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Abstract
Multipartite Einstein-Podolsky-Rosen (EPR) steering is an essential resource for secure one-sided device-independent quantum secret sharing. Here, we analyze the EPR steering properties exhibited in three-mode Gaussian states created by four-wave mixing (FWM) in Rubidium atoms combined with a linear beamsplitter and a nonlinear beamsplitter (second FWM), respectively. By quantifying Gaussian steerability based on a measure determined by the covariance matrix of the produced states, we compare the performance of two schemes to achieve one-way, collective, and genuine tripartite steering, as well as the monogamy constraints for distributing steering among three parties. We show that the scheme with nonlinear beamsplitter is feasible to create stronger bipartite steering and genuine tripartite steering and has more flexibility to manipulate the monogamy relation through the cooperation of the two cascaded FWM processes.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Entanglement plays an important role in quantum information processing and quantum metrology [1–3]. It has been well studied since the nonlocality of entangled states was proposed by Einstein, Podolsky and Rosen (EPR) [4]. Schrödinger put forward the term of “steering” to describe the freaky nonlocal correlation presented in EPR paradox [5]. Reid introduced the experimental criterion to demonstrate EPR paradox based on Heisenberg uncertainty relation [6]. Wiseman et al. formalized the concept of steering in terms of the asymmetric local hidden state (LHS) model [7] and revealed that it is an intermediate type of quantum correlation between state inseparability [1] and Bell nonlocality [8,9]. Due to its inherent asymmetric property [10–14] and the ability to verify entanglement without the assumption of trustworthy on one side [7,8], EPR steering has been identified as useful resource for varied quantum cryptography applications [15–23].
Motivated by constructing quantum networks in real-world applications, EPR steering was extended to multipartite scenarios [24–31]. Continuous-variable (CV) multipartite quantum steering has been generated and distributed to multi-users by employing optical parametric oscillators (OPO) and linear optical beamsplitters (BSs). For instance, Armstrong et al. demonstrated up to eight modes multipartite steering and genuine tripartite entanglement generated by mixing two quadrature-squeezed qumodes and six quantum-noise limited vacuum modes via BSs in a linear optics circuit [32]; Deng et al. investigated the monogamy relations for EPR steering in CV four-mode square Gaussian cluster state [33]. Except for the optical CV systems, multipartite or multidimensional steering was also achieved in photonic qubits system [34–36].
Another powerful technique to generate multipartite entanglement is four-wave mixing (FWM) in a Rubidium (Rb) vapor. In a FWM process, two inputs can be mixed into two output signal and idler beams via a third-order nonlinear process, which is similar to a BS and thus called a nonlinear beamsplitter (NBS) [37–41]. Using single pass FWM based on a Rb vapor doesn’t require optical cavities or phase control, which makes it more efficient for generating a large scale of multipartite quantum entangled states [42]. Moreover, this type of entangled source with narrow bandwidth may be effectively stored in atomic media for quantum memory [43–45]. In 2007, P. D. Lett’s group successfully produced a pair of narrowband, squeezed light by nondegenerate FWM in a Rb vapor [46,47]. Using two cascaded FWM processes, Jing’s group demonstrated the method to produce multiple quantum correlated beams [48–52] and realization of nonlinear interferometer surpassing the standard quantum limit [40,41,53].
It is therefore interesting to analyze the performance of two schemes which contain a FWM combined with a linear BS and a NBS (second FWM) respectively in the generation of multipartite EPR steering. In this paper, we first analyze the eigenmodes of the achieved tripartite entangled states through diagonalizing their covariance matrices (CMs) and find that both schemes give two squeezed eigenmodes and a vacuum eigenmode, yet with different combinations of the output modes. By quantifying the achieved bipartite and tripartite EPR steering, we find that both schemes can generate one-way steering, collective steering, and genuine tripartite steering, but with NBS one can achieve stronger steerability. Moreover, we find that the monogamy relation in the scheme with a linear BS is not relevant to the gain of the first FWM process, however, with NBS it is determined by both two FWM processes. Our results suggests that the cascaded FWM processes is a good candidate to generate multipartite EPR steering that is useful for constructing a scalable quantum secure communication network.
