3-4 D)^2 \cos ^2(2 \theta )-9\right)}{6(3-4 D)^2}\sqrt{(H+ J- K)}, \end{aligned}$$
where
$$\begin{aligned} & F=\sin (2 \theta )\sqrt{\frac{(3-4 D)^2 p^2 \left(p^4 (-\cos (4 \theta ))+p^4+2 \sec ^2(2 \theta ) \left({-}4 p^2+\sec ^2(2 \theta )+1\right)+8 p^2-4\right)}{2\left(p^2-\sec ^2(2 \theta )\right)^2}},\\ &G=\sqrt{\frac{(3-4 D)^2 p^2 \left(p^4 \sin ^2(2 \theta )-2 p^2 \tan ^2(2 \theta )+\sec ^2(2 \theta ) \left({-}2 p^2+\tan ^2(2 \theta )+2\right)+2 p^2-2\right)}{\left(p^2-\sec ^2(2 \theta )\right)^2}},\\ & H=\frac{9 (3-4 D)^4 \cos ^2(2 \theta )}{\left(3 \cos ^2(\theta )-2 D \cos (2 \theta )\right)^2 \left(2 D \cos (2 \theta )+3 \sin ^2(\theta )\right)^2},\\ & J=\frac{8 (3-4 D)^4 p \sin ^2(2 \theta ) \left((3-4 D)^2 \cos (4 \theta )+8 D (2 D-3)+27\right)}{\left((3-4 D)^2 \cos ^2(2 \theta )-9\right)^2}, \end{aligned}$$
$$K= \frac{2 (4 D-3)^4 \sin (2 \theta ) \sqrt{\frac{9 D (3-2 D) \sin ^2(4 \theta )}{\left(3 \cos ^2(\theta )-2 D \cos (2 \theta )\right)^2 \left(2 D \cos (2 \theta )+3 \sin ^2(\theta )\right)^2}+4 \sin ^2(2 \theta )}}{\left(3 \cos ^2(\theta )-2 D \cos (2 \theta )\right) \left(2 D \cos (2 \theta )+3 \sin ^2(\theta )\right)}.$$

7. Appendix C: more experimental results

Figure 6(a-b) and (c-d) show the effects of DC on the reduced states of Alice and Bob in case of manipulation scheme I and manipulation scheme II, respectively. Experimentally, the decoherence degree $D$ is increased from $0$ to $0.7$ in steps of $0.1$ and the average fidelity of these reconstructed states (blue dots) is $0.998 \pm 0.0012$. Clearly, all experimental reduced states (blue dots) are close to their corresponding target states (red dots).

 figure: Fig. 6.

Fig. 6. The distributions of experimental reduced states and theoretical reduced states when DC is added in different manipulation schemes. (a) and (b) are the distributions of experimental reduced states (blue dots) and theoretical reduced states (red dots) of Alice with different decoherence degrees in manipulation scheme I and manipulation scheme II, respectively. (c) and (d) are the distributions of experimental reduced states (blue dots) and theoretical reduced states (red dots) of Bob with different decoherence degrees in manipulation scheme I and manipulation scheme II, respectively. Theoretically, the reduced states of Alice in manipulation scheme II and that of Bob in manipulation scheme I are independent of the decoherence degree.

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Figures 7(a) and (b) show the effects of DC on the direction of quantum steering in case of manipulation scheme I and manipulation scheme II, respectively. The blue dots and red squares represent the experimentally measured results of the steering radius $R_{AB}$ and $R_{BA}$, which are in good agreement with the theoretical predictions (red and blue solid curves). Clearly, in both cases, a tunable degree of decoherence allows the steering direction between Alice and Bob to be shifted from a two-way steerable range to a one-way steerable range, and finally, to a range that is unsteerable in both directions. However, these steerable ranges are smaller than those obtained by measuring the critical radii (Fig. 4 in the main text), indicating that the manipulation of quantum steering direction can be more rigorously verified by the critical radius.

 figure: Fig. 7.

