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Abstract
The quantum steering as an intermediate form of quantum correlations, has various peculiarities differing from the Bell nonlocality and the quantum entanglement. In a specific scenario, each side of the observer shares one of the two entangled qubits, steering can be witnessed simultaneously between any two observers from different sides by performing weak or projective measurements successively, with independently chosen and unbiased inputs. Here, we demonstrate this steering scenario experimentally to simultaneously witness quartic Einstein-Podolsky-Rosen steerings of two successive observers between each side with the entangled pair with a state fidelity of 97.6% compared with a maximally-entangled one.
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1. Introduction
The quantum correlation of spatially separated qubits is an essential property of the quantum mechanics which distinguishes quantum systems from any classical ones. The utilization of quantum correlation resource [1] plays a key role in both fundamental quantum physics and quantum information science, such as teleportations, secure key distributions, quantum computations etc. The hierarchical structure of quantum correlations are composed of three inequivalent concepts, quantum entanglement [2], Einstein-Podolsky-Rosen (EPR) steering [3] and Bell nonlocality [4]. To classify quantum correlations, several methods [5] such as the Bell-like inequalities and the quantum tomography are introduced. In an EPR steering scenario, the measurement devices of partial sides need to be fully characterized, which is also known as a semi-device-independent scenario [6].
Though many researches [7,8] have been done in the standard steering scenario, the implementation of the weak measurement by sequences of observables allows several multi-observer steering scenarios to become true. The weak value is first introduced [9] as the measured observable takes when the interaction between the quantum system and probe is sufficiently weak and a proper post-selected state is given to the quantum system after interaction, which is applied in signal amplifications [10], state tomographies, unbounded randomness certifications and quantum communication networks [11]. Measurements on quantum systems can be described with positive operator valued measurements (POVMs) with Kraus operators. By adjusting the strength of measurement from a projective one to no disturb on the system, arbitrary strength of measurement can be performed in two-level quantum systems. The strength parameter of measurement is the function of coupling parameter of the probe and measured quantum systems. In other words, the weak measurements we implemented are non-orthogonal dichotomic POVMs of all intermediate strengths focusing on the trade-off between the degree of disturbance and the amount of information we gain about the system. Pryde et al.[12] determined weak values for a single photon’s polarization and set the polarization of auxiliary photons as the probe. In our scheme, the polarization information of photons is first encoded into the path degree of freedom (DOF) and then set as the probe. After the weak interaction with the polarization DOF of the measured system, the measurement results are finally encoded into the path DOF for reading out. This process is realized by two beam displacers and half-wave plates and described as kraus operations in the form of $cos\theta \cdot (|H\rangle \langle H|)\pm sin\theta \cdot (|V\rangle \langle V|)$. Silva et al.[13] demonstrated that multiple Bobs performing sequential optimal weak measurements and one Alice share Bell nonlocality with an entangled source. This sequential weak measurements take large advantages in theoretical calculations and experimental realization when successive observers implement our non-orthogonal dichotomic POVMs compared with others [12]. To be specific, in a standard steering scenario, the measured subsystem collapses into one of the eigenstates of the projective measurements acquired by the steering inequalities, which forbids obtaining information about joint properties by successive observables of the subsystem. Compared with the projective measurements, the weak measurements can be less destructive and remain some original features of the initial system for the successive observers [14]. Several researches on the multi-observer scenario are introduced as follows. Schiavon et al. [15] showed the violations of twice Clauser-Horne-Shimony-Holt (CHSH) inequalities of two Bobs and one Alice with a pair of maximally-entangled photons. Choi et al. [16] experimentally demonstrated this violating by accomplishing a sequential steering for three Bobs with photonic systems. Foletto et al. [17] demonstrated that one Alice and a tree-like structure of infinite Bobs violate the CHSH inequalities with the appropriate weak measurements based on the history of previously performed measurements and observed outcomes. Brown et al. [18] showed that there is no limit on the number of Bobs that can violate the CHSH inequality with one Alice with the biased inputs and optimal weak measurements. Shenoy et al. [19] demonstrated that the number of Alices that can steer one Bob submits to $N_{Alice}\sim d/\log d$ when the measurements are unbiased.
