Experimental verification of time-order-dependent correlations in three-qubit Greenberger-Horne-Zeilinger-class states

TL;DR

This study experimentally verifies that generalized GHZ states exhibit genuine three-way nonlocality through Bell-type inequalities, even without Svetlichny inequality violation, highlighting the robustness of tripartite quantum correlations.

Contribution

It demonstrates that gGHZ states can show genuine three-way nonlocality under time-order-dependent principles regardless of Svetlichny inequality violation, both theoretically and experimentally.

Findings

gGHZ states violate tripartite Bell inequalities at all entanglement levels

gGHZ states exhibit nonlocality without Svetlichny inequality violation

maximal noise tolerance for maintaining nonlocality identified

Abstract

In this paper, we investigate the genuine three-way nonlocality which is recognized as the strongest form of tripartite correlations. We consider theoretically and experimentally a series of suitable Bell-type inequalities a violation of which is sufficient for the detection of three-way nonlocality. For the generalized GHZ (gGHZ) states, it is demonstrated that they do violate tripartite Bell-type inequalities for any degree of tripartite entanglement even if they do not violate Svetlichny inequality. It implies that three-way entangled gGHZ can always exhibit genuine three-way nonlocality under the requirement of time-order-dependent principle. Furthermore, we have determined the maximal amount of noise admissible for the gGHZ states to still remain genuine three-way nonlocal.

Tables

Table 1. Table 1: Characteristic of experimentally generated states. First column refers to angle θ 𝜃 \theta defined in Eq. ( 17 ) while θ o ​ p ​ t subscript 𝜃 𝑜 𝑝 𝑡 \theta_{opt} (second column) corresponds to angle which maximize fidelity F 𝐹 F given in Eq. ( 18 ). P 𝑃 P depicts purity, F max subscript 𝐹 F_{\max} stands for an upper bound on fidelity. N tri exp subscript superscript 𝑁 exp tri N^{\textrm{exp}}_{\textrm{tri}} and N tri ​ ( θ , P ) subscript 𝑁 tri 𝜃 𝑃 N_{\textrm{tri}}(\theta,P) denote tripartite negativity for experimentally generated states and the theoretical model discused in Appendix C , respectively.

$\theta$	${\theta }_{opt}$	$F$	$P$	$F_{\max }$	$N_{\text{tri}}^{\text{exp}}$	$N_{\text{tri}}(\theta ,P)$
$45$	$44.39$	$0.977$	$0.967$	$0.983$	$0.970$	$0.976$
$40$	$40.46$	$0.972$	$0.966$	$0.983$	$0.958$	$0.964$
$36$	$35.56$	$0.968$	$0.955$	$0.977$	$0.907$	$0.921$
$30$	$29.84$	$0.975$	$0.965$	$0.982$	$0.837$	$0.851$
$25$	$24.33$	$0.976$	$0.971$	$0.986$	$0.735$	$0.753$
$21$	$20.35$	$0.978$	$0.961$	$0.986$	$0.645$	$0.662$
$17$	$16.37$	$0.984$	$0.971$	$0.986$	$0.528$	$0.548$
$10$	$9.33$	$0.983$	$0.981$	$0.990$	$0.347$	$0.369$
$5$	$4.30$	$0.989$	$0.984$	$0.992$	$0.164$	$0.179$
$0$	$0.4$	$0.995$	$0.994$	$0.997$	$0.035$	$0.013$

Table 2. Table 2: Theoretical and experimental values of four Bell-type inequalities. p ~ ​ I i max ≡ p ~ × I i max ​ ( | 𝒢 ⟩ ) ~ 𝑝 subscript superscript 𝐼 𝑖 ~ 𝑝 subscript superscript 𝐼 𝑖 ket 𝒢 \tilde{p}I^{\max}_{i}\equiv\tilde{p}\times I^{\max}_{i}(|\mathcal{G}\rangle) corresponds to theoretical prediction which includes presence of depolarization p ~ = 8 ​ P − 1 7 ~ 𝑝 8 𝑃 1 7 \tilde{p}=\sqrt{\frac{8P-1}{7}} and I i exp ≡ I i max ​ ( ρ exp ) subscript superscript 𝐼 exp 𝑖 subscript superscript 𝐼 𝑖 subscript 𝜌 exp I^{\textrm{exp}}_{i}\equiv I^{\max}_{i}(\rho_{\textrm{exp}}) denotes the maximal attainable values of inequalities I i subscript 𝐼 𝑖 I_{i} for the reconstructed state ρ exp subscript 𝜌 exp \rho_{\textrm{exp}} .

