## Abstract

Einstein-Podolsky-Rosen steering is an intermediate relationship between entanglement and Bell nonlocality in the hierarchical structure of quantum nonlocality. To certify the steerability of the entangled state, Mermin steering inequality is supposed to be violated by exceeding the inequality bound of 2. We present an experimental generation of post-selected three-photon entangled states and witness a maximal violation of the inequality up to 3.50±0.05. In the context of observing the maximal violation of Mermin steering inequality which requires measuring on the GHZ state, we derive a tight lower bound on the GHZ-fidelity that can be certified from the Mermin steering inequality violation. From this bound, it follows that the violation of Mermin steering inequality by 3.5 certifies the GHZ-fidelity of 78.66% at least. On the other hand, the above maximal violation of Mermin steering inequality observed in our experimental setup is produced by a post-selected entangled state having the GHZ-fidelity of 87.25 ± 0.34% through quantum tomography.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Quantum correlation among multipartite systems is important not only for fundamental quantum physics but also for quantum information processing, such as quantum computation [1] and quantum communication [2]. It is now clear that quantum entanglement, Einstein-Podolsky-Rosen (EPR) steering [3] and Bell nonlocality [4] forms a hierarchical structure of quantum nonlocality with the correlation strength increases. As the intermediate case, EPR steering in multipartite scenario is formalized as a quantum information task that Alice can convince others that she is able to steer their particles through her untrusted measurement device and corresponding outcomes. The steerability relies on that there exists no local hidden state model probably found by those know their own fully-characterized states and outcomes from Alice [3,5]. In addition, the intrinsic characteristic of asymmetry of EPR steering makes it an essential resource for semi-device-independent quantum information processing, which would be crucial to asymmetric quantum networks [6].

It was not until Bell inequality [7] derived from local hidden variable (LHV) theories that Bell nonlocality could be quantified experimentally. Following this breakthrough work, many efforts were made, such as performing tomography [8], constructing entanglement witness [9] and verifying the violation of various Bell-type inequalities [10–12]. Among these methods, observing the violation of Mermin inequality [13] is widely used in multipartite systems [14], for example, the tripartite Greenberger-Horne-Zeilinger (GHZ) state [15], etc. To be specific, we can certify genuine *n*-partite entanglement by observing the violation of Mermin inequality bound of 2^{n/2} for *n* even and 2^{(n−1)/2} for *n* odd. In addition, quantum mechanics allows an upper bound of Mermin inequality at 2^{n−1}.

As a counterpart, several criteria have been derived to detect EPR steering [16]. For instance, the violation of steering inequalities based on moments matrix [17], all-versus-nothing proofs [18], Heisenberg uncertainty principle [19,20], entropy uncertainty principle [21,22], fine-grained uncertainty principle [23], and geometric Bell-like inequalities [24] leads to proof of existence of EPR steering. It has been demonstrated that the violation of tripartite Mermin inequality detects tripartite EPR steering in a two-sided device-independent way [25, 26] i.e., in the presence of two untrusted parties and the trusted party performing measurements in qubit mutually unbiased bases.

The genuine demonstration of EPR steering is still challenging even with the help of criteria above because of the detection loophole. Suffering from the low detection efficiency, all the events where at least one of the particles is not detected are rejected, which means the fair-sampling assumption is made in these cases. Take the Mermin steering inequality as an example, the minimum detection efficiency required for a loophole-free test is 75 % on each side of the tripartite system [27]. Unfortunately, the efficiencies of our single-photon detectors are around 63 % and the overall detector efficiency (defined as the ratio of detected to produced particles) of our setup is 18.2 % with 3 nm interference filter, which is far from closing the detect loophole. As a result, the violation of Mermin steering inequality in our experiment actually proves only the steerability of post-selected GHZ state, not the genuine demonstration of EPR steering.

In this paper, we present an experimental investigation of Mermin steering inequality on tripartite GHZ states. This article is organized as follows. Mermin steering inequality is theoretical derived in [25], and for completeness we introduce it briefly in Sec. II and discuss our motivations in this section. Tight lower bound on the GHZ-fidelity which can be certified by Mermin steering inequality is also derived in this section. We demonstrate the experimental setup in Sec. III, composed with two independent SPDC bipartite entangled sources. In Sec. IV, we describe the experimental process in detail and present analysis on the corresponding experimental results. Finally, we conclude with a summary and outlooks in Sec. V.