2. The covariance matrix and eigenmodes of two schemes
Applying a single-pass FWM process, a signal beam is amplified and an idler beam is generated simultaneously with a strong pump beam, as illustrated in Fig. 1. The FWM1 process occurs by satisfying the energy conservation and phase-matching condition. In Heisenberg picture, the input-output relation of the FWM1 process is represented by ${{\hat a}_{s1}} = {G_1}{{\hat a}_{s0}} + {g_1}\hat a_{v0}^\dagger$ and ${{\hat a}_{i1}} = {g_1}\hat a_{s0}^\dagger + {G_1}{{\hat a}_{v0}}$, where $G_1$ is the amplitude gain in the FWM1 process; $G_1^2-g_1^2=1$; $\hat a_{s0}^\dagger ,\hat a_{s0}$ and $\hat a_{v0}^\dagger , \hat a_{v0}$ are the creation and annihilation operators of the seed and vacuum inputs, respecitively; $\hat a_{i1}$ and $\hat a_{s1}$ are the annihilation operators of the output idler and signal beams. The amplitude and phase quadrature operators are defined as $\hat X = \hat a + {{\hat a}^\dagger }$ and $\hat P = i({{\hat a}^\dagger } - \hat a)$, where the corresponding shot noise limit is unity.
Fig. 1. The schematic of the optical setups to generate the tripartite entangled states using a linear BS (a) and an NBS (b). $\hat a_{s0}$ and $\hat a_{v0}$ are the seed and vacuum inputs. $\hat a_{i1}$ and $\hat a_{s1}$ are the output idler and signal fields, and $\hat a_{i1}$ is labeled by $\hat A_{1}$ or $\hat B_{1}$ in two schemes. $\hat a_{v1}$ is the vacuum introduced by the second stage. In (a), the beam $\hat a_{s1}$ is split by a BS into $\hat a_{1,out}$ and $\hat a_{2,out}$ labeled by $\hat A_{2}$ and $\hat A_{3}$. In (b), the beam $\hat a_{s1}$ is also split by NBS into $\hat a_{i2}$ and $\hat a_{s2}$ labeled by $\hat B_2$ and $\hat B_{3}$. LO denotes the local oscillator for homodyne detection.
As shown in Fig. 1(a), the three-mode Gaussian state is generated by using a FWM process and a BS. Firstly two quantum correlated modes (${\hat a}_{i1}$ and ${\hat a}_{s1}$) are produced with the FWM1; then one of the two correlated modes (${\hat a}_{s1}$) is mixed with a vacuum mode (${\hat a}_{v1}$) at the BS with a variable reflectivity $R\in (0,1)$. Thereby, three output modes ${\hat a}_{i1}$, ${\hat a}_{1,out}$ and ${\hat a}_{2,out}$ (labeled by $\hat A_1$, $\hat A_2$ and $\hat A_3$) are created, whose quadratures could be locally measured by homodyne detection [54]. Applying the FWM1 process and BS operation to the input modes yields for the following output modes $\vec {X}^T=U_{X,BS}\vec {X}^T_{in}$ and $\vec {P}^T=U_{P,BS}\vec {P}^T_{in}$, where $\vec {X}^T=(\hat X_{i1},\hat X_{1,out},\hat X_{2,out})^T$, $\vec {X}^T_{in}=(\hat X_{s0},\hat X_{v0},\hat X_{v1})^T$, $\vec {P}^T=(\hat P_{i1},\hat P_{1,out},\hat P_{2,out})^T$, $\vec {P}^T_{in}=(\hat P_{s0},\hat P_{v0},\hat P_{v1})^T$, and the symplectic transform matrix $U_{X,BS}$, $U_{P,BS}$ are given as
By the Bloch Messiah decomposition, a multipartite Gaussian state can be decomposed into a set of uncorrelated squeezed vacuum states in an appropriately chosen mode basis of annihilation operators [56,57]. We obtain the eigenmodes and eigenvalues of the output modes through diagonalizing their CMs. The amplitude and phase quadrature share the same eigenmodes, but with inverse eigenvalues for pure states. Figures 2(a)-(c) show that the eigenmodes of the first scheme with the gain value $G_1= 2$ and a balanced BS with $R=0.5$. We find the following levels of squeezing for the $\hat X$ quadrature: $\{0$dB, $-11.44$dB, 11.44dB$\}$. The eigenmode 3 has the same level of squeezing as eigenmode 2 on the phase quadrature. For comparison, in Figs. 2(d)-(f) we make the second scheme using two FWM processes at the same total gains ($G_1=G_2=\sqrt 2$) and the same level of squeezing. In practice, we can achieve greater squeezing level by increasing the gain of the FWM2.
Fig. 2. Eigenmodes of the tripartite states, which are decomposed in the system output mode basis for two schemes with the same pump power. (a)(b)(c) The bars represent the relative weight of modes $\hat A_{1}$, $\hat A_{2}$, $\hat A_{3}$ for the case using a BS, respectively. (d)(e)(f) The bars represent the relative weight of modes $\hat B_1$, $\hat B_2$, $\hat B_{3}$ for the case applying an NBS, respectively.