$\theta$	$\tilde{p}{I}_{10}^{\max }$	$I_{10}^{\text{exp}}$	$\tilde{p}{I}_{96}^{\max }$	$I_{96}^{\text{exp}}$	$I_{99}^{\text{exp}}$	$\tilde{p}{I}_{185}^{\max }$	$I_{185}^{\text{exp}}$
$45$	$0.990$	$0.995$	$1.252$	$1.258$	$1.258$	$1.387$	$1.372$
$40$	$1.040$	$1.032$	$1.246$	$1.252$	$1.252$	$1.369$	$1.355$
$36$	$1.064$	$1.068$	$1.221$	$1.230$	$1.230$	$1.311$	$1.283$
$30$	$1.089$	$1.096$	$1.187$	$1.199$	$1.199$	$1.204$	$1.185$
$25$	$1.102$	$1.101$	$1.155$	$1.158$	$1.158$	$1.071$	$1.040$
$21$	$1.087$	$1.086$	$1.115$	$1.117$	$1.117$	$0.945$	$0.913$
$17$	$1.073$	$1.071$	$1.080$	$1.080$	$1.080$	$0.814$	$0.823$
$10$	$1.049$	$1.037$	$1.041$	$1.031$	$1.031$	$0.926$	$0.925$
$5$	$1.014$	$1.000$	$1.008$	$0.999$	$0.999$	$0.980$	$0.981$
$0$	$0.996$	$1.000$	$0.997$	$0.999$	$0.999$	$0.997$	$0.999$

Equations

$$P(abc|XYZ)=\sum _{\lambda }q(\lambda )P_{\lambda }(a|X)P_{\lambda }(b|Y)P_{\lambda }(c|Z),$$

$$P(abc|XYZ)=\sum _{i=1}^3\sum _{{\lambda }_i}q({\lambda }_i)P_{{\lambda }_i}({\delta }_i|{\Delta }_i)P_{{\lambda }_i}({\delta }_j{\delta }_k|{\Delta }_j{\Delta }_k),$$

$$I_{185}=\frac{1}{4}[$$

$$I(\rho )=\text{Tr}(\hat{I}\rho )=p\text{Tr}(\hat{I}|\psi ⟩⟨\psi |)\le C_{LHV},$$

$$p>p^{\min }(|\psi ⟩)=\frac{1}{I^{\max }(|\psi ⟩)},$$

$$I_{96}$$

$$|\mathcal{G}⟩=\cos \theta |000{⟩}_{ABC}+\sin \theta |111{⟩}_{ABC},$$

$$I_{96}^{\max }(|\mathcal{G}⟩)=\frac{1}{3}[1+2\sqrt{1+{\sin }^2(2\theta )}].$$

$$I_{99}$$

$$I_{10}$$

$$I_{10}^{\max }(|\mathcal{G}⟩)\approx 1+0.0622\theta +1.697{\theta }^2-3.391{\theta }^3+1.442{\theta }^4.$$

$$p^{\min}(|\mathcal{G}\rangle)=\left\{\begin{array}[]{ll}\frac{1}{I^{\max}_{10}(|\mathcal{G}\rangle)}&\textrm{for $0\leq\theta<14.94^{\circ}$}\\ \frac{1}{I^{\max}_{96}(|\mathcal{G}\rangle)}&\textrm{for $14.94^{\circ}\leq\theta<29.5^{\circ}$}\\ \frac{1}{I^{\max}_{185}(|\mathcal{G}\rangle)}&\textrm{for $29.5^{\circ}\leq\theta<45^{\circ}$}.\end{array}\right.$$

$$\cos \theta |0HH⟩+\sin \theta |1VV⟩,$$

$$⟨A_i{B}_j{C}_k⟩=\frac{S_{ABC}^{+}-S_{ABC}^{-}}{S_{ABC}^{+}+S_{ABC}^{-}},$$

$$S_{ABC}^{+}$$

$$S_{ABC}^{-}$$

$$⟨A_i{B}_j⟩=\frac{S_{AB}^{+}-S_{AB}^{-}}{S_{AB}^{+}+S_{AB}^{-}},$$

$$S_{AB}^{+}$$

$$S_{AB}^{-}$$

$$⟨A_i⟩=\frac{S_A^{+}-S_A^{-}}{S_A^{+}+S_A^{-}},$$

$$S_A^{+}$$

$$S_A^{-}$$

$$\theta =\arctan \sqrt{f_{111}/f_{000}},$$

$$F=\underset{\theta }{\max }⟨\mathcal{G}|\rho |\mathcal{G}⟩,$$

$$I_{10}^{\max }({\rho }_{\text{exp}})$$

$$I_{96}^{\max }({\rho }_{\text{exp}})$$

$$I_{99}^{\max }({\rho }_{\text{exp}})$$

$$I_{185}^{\max }({\rho }_{\text{exp}})$$

$$I_{10}^{\max }$$

$$I_{96}^{\max }$$

$$I_{99}^{\max }$$

$$I_{185}^{\max }$$

$${\vec{d}}_0\cdot {\vec{d}}_1=\cos {\theta }_0^d\cos {\theta }_1^d+\sin {\theta }_0^d\sin {\theta }_1^d\cos ({\varphi }_0^d-{\varphi }_1^d)=0.$$

$$I_{96}$$

$$⟨A_0{D}_1⟩+⟨B_0{D}_1⟩=-(\cos {\theta }_0^a+\cos {\theta }_1^b)\cos {\theta }_1^d$$

Full text

I Introduction

A fundamental problem in physics is to identify which correlations among different events can be observed within the description based on quantum mechanics. It is known that quantum theory allows correlations between spatially separated systems that are fundamentally different from classical counterparts. Moreover, even a very general quantum theory imposes nontrivial constraints on the allowed correlations among remote observers and hence, the answer for this question becomes not trivial.