Fig. 7. The effect of DC on the direction of quantum steering is quantified by steering radius. (a). The steering radius $R_{AB}$ (blue dots) and $R_{BA}$ (red squares) as a function of decoherence degree $D$ when DC is only added to Alice’s side. (b). The steering radius $R_{AB}$ (blue dots) and $R_{BA}$ (red squares) as a function of decoherence degree $D$ when DC is only added to Bob’s side. Theoretical predictions in (a)-(b) are represented as solid curves with the corresponding colors. The dashed black lines denote the steerable boundaries. Error bars are estimated by the Poissonian statistics of two-photon coincidences, which are too small to be visible.

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Funding

National Natural Science Foundation of China (12004358); National Natural Science Foundation Regional Innovation and Development Joint Fund (U19A2075); Fundamental Research Funds for the Central Universities (202041012, 841912027); Natural Science Foundation of Shandong Province (ZR2021ZD19); Young Talents Project at Ocean University of China (861901013107).

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. Schemes for manipulating the direction of quantum steering. A two-qubit quantum state $\rho _{AB}$ is distributed to Alice and Bob, with a noise channel, on the way to Alice or Bob. The steerablity of the state shared between Alice and Bob $\varepsilon (\rho _{AB})$ is quantified by steering radius or critical radius. Two different types of noise channels, phase damping channel (PDC) and depolarizing channel (DC), are considered here. The steering direction between Alice and Bob can be manipulated by changing the noise added to Alice’s side (scheme I) or Bob’s side (scheme II).

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Fig. 2.
Fig. 2. The effect of noise channel on the direction of quantum steering. The steering radii (a) $R_{AB}$ and (d) $R_{BA}$ as functions of decoherence degree $D$ and state parameters $\lbrace p, \theta \rbrace$ when PDC is either added to Alice’s side or to Bob’s side. The steering radii (b) $R_{AB}$ and (e) $R_{BA}$ as functions of decoherence degree $D$ and state parameters $\lbrace p, \theta \rbrace$ when DC is only added to Alice’s side. The steering radii (c) $R_{AB}$ and (f) $R_{BA}$ as functions of decoherence degree $D$ and state parameters $\lbrace p, \theta \rbrace$ when DC is only added to Alice’s side.

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Fig. 3.
Fig. 3. (a). Experimental setup. A pair of photons in a state $\vert \psi (\theta )\rangle$ is generated via the spontaneous parametric down-conversion process by pumping a type-II cut PPKTP crystal located in a Sagnac interferometer. A half-wave plate (HWP, H0) is used to control the parameter $\theta$. These down-conversion photons are filtered by interference filters (IFs). One of the photons is send to Bob. The other one passes through an unbalanced interferometer (UI), and then is sent to Alice. In the upper path, the state remains unchanged. In the lower path, a half-wave plate set at 22.5$^{\circ }$ (H4) and two sufficiently long calcite crystals (PC1 and PC2) with the length of the last one being two times larger than the first one to completely destroy the coherence between $\vert H\rangle$ and $\vert V\rangle$. By combining these two paths into one, arbitrary two-qubit states in the form of Eq. (7) can be prepared. Two removable shutters (RSs) are used to control the parameter $p$. Two semi-circular HWPs (H5 and H6) are used to implement the steering direction manipulation schemes. Quarter-wave plates (QWP), HWPs, and polarization beam splitters (PBSs) on both sides of Alice and Bob are used to perform the desired measurements. The photons are detected by single-photon detectors, and the signals are sent for coincidence. (b). Implementation of phase damping channel. $T$ is the repetition time. $\tau$ represents the activation time of $\sigma _{z}$; $\tau =0$ corresponds to noiseless channel ($I$), and $\tau >0$ corresponds to phase damping channel. (c). Implementation of depolarizing channel. $T$ is the repetition time, $\tau _{1}$, $\tau _{2}$ and $\tau _{3}$ represent the time intervals of $\sigma _{x}$, $\sigma _{y}$ and $\sigma _{z}$ activation, respectively. $\tau =\tau _{1}+\tau _{2}+\tau _{3}$ is the total Pauli operator time: $\tau =0$ corresponds to noiseless channel ($I$), and $0<\tau _{1} =\tau _{2} =\tau _{3}<T/3$ corresponds to depolarizing channel.