## 2. Motivation

Consider a tripartite scenario in which Alice, Bob and Charlie are three spatially separated parties. The three observers can perform some local measurements on their subsystems. In particular, each of the observer choose independently between two different dichotomic observables by using a single random bit (respectively denoted by *x*, *y* and *z*) taking values in 0 or 1. Thus the binary outcomes of each observer mark as *a*, *b*and *c* taking values in +1 or −1. The conditional probability distribution *P*(*abc*|*A _{x}B_{y}C_{z}*) describes the probability of obtaining outcomes

*a*,

*b*and

*c*when the three observers perform measurements

*A*,

_{x}*B*and

_{y}*C*, respectively. Hence, Mermin inequality [13] is in the form of

_{z}Though the violation of Mermin inequality does not certify genuine nonlocality, it can be used to certify genuine entanglement in a fully device-independent way whenever its violation is more than $2\sqrt{2}$ [28]. Furthermore, it has been shown that the maximal violation of Mermin inequality provides robust self-testing of the GHZ state [29], i.e., the maximal or near maximal violation of the Mermin inequality certifies the GHZ state up to local isometries. In [30], the author has provided an analytical tight lower bound on the GHZ-fidelity (fidelity of the state used for violating the Mermin inequality compared with the maximally entangled tripartite GHZ state) that can be certified by the Mermin inequality violation of more than $2\sqrt{2}$. When the violation of the Mermin inequality more than $2\sqrt{2}$ is observed by using a tripartite state *ρ _{ABC}* in ℂ

*⊗ ℂ*

^{d}*⊗ ℂ*

^{d}*, our GHZ-fidelity is defined as follows:*

^{d}*, Λ*

_{A}*and Λ*

_{B}*, (completely positive trace-preserving maps) of appropriate input and output registers. It has been demonstrated that for a given violation of Mermin inequality*

_{C}*γ*of more than $2\sqrt{2}$, the GHZ-fidelity satisfies the following tight bound [30] Thus, for the violation of Mermin inequality more than $2\sqrt{2}$, one can certify a nontrivial GHZ-fidelity i.e., GHZ-fidelity of more than 50 % and the certified fidelity can be quantified by using the above lower bound.

Multipartite steering inequalities have been derived to witness multipartite entanglement in asymmetric networks in which some of the parties perform untrusted measurements while the other parties perform the trusted measurements [6, 25]. It has been demonstrated that the violation Mermin inequality can be used to witness tripartite steering in the context of a two-sided device-independent scenario. In this specific tripartite steering scenario, Alice and Bob claim that they can steer the state of the subsystem on Charlie’s side through performing uncharacterized local measurements on their respective subsystems. Hence there exists a LHV-LHS model when Alice and Bob fail to steer:

*λ*with probability distribution

*p*, ${\sum}_{\lambda}{p}_{\lambda}=1$. $P\left(c|{C}_{z},{\rho}_{C}^{\lambda}\right)$ are the distributions arising from local hidden states ${\rho}_{C}^{\lambda}$. In this two-side-device-independent (2SDI) scenario where the untrusted parties’ Alice and Bob try to steer Charlie, there exists a steering inequality [25,26]

_{λ}*AB*|

*C*. Interestingly, the violation of Mermin steering inequality given by Eq. (5) by using a three-qubit state requires the presence of genuine tripartite entanglement [26]. Thus, the Mermin steering inequality can be used to witness tripartite steering in the simplest way.

In [31], robust two-sided device-independent self-testing of the GHZ state has been demonstrated theoretically by using the maximal violation of Mermin steering inequality. We now derive tight analytical self-testing bound, i.e., tight lower bound on the GHZ-fidelity that can be certified by the violation of Mermin steering inequality given by Eq. (5). In the context of our above two-sided device-independent scenario where Alice and Bob perform two black-box dichotomic measurements and Charlie performs measurements in two mutually unbiased bases of qubit dimension, our GHZ-fidelity is defined as follows:

*ρ*is a tripartite state in ℂ

_{ABC}*⊗ ℂ*

^{d}*⊗ ℂ*

^{d}^{2}which is the state that gives rise to the observed violation of Mermin steering inequality. We now demonstrate that the above GHZ-fidelity satisfies the following tight lower bound [32].

**Proposition 1** *In the context of the violation of Mermin steering inequality, a nontrivial GHZ-fidelity is certifiable if and only if the violation* $\gamma >1/\sqrt{2}$ *and the certifiable GHZ-fidelity can be quantified by the following tight bound*

*for the violation*$\gamma \in \left[2\sqrt{2},4\right]$.

**Proof 1** *In [26], it has been demonstrated that there exists a family of tripartite correlations giving rising to the violation of Mermin steering inequality up to* $2\sqrt{2}$ *which can be reproduced by a biseparable state in the context of our two-sided device-independent scenario. On the other hand, genuine tripartite entanglement is necessary for the violation of Mermin steering inequality beyond* $2\sqrt{2}$*. Therefore, a nontrivial GHZ-fidelity is certifiable from the violation of Mermin steering inequality if and only if the violation* $\gamma >1/\sqrt{2}$.