One of the most important and at the same time intriguing classes of quantum correlations is known as nonlocal correlations, which has been recognized as an essential resource for quantum information tasks Brunner et al. (2014), such as quantum key distribution Bennet and Brassard (1984); Ekert (1991), communication complexity Buhrman et al. (2010), and randomness generation Pironio et al. (2010). Specifically, the correlations between outcomes of measurements on two or more spatially separated subsystems are said to be nonlocal, if they cannot be reproduced by any local-hidden variable (LHV) model Bell (1964). In particular, for the tripartite case, disused in this paper, the standard definition of nonlocality according to Bell Bell (1964) can be written as

[TABLE]

where local maps $P_{\lambda}(\delta|\Delta)$ are one-partite probability distribution of obtaining outcome $\delta=\{a,b,c\}$ with respect only to the local measurement $\Delta=\{X,Y,Z\}$ and the past variables $\lambda$. Naturally, the hidden variable $\lambda$ is distributed according to $q(\lambda)$ fulfilling requirements $0\leq q(\lambda)\leq 1$ and $\sum_{\lambda}q(\lambda)=1$. If such factorization of joined probability distribution exists then the correlations described by $P(abc|XYZ)$ can be explained by locally causal theory. Otherwise, they are referred to as nonlocal correlations.

This definition of nonlocality and resulting Bell-type inequalities, although sufficient for bipartite systems Bartkiewicz et al. (2013); Bartkiewicz and Chimczak (2018); Bartkiewicz et al. (2017), do not cover all possible variants of multipartite nonlocal correlations. As in the case of quantum entanglement, nonlocal correlations display a much richer and more complex structure for the multipartite case than the bipartite one. For instance, when restricting our consideration to the case of three observers, it is possible to have a hybrid local-nonlocal scenario in which (any) two parties may share bipartite nonlocal probability distribution. According to Svetlichny Svetlichny (1987), an appropriate hybrid local-nonlocal model is given by

[TABLE]

where $\{i,j,k\}$ is an even permutation of $\{1,2,3\}$, $\delta=\{a,b,c\}$, and $\Delta=\{X,Y,Z\}$. As previously, we expect that $0\leq q(\lambda_{i})\leq 1$ and $\sum^{3}_{i=1}\sum_{\lambda_{i}}q(\lambda_{i})=1$.

It turns out that quantum-mechanical description of nature authorizes the existence of correlations that cannot be explained by these more general LHV models and hence, those correlations exhibit genuine multipartite or n-way nonlocality Collins et al. (2002); Bancal et al. (2011). To test for such type of nonlocality, the Svetlichny inequality (a certain type of Bell-type inequality) is usually used Collins et al. (2002); Bancal et al. (2013)

[TABLE]

where $A$, $B$, and $C$ denote the three particles, $\langle A_{i}B_{j}C_{k}\rangle$ means correlation coefficient and subscripts [math], $1$ denote two different measurements. Note that the inequality $I_{185}$ presented here has been normalized which will be explained later.

The above-described phenomenon has naturally attracted much interest since the genuine multipartite nonlocality represents the strongest form of multipartite nonlocality, in which nonlocal correlations are established among all the parties of the system Lavoie et al. (2009); Lu et al. (2011a, b); Hamel et al. (2014); Adesso and Piano (2014); Chen et al. (2014). In particular, it has been shown Emary and Beenakker (2004); Ghose et al. (2009) that within the three-parameter pure GHZ subfamily, $|\Phi(\theta,\xi,\omega)\rangle=\cos\theta~{}|000\rangle_{ABC}+\sin\theta~{}|1\rangle_{A}\otimes(\cos\xi|0\rangle_{B}+\sin\xi|1\rangle_{B})\otimes(\cos\omega|0\rangle_{C}+\sin\omega|1\rangle_{C})$, the upper and lower bounds of maximal violation of Svetlichny inequality for a given tripartite entanglement monotony (the three-tangle Coffman et al. (2000)) are provided by the maximal slice (MS) states Carteret and Sudbery (2000) and the generalized GHZ (gGHZ) states Dür et al. (2000), respectively. Both states are given by $|\Phi(\pi/4,\pi/2,\omega)\rangle$ and $|\Phi(\theta,\pi/2,\pi/2)\rangle$, respectively. For a general case of pure three-qubit state, the upper bound is given by the tetrahedral states which are recognized as a lower bound of the primary yield of GHZ states from the infinitesimal distillation protocol Barasiński (2018).

The hierarchy which occurs between these three states causes an important consequence in the experimental detection of genuine multipartite nonlocal correlations where the inherent presence of noise must be taken into account, i.e., the more the Bell-type inequality is violated the greater amount of noise is needed to suppress nonlocality. In other words, the more the Bell-type inequality is violated, the more the nonlocality of the state is robust against noise. On the other hand, it is also very interesting that not every pure tripartite entangled state reveals Svetlichny correlations Barasiński (2018). For example, the gGHZ state admits the hybrid local-nonlocal model in Eq. (2) when $\theta\leq 22.5^{\circ}$.

The explanation of the last observation is based on the fact that the hybrid local-nonlocal model proposed by Svetlichny allows for correlations capable of two-way signaling among some parties. As a result, Svetlichny’s notion of genuine nonlocality is inconsistent with the operational approaches Gallego et al. (2012). To overcome this problem an alternative definition of genuine multipartite nonlocality based on time-order-dependent correlation (in literature denoted as TOBL) has been defined Gallego et al. (2012); Bancal et al. (2013). Specifically, it is assumed that $P_{\lambda_{i}}(\delta_{j}\delta_{k}|\Delta_{j}\Delta_{k})$ in Eq. (2) is at most one-way signaling. This model of nonlocality is crucial for the simulation of quantum correlations in all protocols where measurements performed on a particular system may depend on the measurement outcome obtained from another system, e.g. measurement-based computation. In such cases, Svetlichny-type simulation models in which all measurements settings are given at the same time are not relevant.