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Fig. 4.
Fig. 4. The effect of PDC on the direction of quantum steering. (a). The steering radii $R_{AB}$ (blue dots) and $R_{BA}$ (red squares) as a function of decoherence degree $D$. (b). The critical radii $C_{AB}$ (blue dots) and $C_{BA}$ (red squares) as a function of decoherence degree $D$. Theoretical predictions in (a) and (b) are represented as solid curves with the corresponding colors. The dashed black lines denote boundaries between steerable and unsteerable. (c). The distributions of Alice’s experimental reduced states (blue dots) and theoretical reduced states (red dots) with different decoherence degrees. (d). The distributions of Bob’s experimental reduced states (blue dots) and theoretical reduced states (red dots) with different decoherence degrees. Theoretically, the reduced states of Alice and Bob are independent of the decoherence degree. Error bars are estimated by the Poissonian statistics of two-photon coincidences, which are too small to be visible.

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Fig. 5.
Fig. 5. The effect of DC on the direction of quantum steering. (a). The critical radii $C_{AB}$ (blue dots) and $C_{BA}$ (red squares) as a function of decoherence degree $D$ when DC is only added to Alice’s side. (b). The critical radii $C_{AB}$ (blue dots) and $C_{BA}$ (red squares) as a function of decoherence degree $D$ when DC is only added to Bob’s side. (c). The enlarged view of the part inside the yellow box in Fig. 5(a). (d). The enlarged view of the part inside the yellow box in Fig. 5(b). Theoretical predictions in (a)-(d) are represented as solid curves with the corresponding colors. The dashed black lines denote the steerable boundaries. Error bars are estimated by the Poissonian statistics of two-photon coincidences, which are too small to be visible.

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Fig. 6.
Fig. 6. The distributions of experimental reduced states and theoretical reduced states when DC is added in different manipulation schemes. (a) and (b) are the distributions of experimental reduced states (blue dots) and theoretical reduced states (red dots) of Alice with different decoherence degrees in manipulation scheme I and manipulation scheme II, respectively. (c) and (d) are the distributions of experimental reduced states (blue dots) and theoretical reduced states (red dots) of Bob with different decoherence degrees in manipulation scheme I and manipulation scheme II, respectively. Theoretically, the reduced states of Alice in manipulation scheme II and that of Bob in manipulation scheme I are independent of the decoherence degree.

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Fig. 7.
Fig. 7. The effect of DC on the direction of quantum steering is quantified by steering radius. (a). The steering radius $R_{AB}$ (blue dots) and $R_{BA}$ (red squares) as a function of decoherence degree $D$ when DC is only added to Alice’s side. (b). The steering radius $R_{AB}$ (blue dots) and $R_{BA}$ (red squares) as a function of decoherence degree $D$ when DC is only added to Bob’s side. Theoretical predictions in (a)-(b) are represented as solid curves with the corresponding colors. The dashed black lines denote the steerable boundaries. Error bars are estimated by the Poissonian statistics of two-photon coincidences, which are too small to be visible.