Interestingly, by applying a BS or NBS, the constructed three-mode Gaussian states both give two squeezed eigenmodes and a vacuum eigenmode, yet with different combinations of the output modes. Also although the NBS scheme includes two FWM processes, the generated squeezing levels are the same when both schemes apply equal total pump power. This suggests that the system squeezing level mainly depends on pump energy rather than the number of FWM processes. In practice, the second pump can be recycled from the first FWM process without more consumption [48]. Also, the number of squeezers do not directly rely on the amount of FWM processes. Note that, the frequencies of two output modes after BS are same ($\hat {A_2}$ and $\hat {A_3}$ in Fig. 1(a)), but that would be nondegenerate when using NBS ($\hat {B_2}$ and $\hat {B_3}$ in Fig. 1(b)). And for both schemes, we find the vacuum eigenmodes are composed of two output modes with same frequency, such as $\hat {A_2}$ and $\hat {A_3}$ (Fig. 2(a)), or $\hat {B_1}$ and $\hat {B_2}$ (Fig. 2(d)).
3. The tripartite EPR steering
The properties of an $(n_\mathcal {A}+n_\mathcal {B})$-mode Gaussian state of a bipartite system can be fully determined by its CM $\gamma _{\mathcal {AB}}={{\ \mathcal {A}\ \ \mathcal {C}}\choose {\mathcal {C}^{\top }\ \mathcal {B}}}$, where submatrices $\mathcal {A}$ and $\mathcal {B}$ are the CMs correlated to the subsystems of Alice’s and Bob’s reduced states, respectively. The steerability of Bob by Alice’s Gaussian measurements ($\mathcal {A} \to \mathcal {B}$) can be quantified by the steering parameter $\mathcal {G}^{\mathcal {A} \to \mathcal {B}}=\max {\bigg \{}0, \underset {l:\bar { \nu }_{l}^{\mathcal {A}\mathcal {B}\backslash \mathcal {A}}<1}{-\sum }\ln (\bar {\nu }_{l}^{\mathcal {A}\mathcal {B}\backslash \mathcal {A}}) {\bigg \}}$ [26,33], where $\nu _{l}^{\mathcal {A}\mathcal {B}\backslash \mathcal {A}}(l=1,2,\ldots ,n_\mathcal {B})$ are the symplectic eigenvalues of Schur complement of $\mathcal {A}$.
Figure 3 shows the steerability distributed among three output beams $\hat A_1$, $\hat A_2$ and $\hat A_3$. This three-mode Gaussian state is prepared by two phase- and amplitude-squeezed states and one vacuum state injected in one Rb vapor with fixed gain $G_1=2$ and another linear BS with adjustable reflectivity $R$. Figure 3(a) illustrates the bipartite steering for any two modes [i.e. (1+1)-mode partitions] under Gaussian measurements, which is detailed in Figs. 3(b)-(d). First, mode $\hat A_1$ can be steered by mode $\hat A_2$ when $R < 0.5$ or by mode $\hat A_3$ when $R > 0.5$, as shown in Fig. 3(b). Second, modes $\hat A_2$ and $\hat A_3$ cannot be steered by each other but both of them can be steered by mode $\hat A_1$ for any value of $R$, as shown in Figs. 3(c) and (d). These results verify the monogamy constraint for Gaussian steering [58]: two distinct modes cannot steer the third mode simultaneously by Gaussian measurements, while no constraint for one mode to steer the other two modes. Furthermore, the one-way EPR steerability is demonstrated [12]. When the reflectivity $R\in [0.5,1)$, the steering parameter $\mathcal {G}^{\hat A_1 \to \hat A_2}>0$ shown in Fig. 3(c), while $\mathcal {G}^{\hat A_2 \to \hat A_1}=0$ shown in Fig. 3(b), which means that mode $\hat A_1$ can steer $\hat A_2$ but $\hat A_2$ cannot steer $\hat A_1$ by Gaussian measurements. Similarly, when the reflectivity $R\in (0,0.5]$, there is one-way steering between modes $\hat A_1$ and $\hat A_3$, as shown in Figs. 3(b) and (d). The steerability for (2+1)-mode partitions are also shown in Fig. 3. The blue dotted lines present any two modes can jointly steer the remaining one. Note that, a collective steering [25] in the direction $\hat A_2\hat A_3\to \hat A_1$ is created when $R=0.5$, where neither $\hat A_2$ nor $\hat A_3$ can individually steer $\hat A_1$, but collectively they can, as shown in Fig. 3(b). The joint steerability $\mathcal {G}^{\hat A_1,\hat A_3\rightarrow \hat A_2}$ is significantly higher $\mathcal {G}^{ \hat A_1 \rightarrow \hat A_2}$ ( $\mathcal {G}^{\hat A_1,\hat A_3\rightarrow \hat A_2}-\mathcal {G}^{ \hat A_1 \rightarrow \hat A_2}>0$ ) in Fig. 3(c), indicating that although the mode $\hat A_3$ cannot steer $\hat A_2$ by itself, its role in assisting joint steering with mode $\hat A_1$ is nontrivial.