In this letter, we demonstrate EPR steering between several observers in a maximally entangled two-photon system, where each qubit is accessible by multiple observers sequentially. The theory of the multi-observer EPR steering scenario using weak measurements and linear steering inequalities are presented as the background in section II. The experimental realization of the steering protocol for two Alices and two Bobs and the detailed discuss on experimental results of steering inequalities are listed in section III. We summarize this letter and discuss its potential applications in section IV.
2. Multi-observer steering via sequential weak measurement
In our protocol represented in Fig. 1, the maximally entangled two subsystems $A$ and $B$, are sequentially accessible by two observers, $Alice_{1}, Alice_{2}$ for subsystem $A$, $Bob_{1}, Bob_{2}$ for subsystem $B$, respectively. In subsystem $A$, one of the entangled subsystems is sent to $Alice_{1}$ firstly. Then, $Alice_{1}$ randomly selects its input $x_{1}\in \{1,2,\ldots,n\}$ to perform her weak measurement $A_{1}(x_{1},\eta _{A_{1}})$ with the sharpness parameter $\eta _{A_{1}}\in [0,1]$. $Alice_{1}$ obtains the outcome $a^{A_1}_{x_1}\in \{-1,+1\}$ and delivers the post-measurement state to $Alice_{2}$, who has no knowledge of the measurement choice $x_{1}$ and the outcome. Furthermore, with random choice of $x_{2}\in \{1,2,\ldots,n\}$, $Alice_{2}$ performs her measurement $A_{2}(x_{2},\eta _{A_{2}})$ and turns out $a^{A_2}_{x_2}\in \{-1,+1\}$. In the two-Alice-two-Bob steering scenario, the measurements performed by $Alice_{2}$ are projective ones, with $\eta _{A_{2}}=1$, for maximally extracting information from the subsystem. The process is similar on the other side in subsystem $B$. The black and white boxes in Fig. 1 denote that each of the four observers tries to steer the other side observers with their untrusted apparatus.
Fig. 1. The scenario of the two-Alice-two-Bob quantum steering network where each observer tries to steer ones on the other side. Two subsystems $A$ and $B$ share the state $\rho$. $Alice_1$ and $Bob_1$ implement weak measurements and the sharpness parameters of their measurements are $\eta _{A_1}$, $\eta _{B_1}\in [0,1]$. $Alice_2$ and $Bob_2$ implement projective measurements with $\eta _{A_2}=\eta _{B_2}=1$.
To verify the steerability, the linear steering inequality is introduced by Saunders et al. [20]. In a standard steering scenario, Alice broadcasts her result $a_{k}\in \{-1,+1\}$, $k\in \{1,2,\ldots,n\}$ of the corresponding measurement, and Bob makes the corresponding dichotomic ${B}_{k}$ from the total $n$ setting of his measurements. The linear steering inequality reads
$C_n$ is the upper bound of the inequality allowed by the LHS model andMoreover, we bring the linear steering inequality into the sequential multi-observer steering scenario. Two qubits prepared in $\rho ^{i'j'}$ are delivered to pairs of observers, $A$ to $Alice_{i}$ and $B$ to $Bob_{j}$, for $i'=i-1$ and $j'=j-1$. In this case, the initial state is denoted by $\rho ^{00}$ while the post-$Alice_1$-measurement state held by $Alice_2$ and $Bob_1$ is marked as $\rho ^{10}$. With the weak measurements, the observables represented by Kraus operators can be defined as:
Here, $k$ is $k$-th measurement, $\pm$ are the outcomes of dichotomic measurements, $I$ means the identity matrix, $\sigma$ represents three Pauli matrixes $(\sigma _{x}, \sigma _{y}, \sigma _{z})$, $\eta = cos2\theta$ is the sharpness parameter of observables. $m_{k}$ is a unit vector on the Bloch sphere representing the measurement direction. Then, the observable can be written as [19]:
whereTo witness the quartic steering, four inequalities should be violated. When four observers choose the same number of measurement settings, the right hand side of four inequalities share the same $C_n$. In contrast, the left hand side four inequalities are respectively listed as $S^{11}_n$, $S^{12}_n$, $S^{21}_n$, $S^{22}_n$ corresponding to different Alice or Bob. $Alice_{i}$ performs the Kraus operators $\{K_{+|k}^{\eta _{A}},K_{-|k}^{\eta _{A}} \}$ with the random input $k$ and gets an output $a\in \{+,-\}$, while $Bob_{j}$ performs the Kraus operators $\{K_{+|l}^{\eta _{B}},K_{-|l}^{\eta _{B}} \}$ with random input $l$ and gets an output $b\in \{+,-\}$. The unnormalized conditional post-measurement two-qubit state would be expressed as
As a result, $Alice_{i}$’s and $Bob_{j}$’s average post-measurement state is
which is delivered to the next observers for further investigation. If $S^{ij}_{n} > C_{n}$, the bidirectional steerability can be declared between $Alice_{i}$ and $Bob_{j}$ by measuring the state $\rho ^{i'j'}$, for $i'=i-1$ and $j'=j-1$.3. Experimental realization and result of sequential multi-observer steering
The schematic of our experimental setup is shown in Fig. 2, which mainly consists of five parts, one is a Bell state preparation and the others are four observers. The initial state is prepared in $\rho ^{00}=|\Psi ^{-}\rangle \langle \Psi ^{-}|$, where $|\Psi ^{-}\rangle =1/\sqrt 2(|HV\rangle -|VH\rangle )$. At the beginning, the $390$ nm laser is pumping on a sandwich-type BBO crystal and generates the photon-pairs in the maximal entanglement via a spontaneous parametric down-conversion (SPDC) process. Here the sandwich-type BBO crystal consists of two pieces of beamlike-phase-matching type-II BBO crystals and an intermediate half-wave plate (HWP) at $780$ nm. A pair of $LiNbO_3$ crystals are used for spatial compensations and a pair of YVO$_4$ crystals for temporal compensations. Accordingly, the down-conversion photon pairs from the sandwich-type BBO crystal are indistinguishable, and the polarization-maximally-entangled state $\left | {{\Psi ^ - }} \right \rangle$ is generated. Here we employ the quantum tomography to achieve a state fidelity of $98.3\%$.
Fig. 2. The experimental realization of a multi-observer steering scenario. The setup consists of state preparation, two observers ($Alice_{1}$ and $Bob_{1}$) performing weak measurements and two observers ($Alice_{2}$ and $Bob_{2}$) performing projective measurements. C-BBO: sandwich-type BBO+HWP+BBO combination; QWP: quarter-wave plate; HWP: half-wave plate; PBS: polarzing beam-splitter; BD: beam displacer; IF3: interference filter with a full width at half maximum (FWHM) of $3$ nm at $780$ nm. SPD: single-photon detector
Following the state preparation module, the pairs of EPR photons are separately distributed to $Alice_{1}$ and $Bob_{1}$, each employing two beam displacers (BDs) forming interferometers to perform weak measurements in the form of $cos\theta \cdot (|H\rangle \langle H|)\pm sin\theta \cdot (|V\rangle \langle V|)$. Two pairs of the half-wave plates (HWPs) are placed on both sides of the fibers to preserve the polarizations of photons. Behind the polarization-maintaining HWP, a suit of the quarter-wave plate (QWP) and the HWP is set to realize the measurements when $n=6$. Half