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$$\varepsilon(\rho_{AB}) =(1-\dfrac{D}{2})\rho_{AB}+\dfrac{D}{2} (\sigma_{z}\otimes \mathbb{I})\rho_{AB}(\sigma_{z}\otimes I),$$
$$\varepsilon(\rho_{AB})=(1-D)\rho_{AB}+\dfrac{D}{3}((\sigma_{x}\otimes \mathbb{I})\rho_{AB}(\sigma_{x}\otimes \mathbb{I})+(\sigma_{y}\otimes \mathbb{I})\rho_{AB}(\sigma_{y}\otimes \mathbb{I})+(\sigma_{z}\otimes \mathbb{I})\rho_{AB}(\sigma_{z}\otimes \mathbb{I}).$$
$$\begin{array}{c}\varepsilon (\rho_{AB})=(1-\dfrac{D}{2})\rho_{AB}+\dfrac{D}{2} (\mathbb{I}\otimes \sigma_{z})\rho_{AB}(\mathbb{I}\otimes\sigma_{z}),\\ \varepsilon (\rho_{AB})=(1-D)\rho_{AB}+\dfrac{D}{3}((\mathbb{I}\otimes\sigma_{x})\rho_{AB}(\mathbb{I}\otimes\sigma_{x})+(\mathbb{I}\otimes\sigma_{y})\rho_{AB}(\mathbb{I}\otimes\sigma_{y})+(\mathbb{I}\otimes \sigma_{z})\rho_{AB}(\mathbb{I}\otimes\sigma_{z})). \end{array}$$
$$R_{AB}=\max_{\lbrace \vec{n}_{1},\vec{n}_{2},\ldots\vec{n}_{N}\rbrace} \lbrace \min _{\lbrace p_{i}\rho_{i}\rbrace} \lbrace {\rm max} \lbrace \vert \vec{L}_{i} \vert \rbrace \rbrace \rbrace.$$
$$C_{AB}=\max_{\eta}\lbrace \eta>0:\rho^{\eta}_{AB} \ \text{is unsteerable} \rbrace,$$
$$\rho_{AB} =p \vert \psi(\theta)\rangle\langle\psi(\theta)\vert+(1-p)\mathbb{I}/2\otimes\rho_{B}^{\theta},$$
$$\rho_{AB}=p\vert \psi(\theta)\rangle\langle\psi(\theta)\vert+(1-p) \mathbb{I} /2\otimes\rho_{B}^{\theta},$$
$$\varepsilon (\rho_{AB})=\frac{1}{2}\left( \begin{array}{cccc} (p+1) \cos ^2\theta & 0 & 0 & (1-D) p \sin (2\theta ) \\ 0 & (1-p) \sin ^2\theta & 0 & 0 \\ 0 & 0 & (1-p) \cos ^2\theta & 0 \\ (D-1) p \sin (2\theta ) & 0 & 0 & (p+1) \sin ^2\theta \\ \end{array} \right),$$
$$\varepsilon (\rho_{AB})=\frac{1}{6}\left( \begin{array}{cccc} (t+3) \cos ^2\theta & 0 & 0 & t \sin (2\theta )\\ 0 & (3-t) \sin ^2\theta & 0 & 0 \\ 0 & 0 & (3-t) \cos ^2\theta & 0 \\ t \sin (2\theta ) & 0 & 0 & (t+3) \sin ^2\theta \\ \end{array} \right),$$
$$\scalebox{0.9}{$\displaystyle\varepsilon (\rho_{AB})=\frac{1}{6} \left( \begin{array}{cccc} 2 D (1\!-\!p)\!+\!m \cos^2\theta & 0 & 0 & t \sin(2\theta)\\ 0 & 2 D (1\!+\!p)\!-\!n \sin^2\theta & 0 & 0 \\ 0 & 0 & 2 D (1\!+\!p)\!-\!n\cos ^2\theta & 0 \\ t \sin(2\theta) & 0 & 0 & 2 D (1\!-\!p)\!+\!m \sin^2\theta\\ \end{array} \right),$}$$
$$\varepsilon (\rho_{AB})=\frac{1}{4}\sum_{\mu ,\nu =0}^{3}\Theta _{\mu \nu }\sigma _{\mu}\otimes \sigma _{\nu},$$
$$Q_A =\frac{1}{1-b^2}\left( T-\boldsymbol{a}\boldsymbol{b} ^{T}\right) \left( \mathbb{I}+\frac{\boldsymbol{ b}\boldsymbol{b}^{T}}{ 1-b^{2}}\right) \left( T^{T}-\boldsymbol{b}\boldsymbol{a}^{T}\right).$$
$$R_{AB}=\max_{\lbrace \vec{n}_{1},\vec{n}_{2},\vec{n}_{3}\rbrace} \lbrace \min _{\lbrace p_{i}\rho_{i}\rbrace} \lbrace {\rm max} \lbrace \vert \vec{L}_{i} \vert \rbrace \rbrace \rbrace,$$