The hybrid local-nonlocal model can be further extended to the nonsignaling principle Barrett et al. (2005); Almeida et al. (2010); Bancal et al. (2013), in which the correlations are nonsignaling for all observers. Consequently, the existence of three-way nonlocality is certified by the violation of at least one of 185 inequalities Bancal et al. (2013), where the Svetlichny’s original inequality is only one of them.

For that reason an important question arises whether previous observations are still valid if one considers other types of LHV models. In particular, (i) what kind of correlations (i.e., Svetlichny correlations, TOBL, nonsignalling correlations) are needed to reveal three-way nonlocal correlations for all gGHZ states and hence, to satisfy the Gisin’s theorem Gisin (1991); Gisin and Peres (1992); (ii) Is the relation between gGHZ state and MS states still valid if one assumes more general LHV model?; (iii) What is the maximum value of noise required for the non-maximally entangled gGHZ states to produce each type of three-way nonlocality.

Note that partial answers for the first and the last questions are given in Refs. Mukherjee et al. (2015) and Bancal et al. (2013), respectively. Here, we present an extensive theoretical and experimental analysis in that field, taking into consideration all 185 Bell-type inequalities. To confirm our predictions, we use experimental setup based on correlated photon pairs generation in cascade of two BBO crystal via the process of spontaneous parametric down-conversion Kwiat et al. (1999); White et al. (1999); Halenková et al. (2012). In our experiment, two qubits are encoded into polarization and one qubit is encoded into propagation path. We demonstrate the ability of fast and efficient detection of three-way nonlocality, a task that was not possible with previous three-photon sources Lavoie et al. (2009); Lu et al. (2011a, b). Therefore, our experiment opens the door for future investigations of genuine multipartite nonlocal correlations based on the recent concept of nonlocal fraction Fonseca and Parisio (2015); de Rosier et al. (2017); Lipinska et al. (2018); Barasiński and Nowotarski (2018); Fonseca et al. (2018), where accumulation of a large amount of data is required.

II Nonlocality of GHZ states under the operational principle

In this paper we discuss the following problem: Suppose we have three observers Alice (A), Bob (B), and Charlie (C) who perform dichotomic measurements on a quantum state $\rho$ and each party has a choice of two measurement settings. Moreover, we assume that $\rho=p|\psi\rangle\langle\psi|+\frac{1-p}{8}{\rm 1\kern-2.20001ptl}$, where $|\psi\rangle$ denotes a tripartite pure state, ${\rm 1\kern-2.20001ptl}$ stands for identity matrix and $0\leq p\leq 1$. Suppose also that we have a given Bell-type operator $\hat{I}$ (e.g., Svetlichny operator Ghose et al. (2009)). Then, the nonlocal correlations of such a system can be generated if the expectation value for $\rho$ exceeds the Bell-type inequality

[TABLE]

where the second equality holds because the operator $\hat{I}$ is traceless. The parameter $C_{LHV}$ depicts the upper threshold of one inequality for local realism Bancal et al. (2013). To render the discussion more transparent, in the rest of the paper we use Bell-type operator $\hat{I}$ normalized in such a way that $C_{LHV}\equiv 1$. On the basis of that result we can determine the minimal value of the $p$ parameter (maximal strength of noise) required for the quantum state to exhibit genuine multipartite nonlocality, namely

[TABLE]

where $I^{\max}(|\psi\rangle)=\max_{\mathcal{M}}\textrm{Tr}(\hat{I}|\psi\rangle\langle\psi|)$ is the maximal violation of the Bell-type inequality with respect to $|\psi\rangle$ and the maximum is taken over all possible measurement settings. Naturally, for various $I$, different values of $p^{\min}$ can be reached. Therefore, in order to investigate the maximal extent to which the gGHZ and MS states can produce each type of multipartite nonlocality, we consider all $185$ Bell-type inequalities which describe the tripartite polytope of three-way correlations formulated in Ref. Bancal et al. (2013).

As an example of Bell-type inequality, let us now consider the 96th facet inequality, a violation of which implies that nonlocal correlations are time-order dependent Bancal et al. (2013),

[TABLE]

where we assume to have an ensemble of projective measurement $A_{0}=\vec{a}_{0}\cdot\vec{\sigma}_{A}$, or $A_{1}=\vec{a}_{1}\cdot\vec{\sigma}_{A}$ on qubit $A$, $B_{0}=\vec{b}_{0}\cdot\vec{\sigma}_{B}$, or $B_{1}=\vec{b}_{1}\cdot\vec{\sigma}_{B}$ on qubit $B$, $C_{0}=\vec{c}_{0}\cdot\vec{\sigma}_{C}$, or $C_{1}=\vec{c}_{1}\cdot\vec{\sigma}_{C}$ on qubit $C$. All vectors $\vec{a}_{0}$, $\vec{a}_{1}$, $\vec{b}_{0}$, $\vec{b}_{1}$ and $\vec{c}_{0}$, $\vec{c}_{1}$ are of length $1$ and $\vec{\sigma}=\{\sigma_{x},\sigma_{y},\sigma_{z}\}$, where $\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$ denote the Pauli operators associated with three orthogonal directions. For the generalized GHZ state,