Fig. 3. The EPR steering of the BS scheme. Fix the gain value of FWM1 ($G_1=2$), adjust the reflectivity R of the linear BS. (a) The mutual hierarchical relations of the bipartite steering, solid arrow, dotted arrow and solid with a cross represent deterministic, conditional and none steerability, respectively. (b) The mode $\hat A_1$ is steered by mode $\hat A_2$ (black solid), mode $\hat A_3$ (red dashed) separately, and by the collaboration of modes $\hat A_2$ and $\hat A_3$ (blue dotted). (c) The mode $\hat A_2$ is steered by $\hat A_1$ (black solid), $\hat A_3$ (red dashed) and both of them (blue dotted). (d) The mode $\hat A_3$ is steered by $\hat A_1$ (black solid), $\hat A_2$ (red dashed) and the joint of $\hat A_1$ and $\hat A_2$ (blue dotted).
By comparing with the tripartite system generated by a linear BS, we find that the scheme with NBS can significantly enhance the steerability by increasing the gain of NBS process. In this case, the three-mode Gaussian state is prepared by the same fixed gain $G_1=2$, and an adjustable NBS with $G_2$ from 1 to 5. As illustrated in Fig. 4(a), we find that the mode $\hat B_3$ can steer $\hat B_1$ and $\hat B_2$ deterministically (red dashed in Figs. 4(b) amd (c)), while steering in the opposite direction $\hat B_1\rightarrow \hat B_3$ and $\hat B_2 \rightarrow \hat B_3$ only happen when $G_2<1.32$ (black solid in Fig. 4(d)) and $G_2>1.32$ (red dashed in Fig. 4(d)), respectively. Thus, one-way EPR steering is achieved between $\hat B_1$ and $\hat B_3$ when $G_2>1.32$, and between $\hat B_2$ and $\hat B_3$ when $G_2<1.32$. Modes $\hat B_1$ and $\hat B_2$ cannot steer each other indicated by the black solid curve in Figs. 4(b) and (c). This is mainly due to that the frequency of beams $\hat B_1$ and $\hat B_2$ are the same and they are not coupled with each other in the Hamiltonian of whole system [59]. The joint steerability (by Gaussian measurements) of two modes to the third mode indicated by the blue dotted curves is always higher than the sum of the degrees of steerability exhibited by the individual pairs (black solid and red dashed curves). For instance, $\mathcal {G}^{\hat B_2,\hat B_3 \rightarrow \hat B_1}\geqslant \mathcal {G}^{ \hat B_2 \rightarrow \hat B_1}+\mathcal {G}^{ \hat B_3 \rightarrow \hat B_1}$ shown in Fig. 4(b), $\mathcal {G}^{\hat B_1,\hat B_3 \rightarrow \hat B_2}\geqslant \mathcal {G}^{ \hat B_1 \rightarrow \hat B_2}+\mathcal {G}^{ \hat B_3 \rightarrow \hat B_2}$ shown in Fig. 4(c), and $\mathcal {G}^{\hat B_1,\hat B_2 \rightarrow \hat B_3}\geqslant \mathcal {G}^{ \hat B_1 \rightarrow \hat B_3}+\mathcal {G}^{ \hat B_2 \rightarrow \hat B_3}$ shown in Fig. 4(d). This verifies the Coffman-Kundu-Wootters (CKW)-type monogamy relation, i.e., $\mathcal {G}^{ i, j\rightarrow k}-\mathcal {G}^{ i \rightarrow k}-\mathcal {G}^{j \rightarrow k} \geqslant 0$, which quantifies how the steerability is distributed among the output modes [28]. Moreover, there exist quantum states exhibiting collective steering when $G_2=1.32$ where $\mathcal {G}^{\hat B_1,\hat B_2 \rightarrow \hat B_3}>0$ but $\mathcal {G}^{ \hat B_1 \rightarrow \hat B_3}=\mathcal {G}^{ \hat B_2 \rightarrow \hat B_3}=0$.
Fig. 4. The EPR steering of the NBS scheme. Fix the gain value of FWM1 at $G_1=2$, and adjust $G_2$. (a) The mutual hierarchical relations of the bipartite steering, solid arrow, dotted arrow and solid with a cross represent deterministic, conditional and none steerability, respectively. (b) The mode $\hat B_1$ is steered by $\hat B_2$ (black solid), by $\hat B_3$ (red dashed) separately, and by their joint (blue dotted). (c) and (d) are same as (b) but for mode $\hat B_2$ and mode $\hat B_3$, respectively.