$$\scalebox{0.9}{$\displaystyle R_{AB}=\sqrt{(D-1)^2 p^2 \sin ^2(2 \theta )+1-\frac{(D-1) p^2 \sin (2 \theta )}{2}*A+\frac{(D-1) \tan (2 \theta ) }{p}*B+(D-1)^2 \tan ^2(2 \theta )},$}$$
$$\scalebox{0.96}{$\begin{aligned} & A=\sqrt{\frac{ \left((D-1)^2 p^4 \sin ^4(4 \theta ) \csc ^2(2 \theta )-8 \left(((D-2) D+2) p^2-1\right) \sin ^2(4 \theta )+16 (D-1)^2 \sin ^2(2 \theta )\right)}{\left(p^2 \cos (4 \theta )+p^2-2\right)^2}},\\ &B=\sqrt{\frac{\sec^2 (2 \theta )((D-1)^2 p^6 \sin ^2(2 \theta )+p^2 \tan ^2(2 \theta ) \left({-}2 ((D-2) D+2) p^2+(D-1)^2 \sec ^2(2 \theta )+2\right))}{\left(p^2-\sec ^2(2 \theta )\right)^2}}. \end{aligned}$}$$
$$R_{BA}=\sqrt{p^2 \left(2 (D-1)^2 \sin ^2(2 \theta )+1\right)}.$$
$$\begin{aligned} R_{AB}&=\sqrt{\sec ^2(2 \theta )+\frac{\tan (2 \theta ) \sec (2 \theta )}{(4 D-3) p}* C+ \frac{((4 D-3) p \sin (2 \theta ))^2}{9}*E},\\ R_{BA}&=\frac{1}{3} \sqrt{-(3-4 D)^2 p^2 (\cos (4 \theta )-2)}, \end{aligned}$$
$$\begin{aligned}& C= \sqrt{\frac{(3-4 D)^6 p^6 \sin ^2(2 \theta )+9 (3-4 D)^2 p^2 \tan ^2(2 \theta ) \left({-}4 (3-4 D)^2 p^2+9 \sec ^2(2 \theta )+18\right)}{\left((3-4 D)^2 p^2-9 \sec ^2(2 \theta )\right)^2}},\\ & E= 1-\sqrt{\frac{(3-4 D)^6 p^6 \sin ^2(2 \theta )+9 (3-4 D)^2 p^2 \tan ^2(2 \theta ) \left({-}4 (3-4 D)^2 p^2+9 \sec ^2(2 \theta )+18\right)}{\left((3-4 D)^2 p^2-9 \sec ^2(2 \theta )\right)^2*((4 D-3) p \sin (2 \theta ))^2}}. \end{aligned}$$
$$\begin{aligned} R_{AB}&=\frac{1}{3}\sqrt{\frac{4 D-3}{p}((4 D-3) p^3 \sin ^2(2 \theta )+\sec ^2(2 \theta ) *(F+(4 D-3) p)- p^2 \sin (2 \theta )* G)},\\ R_{BA}&=\frac{p\sin(2 \theta ) \csc(4 \theta ) \left((3-4 D)^2 \cos ^2(2 \theta )-9\right)}{6(3-4 D)^2}\sqrt{(H+ J- K)}, \end{aligned}$$
$$\begin{aligned} & F=\sin (2 \theta )\sqrt{\frac{(3-4 D)^2 p^2 \left(p^4 (-\cos (4 \theta ))+p^4+2 \sec ^2(2 \theta ) \left({-}4 p^2+\sec ^2(2 \theta )+1\right)+8 p^2-4\right)}{2\left(p^2-\sec ^2(2 \theta )\right)^2}},\\ &G=\sqrt{\frac{(3-4 D)^2 p^2 \left(p^4 \sin ^2(2 \theta )-2 p^2 \tan ^2(2 \theta )+\sec ^2(2 \theta ) \left({-}2 p^2+\tan ^2(2 \theta )+2\right)+2 p^2-2\right)}{\left(p^2-\sec ^2(2 \theta )\right)^2}},\\ & H=\frac{9 (3-4 D)^4 \cos ^2(2 \theta )}{\left(3 \cos ^2(\theta )-2 D \cos (2 \theta )\right)^2 \left(2 D \cos (2 \theta )+3 \sin ^2(\theta )\right)^2},\\ & J=\frac{8 (3-4 D)^4 p \sin ^2(2 \theta ) \left((3-4 D)^2 \cos (4 \theta )+8 D (2 D-3)+27\right)}{\left((3-4 D)^2 \cos ^2(2 \theta )-9\right)^2}, \end{aligned}$$
$$K= \frac{2 (4 D-3)^4 \sin (2 \theta ) \sqrt{\frac{9 D (3-2 D) \sin ^2(4 \theta )}{\left(3 \cos ^2(\theta )-2 D \cos (2 \theta )\right)^2 \left(2 D \cos (2 \theta )+3 \sin ^2(\theta )\right)^2}+4 \sin ^2(2 \theta )}}{\left(3 \cos ^2(\theta )-2 D \cos (2 \theta )\right) \left(2 D \cos (2 \theta )+3 \sin ^2(\theta )\right)}.$$
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