*Let us derive the bound on the GHZ-fidelity certifiable from the violation of Mermin steering inequality beyond* $2\sqrt{2}$*. For this purpose, we note that the violation of Mermin steering inequality up to* $2\sqrt{2}$ *can be achieved by the state* |*ν*〉* _{ABC}* = |Φ〉

*|0〉*

_{AB}*|Φ〉*

_{C}(where*$2\sqrt{2}$*

_{AB}is the singlet state up to local unitaries). This implies that the violation between*and*4

*can be realized by a family of tripartite correlations which are a convex mixture of the two correlations arising from*|

*ν*〉

*and ${|\mathrm{\Upsilon}\u3009}_{\mathit{ABC}}=\left(|000\u3009+|111\u3009\right)/\sqrt{2}$*

_{ABC}*in the context of our two-sided device-independent scenario. In [30], it was shown that the above realization of violation of Mermin inequality between*$2\sqrt{2}$

*and*4

*implies the tight bound on the GHZ-fidelity given by Eq. (3) that can be certified in a fully device-independent way. From this, it follows that in the context of our two-sided device-independent scenario as well, this tight bound on the GHZ-fidelity holds.*

## 3. Experimental setup

We implement an experimental demonstration of the Mermin steering inequality for a class of three-photon state shown in Fig. 1, i.e., the GHZ states

*H*〉 and |

*V*〉 represent horizontal and vertical polarizations, which are two directions of

*z*axis in the Bloch sphere. The state source can be divided into three parts, two independent spontaneous parametric down-conversion (SPDC) sources. Firstly, generated by a mode-locked Ti:sapphire laser, a beam of light with a duration of less than 100 fs, a repetition rate of 80 MHz, and a central wavelength of 780 nm passes through a frequency doubler. Then, the ultraviolet pulsed light from the doubler pumps two SPDC sources in sequence.

In the former SPDC source, two pieces of 1-mm beamlike-phase-matching type-II BBO crystals (BBO1 and BBO2), with one true-zero-order half-wave plate (THWP) inserted between them, form a sandwich-like structure [33]. These sandwich-like crystals are used for photon pair generation through the SPDC process. In this process, |*H*_{e,1}〉 denotes the extraordinary down-converted photons in the horizontal polarization from BBO1 and |*V*_{o,2}〉 denotes the ordinary down-converted photons in the vertical polarization from BBO2. The role of the inserted THWP is to rotate the down-converted photons |*H*_{e,1}〉 |*V*_{o,1}〉 from BBO1 to its orthogonal polarization |*V*_{e,1}〉 |*H*_{o,1}〉 while keeping the pump light polarization state nearly unchanged. Naturally, |*V*_{e,1}〉 (|*H*_{o,1}〉) and |*H*_{e,2}〉 (|*V*_{o,2}〉) remain spatial and temporal differences, which are compensated by a pair of LiNbO_{3} crystals and a pair of YVO_{4} crystals respectively. As a result, this pair of photons become indistinguishable with any information except polarization and is projected into the Bell state $\left(1/\sqrt{2}\right)\left(|HH\u3009+|VV\u3009\right)$. We perform quantum state tomography on the generated state and the fidelity is up to 98.45 ± 0.24 %. The only difference between the latter SPDC source and the former one is that the photons in mode *D* is projected in horizontal polarization, which heralds the photons in mode *C′* with the same polarization.

To produce the maximally entangled GHZ state, photons in mode *A′* are transmitted directly to the PBS, yet photons in mode *C′* are rotated from |*H*〉 to |*D*〉 through the HWP1 formed an angle of 22.5° with its fast axis. They finally enter opposite ports of the PBS1 through two fibers. However, resulting mainly from the distance between two SPDC sources, photons pumped by the same pulse in mode *A′* and *C′* exist temporal difference. The PBS1 transmits horizontally-polarized component of photons in mode *C′*, and reflects vertically-polarized component of photons in mode *A′*. These photons emit through mode *C* without spatial difference. After the HWP before PBS2 rotated by 22.5°, HOM-like interference could be observed by APD1 and APD2 in mode *C* (triggering by APDs in mode *B*and *D*).