[TABLE]

where $0\leq\theta\leq\frac{\pi}{4}$, the maximum value of $I_{96}$ in Eq. (6) becomes

[TABLE]

To achieve $I^{\max}_{96}$, one can measure with the following set of unit vector: $\vec{a}_{0}$ and $\vec{b}_{0}$ are aligned along the $-\vec{z}$ direction, $\vec{a}_{1}$ and $\vec{b}_{1}$ are aligned along $\vec{x}$, and vectors $\vec{c}_{0}$ and $\vec{c}_{1}$ lie in the $x-z$ plane. Specifically, $\vec{c}_{i}=\{-\sin\alpha,0,(-1)^{i}\cos\alpha\}$, where $\alpha=\arctan[\sin(2\theta)]$ (see Appendix A).

Interestingly, the same boundary can be found for the 99th facet inequality

[TABLE]

where all symbols have the same meaning as before. Indeed, when $\vec{a}_{0}$ and $\vec{b}_{0}$ are aligned along the $\vec{x}$ direction, and $\vec{a}_{1}$ and $\vec{b}_{1}$ are aligned along $\vec{z}$ and $\vec{c}_{i}=\{(-1)^{i}\sin\alpha,0,\cos\alpha\}$ then, one has $I^{\max}_{96}(|\mathcal{G}\rangle)\equiv I^{\max}_{99}(|\mathcal{G}\rangle)$ (see also Ref. Mukherjee et al. (2015)).

As we see in Fig. 1, when $\theta<29.5^{\circ}$ the normalized 96th facet inequality can be violated stronger than the normalized Svetlichny inequality I^{\max}_{185}(|\mathcal{G}\rangle)=\max\big{\{}\sqrt{1-\sin^{2}(2\theta)},\sqrt{2\sin^{2}(2\theta)}\big{\}} (discussed in details in Ref. Ghose et al. (2009)). This means that for nonmaximally entangled gGHZ states, there exists a range of $p$ values for which these states are too noisy to exhibit Svetlichny nonlocality, but can still generate the time-order-dependent correlations. This result can be enhanced even more if one takes the 10th facet inequality (also TOBL class inequality) into consideration:

[TABLE]

Then, the maximal violation of $I_{10}$ with respect to the gGHZ state can be approximated as

[TABLE]

Note that the analytical solution, although attainable for a given $\theta$, is too complicated to be presented here (see Appendix B). Direct comparison of $I^{\max}_{10}(|\mathcal{G}\rangle)$ and $I^{\max}_{96}(|\mathcal{G}\rangle)$ implies that the first one corresponds to the slightly higher threshold visibility $p^{\min}$ than the former inequality when $\theta<14.94^{\circ}$ (see Fig. 1). Furthermore, numerical optimization performed on the remaining Bell-type inequalities given in Ref. Bancal et al. (2013) reveals that the above-described results are sufficient to estimate the threshold visibility of the gGHZ states. Consequently, the higher threshold visibility $p^{\min}$ for the gGHZ states can be summarized as

[TABLE]

At the end of this section, let us mention that similar calculations have been made for the MS state. In that case we have found that Svetlichny correlations are sufficient to determine minimal values of $p$, i.e., $I^{\max}_{i}(|MS\rangle)<I^{\max}_{185}(|MS\rangle)=\sqrt{1+\sin^{2}\omega}$ for $i=1,2,\dots,184$. In order to compare the threshold visibility achieved for gGHZ and MS state, we use a quantity called three-tangle $\tau$ Coffman et al. (2000) which refers to tripartite entanglement, where $\tau(|\mathcal{G}\rangle)=\sin^{2}(2\theta)$ and $\tau(|MS\rangle)=\sin^{2}\omega$. Then, one can find $p(|MS\rangle)>p(|\mathcal{G}\rangle)$ if and only if $\tau>0.0434$. In other words, the three-way nonlocal correlations generated with gGHZ states are more fragile against noise then correlation produced with MS states almost in the entire range of tripartite entanglement.

III Experimental Setup

We use a standard configuration for generation of photon pairs in the process of spontaneous parametric down-conversion; see Fig. 2. A cascade of two BBO crystals is used in a so-called Kwiat configuration to generate polarization-correlated photon pairs Kwiat et al. (1999); White et al. (1999); Halenková et al. (2012). The beam displacer (BD) in the state preparation part allows one to split horizontal ($H$) and vertical ($V$) components of the first photon state to two spatial modes ($0,1$):

[TABLE]

where $\{0,1\}$ labels the first photon’s spatial mode and $\{H,V\}$ the polarization state of the first and second photon, respectively. Associating polarization states $\{H,V\}$ with logical state $\{0,1\}$ immediately identifies the state as $|\mathcal{G}\rangle$ according to Eq. (7). The angle $\theta$ and zero phase between the two components of the state (13) are set using rotation and tilt of two wave plates in the pump laser beam. To measure any correlation coefficient $\langle A_{i}B_{j}C_{k}\rangle$ we set accordingly the six wave plates in the projection part of the setup and measure coincidence counts on the two detectors, denoted $f^{+++}$. For normalization we have to measure also all seven orthogonal projections, i.e., $f^{-++}$, which means orthogonal projection in qubit $A$ and likewise for other terms. Correlation function is then calculated as