Moreover, it is clearly seen from Fig. 5 that the scheme with NBS provides more flexibility to manipulate the monogamy of Gaussian steering. In the scheme with linear BS shown in Fig. 5(a) the mode $\hat A_1$ is steered by $\hat A_2$ with the reflectivity $R\in (0,0.5)$, steered by $\hat A_3$ with the reflectivity $R\in (0.5, 1)$, and steered by neither of them at $R=0.5$. This is true for any value of $G_1$, i.e., the distribution of steering among three users is only adjusted by the reflectivity of beamsplitter. Increasing the value of $G_1$ enhances the steerability of the entangled state achieved by FWM1, but doesn’t change the monogamy relation. While in the scheme with NBS, both $G_1$ and $G_2$ affect how Gaussian steering are distributed between three parties. As shown in Fig. 5(b), mode $\hat B_1$ can steer $\hat B_3$ when $G_1>\frac {1}{{\sqrt {2 - G_2^2} }}$, mode $\hat B_2$ can steer $\hat B_3$ with a restricted condition for $G_1<\frac {1}{{\sqrt {2 - G_2^2} }}$ or $G_2\ge \sqrt {2}$, but neither of them can steer $\hat B_3$ when $G_1 =\frac {1}{{\sqrt {2 - G_2^2} }}$. This results from the role of FWM2 further creates entanglement between modes $\hat B_2$ and $\hat B_3$ except for simply distributing the entanglement generated from FWM1 to them.
Fig. 5. (a) The mode $\hat A_1$ is steered by $\hat A_2$ (left light green region) or by $\hat A_3$ (right light red region) with the reflectivity $R$. (b) The mode $\hat B_3$ is steered by $\hat B_1$ (left light green region) or by $\hat B_2$ (right light red region) varying with the gain values $G_1$ and $G_2$.
So far, we have studied the bipartite steering exhibited by the states generated by the linear and nonlinear schemes. It is interesting to examine whether the present schemes are useful to create genuine tripartite EPR steering, in which a steering nonlocality is necessarily shared by three parties altogether [24,28,60]. For pure three-mode Gaussian states, as demonstrated here, it has been shown that the residual Gaussian steering (RGS) $\mathcal {G}^{ i:j:k}=\min _{\langle i,j,k\rangle }\{\mathcal {G}^{ i,j\rightarrow k}-\mathcal {G}^{ i\rightarrow k}-\mathcal {G}^{ j\rightarrow k}\}$ emerging from the CKW monogamy inequality acts as a quantifier of genuine tripartite steering under Gaussian measurements [28]. The larger the RGS is, the stronger the genuine tripartite steering exists. Figure 6 presents the RGS measure $\mathcal {G}^{ \hat A_1: \hat A_2:\hat A_3}$ for Gaussian states generated by FWM with $G_1$ and linear BS with $R$, and $\mathcal {G}^{ \hat B_1: \hat B_2:\hat B_3}$ for Gaussian states generated by two cascaded FWMs with $G_1$ and $G_2$. Figure 6(a) reveals that for any fixed $G_1$ the RGS $\mathcal {G}^{ \hat A_1: \hat A_2:\hat A_3}$ is maximized when the beamsplitter with reflectivity $R=1/2$ to obtain bisymmetric states for modes $\hat A_2$ and $\hat A_3$, and increasing $G_1$ can further enhance the genuine tripartite steering as one might expect. Figure 6(b) shows that using two cascaded FWM can create stronger genuine tripartite steering, e.g., $\mathcal {G}^{ \hat B_1: \hat B_2:\hat B_3}\approx 0.76$ with experimentally feasible gain factors $G_1=G_2=\sqrt {2}$ [48], while $\mathcal {G}^{ \hat A_1: \hat A_2:\hat A_3}\approx 0.29$ maximized at $R=0.5$ with same level of gain $G_1=\sqrt {2}$. Our results pave the way to experimentally create stronger genuine multipartite steering from cascaded FWM processes, which is useful for implementing directional and secure quantum teleportation between more than two parties [18,61], as well as providing tight bounds on the guaranteed key rate of a partially device-independent quantum secret sharing protocol [29].
Fig. 6. Genuine tripartite steering examined by the residual Gaussian steering $\mathcal {G}^{ \hat A_1: \hat A_2:\hat A_3}$ (a) and $\mathcal {G}^{ \hat B_1: \hat B_2:\hat B_3}$ (b) as a function of $G_1$ and $R$, $G_1$ and $G_2$, respectively.