of the photons are blocked by two black cubes behind the second BDs, which means only one direction of the measurements is detected. The opposite directions are measured in another 6 times by rotating the suit of QWP and HWP. Between the two BDs forming a interferometer, there are two HWPs cut along the slow axis (set at $0^{\circ }$ for $|H\rangle$ path and $45^{\circ }$ for $|V\rangle$ path of the first BD output) and one intermediate HWP forming an angle $\theta$ with its fast axis. These apparatus realize the weak measurement in Eq. (3) when the sharpness parameter $\eta =cos2\theta$. By rotating the middle uncut HWP to vary $\theta \in [0^{\circ },22.5^{\circ }]$ in the BD interferometer, the weak measurement with arbitrary sharpness parameter $\eta$ can be performed by $Alice_{1}$ and $Bob_{1}$. To be specific, when $\theta =0^{\circ }$, the weak measurement is reduced to a projective one, while $\theta =22.5^{\circ }$, the weak measurement is equal to a unitary operator $-$ no measurement is performed on the system. For $\theta _{A}$ and $\theta _{B}$ denoting the corresponding axis of the HWPs of $Alice_{1}$ and $Bob_{1}$, we employ $\theta = \theta _{A} =\theta _{B}$ to achieve $\eta =\eta _{A}=\eta _{B}$ for a symmetric structure at each sample. Finally, the post-measurement photons are delivered to corresponding observers $Alice_{2}$ and $Bob_{2}$, implementing projective measurements realized by the polarization analysis system which consists of a piece of QWP, HWP, polarzing beam-splitter (PBS) and a single-photon detector (SPD). Coincidence counts within 5 seconds from two corresponding SPDs are recorded to calculate the joint probability distributions of the observers with different measurements.
It is worth mentioning that implementing 6 sets of measurements is the most suitable strategy in the scheme. Firstly, more sets of measurements lead to a longer experimental duration. Suffering from the loss of measurements from negative direction by $Alice_{1}$ and $Bob_{1}$, we should measure 864 times for each point (about 1.2 hour for a point in Fig. 3), while only 108 times of measurements are needed for $n=3$. Secondly, since the ideal source can also witness quartic violations of the steering inequalities around $LHS = 0.582$ for $n=3,4$, it is difficult to realize because of the imperfection of entangled sources and the environment disturbance within the long experimental duration. Thirdly, adding the set of measurements $n$ only results in the decrease of value of $C_{n}$, while it remains the LHS $S^{ij}_{n}$ unchanged.
Fig. 3. (a) Experimental points with error bars and theoretical predicted curves of left-hand side of linear steering inequality $S^{ij}$ with imperfect entangled source varying with the sharpness parameter $\eta$, while $\eta =\eta _A=\eta _B$. The blue solid curve represents $S^{11}$ ($Alice_1 \& Bob_1$). The green solid curve represents $S^{12}/S^{21}$ ($Alice_1\&Bob_2$/ $Alice_2\&Bob_1$). The red solid curve represents $S^{22}$ ($Alice_2\&Bob_2$). The dash lines denote right-hand side of the inequality $C_{n}$, while the colors $-$ black, red, purple and blue $-$ refer to the sets $n$ of measurements when $n = 2,$ $n = 3, 4,$ $n = 6$ and $n = 10$, respectively. We measure the corresponding LHS points for six times, when $\eta = 0, 0.588, 0.766, 0.788, 0.951, 1$. (b) A zoom-in picture when $\eta \in (0.75, 0.8)$ is also demonstrated here for a sweeping view of quartic violation of the steering inequalities.