[TABLE]

where

[TABLE]

To obtain the experimental value of $I_{185}$ the three-qubit correlation functions are sufficient. For the other inequalities $I_{10}$, $I_{96}$, $I_{99}$ we have to evaluate also two-qubit and one-qubit correlations. Correlation functions for only two qubits can be calculated summing coincidence counts measured for two orthogonal projections on the third irrelevant qubit, e.g.:

[TABLE]

where

[TABLE]

And similarly, we can measure the one-qubit correlation from the coincidence counts as follows,

[TABLE]

where

[TABLE]

IV Experimental Results

First, we experimentally verify the resulting conclusions presented in Sec. II for the prototype GHZ state. To do this, we adjust the experimental setup in such a way to have $\theta=(45.0\pm 0.5)^{\circ}$. This angle is calculated from the experimental measurements as

[TABLE]

where $f_{i}$ denotes two-photon coincidence rates when projecting all three qubits on logical states $|111\rangle$ and $|000\rangle$, respectively.

To obtain complete information about generated state $\rho_{\textrm{exp}}$, quantum state tomography and maximum-likelihood estimation have been used to reconstruct the output-state density matrix Ježek et al. (2003); Černoch et al. (2008). Results are shown in Fig. 3. We then determine fidelity $F$ of $\rho_{\textrm{exp}}$ with respect to the ideal state in Eq. (7), i.e., we use

[TABLE]

where maximization is done over $\theta\in\langle 0^{\circ},45^{\circ}\rangle$. In the ideal case, the fidelity $F$ shall be equal to $1$. In our case, due to experimental imperfections, the observed output-state fidelity $F(\rho_{exp})=0.977\pm 0.002$ with the optimal angle $\theta_{opt}$ (which provides maximum $F$) equal to $44.39^{\circ}$, which is in line with previous fast estimation of $\theta$. The error bar of the fidelity is determined by Monte Carlo simulations of Poissonian noise distribution.

To investigate whether the reduced fidelity is caused by improper setting of individual components or by depolarization effects, we calculate the state’s purity $P=\textrm{Tr}(\rho^{2})$. The undesired but coherent state transformations can arise, e.g., from imperfect polarization compensation by means of the controllers at the input of the beam displacer (photon in mode $1$) while the main reason of the depolarization is noise arising from photon-pair emission of the source, which leads to a reduction in the purity $P$ of the output state. By straightforward calculations we find that $P(\rho_{\textrm{exp}})=0.967\pm 0.005$. This value allows us to determine an upper bound $F_{\max}(\rho_{\textrm{exp}})=\frac{1}{8}(1+\sqrt{56P(\rho_{\textrm{exp}})-7})=0.983\pm 0.003$ on the observable fidelity (discussed in Appendix C). As we see, the fidelities $F(\rho_{\textrm{exp}})$ is very close to $F_{\max}$ and hence, is mainly limited by the presence of noise.

The influence of parasitic states can be also estimated by means of tripartite negativity $N_{\textrm{tri}}=[N_{A|BC}N_{B|AB}N_{C|AB}]^{1/3}$, where $N_{i|jk}$ indicates the bipartition negativity of qubit $i$ and joined qubits $jk$ Love et al. (2007). We find that $N_{\textrm{tri}}(\rho_{\textrm{exp}})=0.970\pm 0.005$, what is in perfect agreement with theoretical predictions, $N_{\textrm{tri}}(45^{\circ},P)=0.976\pm 0.004$, where $P\equiv P(\rho_{\textrm{exp}})$ depicts the purity of $\rho_{\textrm{exp}}$ (see Appendix C).

Next, using the reconstructed state $\rho_{\textrm{exp}}$ and numerical optimization performed with respect to measurements setting, the maximal attainable values of inequalities discussed in Sec. II have been determined as

[TABLE]

while the results predicted by quantum mechanics for $\theta=45^{\circ}$ are given by $\tilde{p}\times I^{\max}_{10}(|\mathcal{G}\rangle)=0.990\pm 0.003$, $\tilde{p}\times I^{\max}_{96}(|\mathcal{G}\rangle)=\tilde{p}\times I^{\max}_{99}(|\mathcal{G}\rangle)=1.252\pm 0.004$, and $\tilde{p}\times I^{\max}_{185}(|\mathcal{G}\rangle)=1.387\pm 0.004$. Note that theoretical results presented above are corrected by including a small amount of depolarization $\tilde{p}=\sqrt{\frac{8P(\rho_{\textrm{exp}})-1}{7}}$. As we see, in each case the difference between $I_{i}(\rho_{\textrm{exp}})$ and theoretical prediction is not greater than $0.015$, which suggests that our measurement outcomes behave as $\tilde{p}\times I^{\max}_{i}(|\mathcal{G}\rangle)$.