4. Conclusion
In summary, we investigate the steering properties of three-mode Gaussian states generated by two schemes where a FWM of Rb atoms is combined with a linear beamsplitter and a nonlinear beamsplitter (second FWM), respectively. We analyze the eigenmodes of the tripartite entangled states produced by two schemes through diagonalizing their covariance matrices and find that both give two squeezed eigenmodes and a vacuum eigenmode, yet with different combinations of the output modes. In addition, we examine the (one-way, two-way) bipartite and (collective, genuine) tripartite quantum steering properties exhibited in the generated entangled states and show that using NBS can significantly enhance the steerability. Moreover, our results also verify the monogamy relations for Gaussian steering and show that the nonlinear scheme has more flexibility to manipulate the monogamy relation. Therefore, the comparison of two schemes shows that the cascaded FWM processes become a promising candidate to generate various and stronger multipartite EPR steering for secure quantum communications.
Funding
National Natural Science Foundation of China (11604256, 11622428, 11804267, 11904279, 61605154, 61675007, 61975159); Key Technologies Research and Development Program of Guangzhou (2018B030329001); National Key R&D Program of China (2016YFA0301302, 2017YFA0303700, 2018YFA0307500, 2018YFB1107200); Natural Science Foundation of Jiangsu Province (BK20180322); National Postdoctoral Program for Innovative Talents (BX20180015); China Postdoctoral Science Foundation (2019M650291); Natural Science Foundation of Beijing Municipality (Z190005).
Disclosures
The authors declare no conflicts of interest.
References
1. R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Rev. Mod. Phys. 81(2), 865–942 (2009). [CrossRef]
2. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006). [CrossRef]
3. J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011). [CrossRef]
4. A. Einstein, B. Podolsky, and N. Rosen, “Can quantum-mechanical description of physical reality be considered complete?” Phys. Rev. 47(10), 777–780 (1935). [CrossRef]
5. E. Schrödinger, “Discussion of probability relations between separated systems,” Math. Proc. Cambridge Philos. Soc. 31(4), 555–563 (1935). [CrossRef]
6. M. D. Reid, “Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification,” Phys. Rev. A 40(2), 913–923 (1989). [CrossRef]
7. H. M. Wiseman, S. J. Jones, and A. C. Doherty, “Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. Lett. 98(14), 140402 (2007). [CrossRef]
8. E. G. Cavalcanti, S. J. Jones, H. M. Wiseman, and M. D. Reid, “Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox,” Phys. Rev. A 80(3), 032112 (2009). [CrossRef]
9. N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Rev. Mod. Phys. 86(2), 419–478 (2014). [CrossRef]
10. J. Bowles, T. Vértesi, M. T. Quintino, and N. Brunner, “One-way Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 112(20), 200402 (2014). [CrossRef]
11. Q. Y. He, Q. H. Gong, and M. D. Reid, “Classifying directional Gaussian entanglement, Einstein-Podolsky-Rosen steering, and discord,” Phys. Rev. Lett. 114(6), 060402 (2015). [CrossRef]
12. V. Händchen, T. Eberle, S. Steinlechner, A. Samblowski, T. Franz, R. F. Werner, and R. Schnabel, “Observation of one-way Einstein-Podolsky-Rosen steering,” Nat. Photonics 6(9), 596–599 (2012). [CrossRef]
13. S. Wollmann, N. Walk, A. J. Bennet, H. M. Wiseman, and G. J. Pryde, “Observation of genuine one-way Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116(16), 160403 (2016). [CrossRef]
14. K. Sun, X. J. Ye, J. S. Xu, X. Y. Xu, J. S. Tang, Y. C. Wu, J. L. Chen, C. F. Li, and G. C. Guo, “Experimental quantification of asymmetric Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 116(16), 160404 (2016). [CrossRef]
15. C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani, and H. M. Wiseman, “One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering,” Phys. Rev. A 85(1), 010301 (2012). [CrossRef]
16. N. Walk, S. Hosseini, J. Geng, O. Thearle, J. Y. Haw, S. Armstrong, S. M. Assad, J. Janousek, T. C. Ralph, T. Symul, H. M. Wiseman, and P. K. Lam, “Experimental demonstration of Gaussian protocols for one-sided device-independent quantum key distribution,” Optica 3(6), 634–642 (2016). [CrossRef]
17. M. D. Reid, “Signifying quantum benchmarks for qubit teleportation and secure quantum communication using Einstein-Podolsky-Rosen steering inequalities,” Phys. Rev. A 88(6), 062338 (2013). [CrossRef]
18. Q. Y. He, L. Rosales-Zárate, G. Adesso, and M. D. Reid, “Secure continuous variable teleportation and Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 115(18), 180502 (2015). [CrossRef]
19. C. Y. Chiu, N. Lambert, T. L. Liao, F. Nori, and C. M. Li, “No-cloning of quantum steering,” npj Quantum Inf. 2(1), 16020 (2016). [CrossRef]
20. M. Piani and J. Watrous, “Necessary and sufficient quantum information characterization of Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 114(6), 060404 (2015). [CrossRef]
21. D. Cavalcanti and P. Skrzypczyk, “Quantum steering: a review with focus on semidefinite programming,” Rep. Prog. Phys. 80(2), 024001 (2017). [CrossRef]
22. M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, “Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications,” Rev. Mod. Phys. 81(4), 1727–1751 (2009). [CrossRef]