Considering that both $Alice_{1}$ and $Bob_{1}$ perform six sets of independent measurements with the same sharpness parameter $\eta =\eta _{A}=\eta _{B}$, the theoretical predictions of EPR steering correlation $S^{ij}$ ($S^{ij}_{n=6}$) and corresponding experimental sampling points are showed in Fig. 3. The left-hand side (LHS) of the linear steering inequality varies with the sharpness parameter $\eta \in [0,1]$. Due to the symmetry between $Alice$s and $Bob$s, it is found that $S^{12}=S^{21}$. The steerabilities for the observer-pair ($Alice_{1}, Bob_{1}$) and ($Alice_{2}, Bob_{2}$) is a trade-off with $\eta$. For ($Alice_{1}, Bob_{2}$) and ($Alice_{2}, Bob_{1}$), their steerabilities reach the peak around 0.56826 with $\eta \sim 0.81$ . Around $\eta \sim 0.76$, it is found that all the four $S^{ij}$ curves can violate the linear steering inequalities for $n=6$. That means, every observer convinces others in the remote subsystem that their photons can be steered to the corresponding state without any local hidden state model. To be specific, points in Fig. 3 (b) measured with $\eta = 0.766, 0.788$ achieve the left-hand-side inequalities value $\{S^{11},S^{12},S^{21},S^{22}\}$ at $\{0.55481,0.54416,0.5378,0.53096\}$ and $\{0.58282,0.55052,0.54034,0.51196\}$. The error bars calculated via the Monte Carlo method are below 0.00276 in general. Although matched well with the theoretical curves, the LHS values when $\eta = 0.766$ succeed in violating the $C_{10}$ bound but partly fail to violate the $C_6$ bound because of the imperfections of the BD interferometers. Both of the imperfections of the two BD interferometers result in the worst performance of $S_{22}$ when $\eta =0$, which means $Alice_{1}, Bob_{1}$ implement imperfect unitary operators and $Alice_{2}, Bob_{2}$ implement projective measurements.
4. Conclusion
4 In conclusion, a type of quantum steering scenario with a pair of maximally entangled photons is presented in this article, while each of the subsystems is accessible by two successive observers trying to steer two observers on the other side. It is worth mentioning that there are some difference between Zhu et al. [23] and our theoretical part of this article [24]. We demonstrate that every single one of the four observers have the ability to simultaneously steer two observers on the other side when $n = 6$, which is shown in Fig. 3 with experimental results, while Zhu et al. show the steering between two intermediate observers and between two terminal ones when $n = 6$ and steering only between two terminal ones when $n = 10$. And $n$ denotes the number of measurement settings of the linear inequalities. To be specific, we demonstrate not only double EPR steerings ($Alice_{1}-Bob_{1}$, $Alice_{2}-Bob_{2}$), but quartic EPR steerings with experimental verifications ($Alice_{1}-Bob_{2}$, $Alice_{2}-Bob_{1}$ added denoted by green curve in Fig. 3). The steering task is appropriate for the coming quantum network because of the untrusted messengers and the trusted local device. To the best of my knowledge, the sequential multi-observer steering may be applied to a scenario where two customers ($Alice_2$ and $Bob_2$ from different quantum network operators ($Alice_1$ and $Bob_1$) utilize quantum correlation resource. In this usage scenario, operators only implement enough weak measurements for necessary connections to avoid the waste of correlation resources. Witnessing quartic EPR steerings simultaneously plays an important role of testing the stability of whole network system and eliminating the possible fifth observer in this network. Furthermore, it is found that the sequential multi-party quantum random access code can witness the quantum channel [21], verify the unsharp measurement [25,26], and be generalized to characterise the correlations of quantum network in prepare-and-measurement scenarios [27].
Funding
National Key Research and Development Program of China (2017YFA0304100, 2018YFA0306400); National Natural Science Foundation of China (12074194, U19A2075); the Leading-edge technology Program of Natural Science Foundation of Jiangsu Province (BK20192001); Industrial Prospect and Key Core Technology Projects of Jiangsu provincial key R & D Program (BE2022071); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX19_0901).
Acknowledgments
We gratefully acknowledge the financial support from National Key R $\&$ D Program of China (2018YFA0306400, 2017YFA0304100), National Natural Science Foundation of China (12074194, U19A2075), Leading-edge technology Program of Jiangsu Natural Science Foundation (BK20192001), Industrial Prospect and Key Core Technology Projects of Jiangsu provincial key R $\&$ D Program(BE2022071), and Postgraduate Research $\&$ Practice Innovation Program of Jiangsu Province (KYCX19$\_$0901).
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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