Finally, all necessary correlations that enter into $I_{10}$, $I_{96}$, $I_{99}$, and $I_{185}$ are measured directly. To do this we use the correlation function in Eq. (14) and the appropriate measurements settings described in Sec. II. As a result we get

[TABLE]

Note that the differences between results presented in Eqs. (19) and (20) are not greater than $0.025$. It is also important to emphasize that our result for $I^{\max}_{185}$ is much closer to the maximal attainable value for Svetlichny inequality of $\sqrt{2}$ than the outcomes so far presented in literature, where $I^{\max}_{185}$ is approximately equal to $1.128$ Lavoie et al. (2009) or $1.115$ Lu et al. (2011a, b).

We repeat our investigation for other angles $\theta$ (presented in Table 1). As we see in Fig. 4 (a), for each case the fidelity $F$ (purity $P$) fluctuates between $0.968$ and $0.995$ (between $0.955$ and $0.994$), which confirms a good quality of generated states. Moreover, the tripartite negativity $N_{\textrm{tri}}(\rho)$ is also in perfect agreement with theoretical predictions $N_{\textrm{tri}}(\theta,P)$ (see Fig. 4 (b)), implying that the influence of parasitic states is negligible for all analyzed $\theta$.

Further analysis of the generated states confirms that the maximal attainable value of $I_{10}$, $I_{96}$, $I_{99}$, $I_{185}$ as functions of $\theta$ behaves like $\tilde{p}\times I^{\max}_{10}(|\mathcal{G}\rangle)$ (see Table 2). The maximal difference $\Delta I_{i}=|I_{i}-\tilde{p}I^{\max}_{i}(|\mathcal{G}\rangle)|$ is given by: $\Delta I_{10}=0.014$, $\Delta I_{96}=0.012$, $\Delta I_{99}=0.012$, $\Delta I_{185}=0.032$. The same conclusion can be found if one takes direct measurements of Bell-type inequalities into considerations. Our results are presented in Fig. 5. In this case, the highest difference between measurements and theoretical predictions $\tilde{p}\times I^{\max}_{i}(|\mathcal{G}\rangle)$ is given by: $\Delta I_{10}=0.028$, $\Delta I_{96}=0.035$, $\Delta I_{99}=0.030$, $\Delta I_{185}=0.045$. Note that our measurements of $I_{185}$ are of better quality than the results already published Lu et al. (2011a), which clearly demonstrate the efficiency of our experimental setup for detection of three-way nonlocality.

Finally, we verify the relation between $p^{\min}$ and angle $\theta$ presented in Eq. (12). As we see in Fig. 6, due to experimental imperfections all measured values of $p^{\min}$ are greater than theoretical predictions. However, when we include the correction $\tilde{p}\times I^{\max}_{i}(|\mathcal{G}\rangle)$ in Eq. (12), our results are in perfect agreement with theory (dotted line in Fig. 6).

V Conclusions

In conclusion, we have theoretically and experimentally investigated the violation of four Bell-type inequalities which refer to the existence of three-way nonlocality. In particular, series of gGHZ states with high fidelity ($F\geq 0.968$) have been prepared experimentally and we have demonstrated the test of the above mentioned Bell-type inequality on these states. Our results clearly demonstrate that the gGHZ states exhibit nonlocal correlations in the entire range of tripartite entanglement (for any $\theta>0$), although these states cannot violate the standard Svetlichny inequality, i.e., the three-way nonlocal correlations obtained from these states violate the newly derived Bell-type inequalities based on time-order-dependent local realism models. On the basis of that result the minimal value of the $p$ parameter (maximal strength of noise) admissible for the gGHZ states to exhibit genuine multipartite nonlocality has been determined.

Our experiment demonstrates the ability of fast and efficient detection of three-way nonlocality and hence, it opens the door for future investigations of genuine multipartite nonlocal correlations based on the recent concept of nonlocal fraction Fonseca and Parisio (2015); de Rosier et al. (2017); Lipinska et al. (2018); Barasiński and Nowotarski (2018); Fonseca et al. (2018).

VI Acknowledgements

The authors thank Cesnet for providing data management services. A.B. was supported by GA ČR Project No. 17-23005Y, A.Č., J.S. and K.L. by Project No. 17-10003S. Authors also thank MŠMT ČR for support by the project CZ.02.1.01/0.0/0.0/16_019/0000754.

Appendix A

In order to derive Eq. (8) we adapt the technique used in Popescu and Rohrlich (1992). First, we represent all unit vectors in spherical coordinates, i.e. $\vec{a}_{0}=(\sin\theta^{a}_{0}\cos\phi^{a}_{0},\sin\theta^{a}_{0}\sin\phi^{a}_{0},\cos\theta^{a}_{0})$ and likewise for similarly defined terms. Our aim is to find a set of optimal spherical angles for all measurements $A_{0}$,…,$C_{1}$. Due to the experimental interpretation of measurements $A_{0}$,…,$C_{1}$ (see main text) and without lose of generality, we restrict our calculation to azimuthal angles in the interval $[0,\pi]$ and polar angles in $[0,2\pi]$.