23. R. Uola, A. C. S. Costa, H. C. Nguyen, and O. Gühne, “Quantum steering,” arXiv:1903.06663 (2019).
24. Q. Y. He and M. D. Reid, “Genuine multipartite Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 111(25), 250403 (2013). [CrossRef]
25. M. Wang, Q. Gong, and Q. He, “Collective multipartite Einstein-Podolsky-Rosen steering: more secure optical networks,” Opt. Lett. 39(23), 6703–6706 (2014). [CrossRef]
26. I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, “Quantification of Gaussian quantum steering,” Phys. Rev. Lett. 114(6), 060403 (2015). [CrossRef]
27. A. Riccardi, C. Macchiavello, and L. Maccone, “Multipartite steering inequalities based on entropic uncertainty relations,” Phys. Rev. A 97(5), 052307 (2018). [CrossRef]
28. Y. Xiang, I. Kogias, G. Adesso, and Q. He, “Multipartite Gaussian steering: Monogamy constraints and quantum cryptography applications,” Phys. Rev. A 95(1), 010101 (2017). [CrossRef]
29. I. Kogias, Y. Xiang, Q. He, and G. Adesso, “Unconditional security of entanglement-based continuous-variable quantum secret sharing,” Phys. Rev. A 95(1), 012315 (2017). [CrossRef]
30. J. L. Chen, X. J. Ye, C. Wu, H. Y. Su, A. Cabello, L. C. Kwek, and C. H. Oh, “All-versus-nothing proof of Einstein-Podolsky-Rosen steering,” Sci. Rep. 3(1), 2143 (2013). [CrossRef]
31. D. Cavalcanti, P. Skrzypczyk, G. H. Aguilar, R. V. Nery, P. H. Ribeiro, and S. P. Walborn, “Detection of entanglement in asymmetric quantum networks and multipartite quantum steering,” Nat. Commun. 6(1), 7941 (2015). [CrossRef]
32. S. Armstrong, W. Meng, R. Y. Teh, Q. Gong, Q. He, J. Janousek, H. A. Bachor, M. D. Reid, and K. L. Ping, “Multipartite Einstein-Podolsky-Rosen steering and genuine tripartite entanglement with optical networks,” Nat. Phys. 11(2), 167–172 (2015). [CrossRef]
33. X. Deng, Y. Xiang, C. Tian, G. Adesso, Q. He, Q. Gong, X. Su, C. Xie, and K. Peng, “Demonstration of monogamy relations for Einstein-Podolsky-Rosen steering in Gaussian cluster states,” Phys. Rev. Lett. 118(23), 230501 (2017). [CrossRef]
34. C.-M. Li, K. Chen, Y.-N. Chen, Q. Zhang, Y.-A. Chen, and J.-W. Pan, “Genuine high-order Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 115(1), 010402 (2015). [CrossRef]
35. Q. Zeng, B. Wang, P. Li, and X. Zhang, “Experimental high-dimensional Einstein-Podolsky-Rosen steering,” Phys. Rev. Lett. 120(3), 030401 (2018). [CrossRef]
36. J. Wang, S. Paesani, Y. Ding, R. Santagati, P. Skrzypczyk, A. Salavrakos, J. Tura, R. Augusiak, L. Mančinska, D. Bacco, D. Bonneau, J. W. Silverstone, Q. Gong, A. Acín, K. Rottwitt, L. K. Oxenløwe, J. L. O’Brien, A. Laing, and M. G. Thompson, “Multidimensional quantum entanglement with large-scale integrated optics,” Science 360(6386), 285–291 (2018). [CrossRef]
37. Y. Fang, J. Feng, L. Cao, Y. Wang, and J. Jing, “Experimental implementation of a nonlinear beamsplitter based on a phase-sensitive parametric amplifier,” Appl. Phys. Lett. 108(13), 131106 (2016). [CrossRef]
38. F. Hudelist, J. Kong, C. Liu, J. Jing, Z. Y. Ou, and W. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5(1), 3049 (2014). [CrossRef]
39. Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85(2), 023815 (2012). [CrossRef]
40. J. Jing, C. Liu, Z. Zhou, Z. Y. Ou, and W. Zhang, “Realization of a nonlinear interferometer with parametric amplifiers,” Appl. Phys. Lett. 99(1), 011110 (2011). [CrossRef]
41. H. Wang, A. M. Marino, and J. Jing, “Experimental implementation of phase locking in a nonlinear interferometer,” Appl. Phys. Lett. 107(12), 121106 (2015). [CrossRef]