In other words, HOM-like interference with a visibility of 94.9 ± 2.2 % based on four-hold coincidences (observed by APD1, APD2 and APDs in mode *B*and *D*) ensures that it is almost impossible to distinguish two SPDC sources with any other information except polarization. With this indistinguishability, the source is projected into the GHZ state observed by APD1 and APDs in mode *B*, *C* and *D*. Finally, by rotating the angle of HWP1, the relative amplitudes of |*HHH*〉 and |*VVV*〉 can be tuned, and we get the GHZ-like state in form of

*θ*is the angle of HWP1 forming with its fast axis. The density matrix of the maximally entangled GHZ state we prepared is shown in Fig. 2.

## 4. Experimental result

Specifically, when the measurements are preformed in the form of *σ _{x}* and

*σ*, the Mermin steering inequality (5) with

_{y}*A*

_{0}= −

*σ*,

_{y}*A*

_{1}=

*σ*,

_{x}*B*

_{0}= −

*σ*,

_{y}*B*

_{1}=

*σ*,

_{x}*C*

_{0}=

*σ*and

_{x}*C*

_{1}=

*σ*can be rewritten as

_{y}*AB*→

*C*indicates that each expectation in the above sum of joint observables is measured when Charlie performs measurements on the conditional states prepared by Alice and Bob’s local measurements. To be specific, for

*σ*and

_{x}*σ*measurements, when we measure |

_{y}*D*〉 or |

*L*〉, the outcome is +1. When we measure |

*A*〉 or |

*R*〉, the outcome is −1. Here $|D\u3009=\left(|H\u3009+|V\u3009\right)/\sqrt{2}$ and $|A\u3009=\left(|H\u3009-|V\u3009\right)/\sqrt{2}$, while $|L\u3009=\left(|H\u3009+i|V\u3009\right)/\sqrt{2}$ and $|R\u3009=\left(|H\u3009-i|V\u3009\right)/\sqrt{2}$. Coincidence counts for different measurements are listed in Table 1. Each item is the four-fold coincidence of mode

*A*,

*B*,

*C*and

*D*, with a duration of 300 seconds per measurement. The value of the inequality calculated by Eq. (6) for the GHZ state is 3.50 ± 0.05.

The Mermin steering inequality can also be violated when some noise is introduced. State prepared in the form of Eq. (5) identifies with a mixture of the maximally entangled state and |*HHH*〉 works as noise. The parameters of Mermin steering inequality for states with different *θ* are measured and plotted in Fig. 3. We measured the parameter every other 5° when *θ* varied from 0° (|*HHH* 〉) to 22.5° (maximally entangled GHZ state). In Fig. 3, the experimental results fits the theoretical curve well. We can witness tripartite EPR steering around *θ* ∈ (9.2°, 22.5°).

## 5. Conclusion

We have experimentally prepared a tripartite GHZ entangled state and witnessed the violation of Mermin steering inequality. By carrying out quantum tomography on the generated states, we have reconstructed the density matrix and calculated the fidelity compared with maximally entangled GHZ state. As a result, we have achieved the state fidelity for GHZ state with 87.25 ± 0.34 %. Based on the reliable source, we measure the parameter of Mermin steering inequality with 3.50 ± 0.05. On the other hand, from the bound on the GHZ-fidelity given by Eq. (7) for the violation of Mermin steering inequality of *γ* = 3.5, it follows that the GHZ-fidelity of 78.66 % can be certified at least. Thus, our experimental setup has certified higher amount of GHZ-fidelity than the bound given by Eq. (3). In addition, we prepare a class of GHZ-like states and see a phase of Mermin parameters increasing with the growth of fidelity compared with the maximally entangled tripartite GHZ state. As soon as the Mermin steering inequality is observed, the GHZ-fidelity of the class of GHZ-like states exceeds 50 %. Thus, our experimental setup confirms tripartite steerability of post-selected entangled states by observing the violation of Mermin steering inequality.

In [6], the authors derived a tripartite steering inequality to detect tripartite steering of three-qubit states in the context of a two-sided device-independent scenario where each party performs three dichotomic measurements on their respective subsystems. The critical robustness to white noise of the GHZ state above which tripartite steering can be detected by using this inequality is 0.63. On the other hand, for the Mermin steering inequality, this bound reads 0.5. Thus, Mermin steering inequality provides a simpler way to detect tipartite EPR steering in the three-qubit states with higher GHZ-fidelity than the steering inequality presented in [6].

## Funding

National Key R&D Program of China (2018YFA0306400, 2017YFA0304100); National Natural Science Foundation of China (61475197, 11774180, 61590932); Priority Academic Program Development of Jiangsu Higher Education Institutions; Postgraduate Research and Practice Innovation Program of Jiangsu Province. Ministry of Science and Technology of Taiwan (108-2811-M-006-501)

## Acknowledgments

CJ thanks Manik Banik and Jedrzej Kaniewski for useful discussion and S. N. Bose Centre Kolkata for the postdoctoral fellowship.

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