42. Y. Cai, J. Feng, H. Wang, G. Ferrini, X. Xu, J. Jing, and N. Treps, “Quantum-network generation based on four-wave mixing,” Phys. Rev. A 91(1), 013843 (2015). [CrossRef]
43. D.-S. Ding, W. Zhang, Z.-Y. Zhou, S. Shi, B.-S. Shi, and G.-C. Guo, “Raman quantum memory of photonic polarized entanglement,” Nat. Photonics 9(5), 332–338 (2015). [CrossRef]
44. C. Shu, X. Guo, P. Chen, M. M. T. Loy, and S. Du, “Narrowband biphotons with polarization-frequency-coupled entanglement,” Phys. Rev. A 91(4), 043820 (2015). [CrossRef]
45. Y. Wang, J. Li, S. Zhang, K. Su, Y. Zhou, K. Liao, S. Du, H. Yan, and S.-L. Zhu, “Efficient quantum memory for single-photon polarization qubits,” Nat. Photonics 13(5), 346–351 (2019). [CrossRef]
46. A. M. Marino, R. C. Pooser, V. Boyer, and P. D. Lett, “Tunable delay of Einstein-Podolsky-Rosen entanglement,” Nature 457(7231), 859–862 (2009). [CrossRef]
47. C. F. Mccormick, V. Boyer, E. Arimondo, and P. D. Lett, “Strong relative intensity squeezing by four-wave mixing in rubidium vapor,” Opt. Lett. 32(2), 178–180 (2007). [CrossRef]
48. Z. Qin, L. Cao, H. Wang, A. M. Marino, W. Zhang, and J. Jing, “Experimental generation of multiple quantum correlated beams from hot rubidium vapor,” Phys. Rev. Lett. 113(2), 023602 (2014). [CrossRef]
49. J. Du, L. Cao, K. Zhang, and J. Jing, “Experimental observation of multi-spatial-mode quantum correlations in four-wave mixing with a conical pump and a conical probe,” Appl. Phys. Lett. 110(24), 241103 (2017). [CrossRef]
50. L. Wang, S. Lv, and J. Jing, “Quantum steering in cascaded four-wave mixing processes,” Opt. Express 25(15), 17457–17465 (2017). [CrossRef]
51. H. Wang, Z. Zheng, Y. Wang, and J. Jing, “Generation of tripartite entanglement from cascaded four-wave mixing processes,” Opt. Express 24(20), 23459–23470 (2016). [CrossRef]
52. S. Lv and J. Jing, “Generation of quadripartite entanglement from cascaded four-wave-mixing processes,” Phys. Rev. A 96(4), 043873 (2017). [CrossRef]
53. S. Liu, Y. Lou, J. Xin, and J. Jing, “Quantum enhancement of phase sensitivity for the bright-seeded SU(1,1) interferometer with direct intensity detection,” Phys. Rev. Appl. 10(6), 064046 (2018). [CrossRef]
54. P. van Loock and A. Furusawa, “Detecting genuine multipartite continuous-variable entanglement,” Phys. Rev. A 67(5), 052315 (2003). [CrossRef]
55. S. L. Braunstein and P. van Loock, “Quantum information with continuous variables,” Rev. Mod. Phys. 77(2), 513–577 (2005). [CrossRef]
56. S. L. Braunstein, “Squeezing as an irreducible resource,” Phys. Rev. A 71(5), 055801 (2005). [CrossRef]
57. Y. Cai, J. Roslund, G. Ferrini, F. Arzani, X. Xu, C. Fabre, and N. Treps, “Multimode entanglement in reconfigurable graph states using optical frequency combs,” Nat. Commun. 8(1), 15645 (2017). [CrossRef]
58. M. D. Reid, “Monogamy inequalities for the Einstein-Podolsky-Rosen paradox and quantum steering,” Phys. Rev. A 88(6), 062108 (2013). [CrossRef]
59. M. Jasperse, L. D. Turner, and R. E. Scholten, “Relative intensity squeezing by four-wave mixing with loss: an analytic model and experimental diagnostic,” Opt. Express 19(4), 3765–3774 (2011). [CrossRef]
60. R. Y. Teh and M. D. Reid, “Criteria for genuine $N$-partite continuous-variable entanglement and Einstein-Podolsky-Rosen steering,” Phys. Rev. A 90(6), 062337 (2014). [CrossRef]
61. Y. Yeo and W. K. Chua, “Teleportation and dense coding with genuine multipartite entanglement,” Phys. Rev. Lett. 96(6), 060502 (2006). [CrossRef]