Next, let us rewrite Eq. (6) in terms of unit vectors $\vec{d}_{0}$ and $\vec{d}_{1}$ defined such that $\vec{c}_{0}+\vec{c}_{1}=2\vec{d}_{0}\cos\eta$ and $\vec{c}_{0}-\vec{c}_{1}=2\vec{d}_{1}\sin\eta$. Note that

[TABLE]

Then, with settings $D_{0}=\vec{d}_{0}\cdot\vec{\sigma}_{C}$ and $D_{1}=\vec{d}_{1}\cdot\vec{\sigma}_{C}$ one obtains

[TABLE]

The expectation value of terms in the first parentheses with respect to the gGHZ state is given by

[TABLE]

and hence, the global maximum of $I_{96}$ with respect to $\theta^{d}_{1}$ is reached when $\theta^{d}_{1}=\{0,\pi\}$. (It should be noted that for $\eta=\{0,\pi\}$ inequality $I_{96}\leq 1$ for any $\theta^{a}_{0}$, $\theta^{a}_{1}$ etc.) Substitution of this result into Eq. (21) implies $\theta^{d}_{0}=\pi/2$. Consequently, when inserting $\{\theta^{d}_{0},\theta^{d}_{1}\}=\{\pi/2,\pi\}$ into Eq. (LABEL:eq:ineq96d) we find

[TABLE]

where $\phi^{abc}_{0}=\phi^{a}_{0}+\phi^{b}_{0}+\phi^{c}_{0}$ and $\phi^{abc}_{1}=\phi^{a}_{1}+\phi^{b}_{1}+\phi^{c}_{0}$. Furthermore, due to the symmetry of indexes $a\leftrightarrow b$ one can assume $\theta^{a}_{0}=\theta^{b}_{0}$, $\theta^{a}_{1}=\theta^{b}_{1}$ and likewise for angles $\phi^{a,b}_{i}$. Then,

[TABLE]

After some standard algebra we find turning points of $I_{96}(|\mathcal{G}\rangle)$ in Eq. (25) at $\theta^{a}_{0}=0$, $\theta^{a}_{1}=\pi/2$ and $\phi^{abc}_{1}=\pi$ and hence, $I_{96}(|\mathcal{G}\rangle)$ is limited by

[TABLE]

where the last inequality comes from $x\cos\alpha+y\sin\alpha\leq\sqrt{x^{2}+y^{2}}$ and is saturated when $\tan\alpha=y/x$. Finally, the set of measurement angles that provides the equality in the above expression is given by $\theta^{a}_{0}=\theta^{b}_{0}=\phi^{c}_{0}=\phi^{c}_{1}=\pi$, $\theta^{a}_{1}=\theta^{b}_{1}=\pi/2$, $\phi^{a}_{0}=\phi^{b}_{0}=\phi^{a}_{1}=\phi^{b}_{1}=0$ and $\theta^{c}_{0}=\arctan[\sin(2\theta)]$, $\theta^{c}_{1}=\pi-\arctan[\sin(2\theta)]$ what ends the proof.

Appendix B

In order to find a global maximum of the 10th facet inequality a similar method as in Appendix A has been applied. In this way the following set of measurements has been determined: $\{\theta^{a}_{0},\phi^{a}_{0}\}=\{\pi/2,\pi/3\}$, $\{\theta^{a}_{1},\phi^{a}_{1}\}=\{\vartheta_{0},-2\pi/3\}$, $\{\theta^{b}_{0},\phi^{b}_{0}\}=\{\vartheta_{1},\pi/3\}$, $\{\theta^{b}_{1},\phi^{b}_{1}\}=\{\vartheta_{1},4\pi/3\}$, $\{\theta^{c}_{0},\phi^{b}_{0}\}=\{\vartheta_{1},\pi/3\}$, $\{\theta^{c}_{1},\phi^{b}_{1}\}=\{\vartheta_{1},4\pi/3\}$, where both $\vartheta_{0}$ and $\vartheta_{1}$ depend on angle $\theta$ and satisfy $\partial\tilde{I}_{10}/\partial\vartheta_{0}=0\wedge\partial\tilde{I}_{10}/\partial\vartheta_{1}=0$ with

[TABLE]

It should be noted that angles $\vartheta_{0}$ and $\vartheta_{1}$ can be easily approximated as: $\vartheta_{0}\approx(0.004\theta^{2}-1.024\theta+180)$ and $\vartheta_{1}\approx\max\{(\frac{1}{1.6992\cdot 10^{-7}+6.25\cdot 10^{-8}\sqrt{\theta}})^{1/3},(\frac{1}{3.1\cdot 10^{-5}+6.168\cdot 10^{-6}\sqrt{\theta}})^{1/2}\}$.

Appendix C

Let $\rho=p|\psi\rangle\langle\psi|+\frac{1-p}{8}{\rm 1\kern-2.20001ptl}$, where $|\psi\rangle$ denotes the tripartite pure state, ${\rm 1\kern-2.20001ptl}$ stands for identity matrix, and $0\leq p\leq 1$. Then, the fidelity $F=\langle\psi|\rho|\psi\rangle$ becomes

[TABLE]

In other words, $F$ increases linearly with $p$. Analogously, the purity is given by

[TABLE]

As we see, when the experimental noise is negligible, both the purity and fidelity should be equal to $1$ and they decrease to $1/8$ if we measure unbiased noise only. Moreover, the fidelity decreases with the purity $P$, the presence of undesired states and with any unwanted qubit’s rotations around the Bloch sphere, that could be due to, e.g., optical misalignment.

To get some indication about whether the less-than-unity $F$ is caused mostly due to the purity reduction (assuming the input state $|\psi\rangle$ is pure) or to imperfect preparation of the state (e.g., qubits’ rotation), we can estimate the maximal fidelity $F_{\max}$. Specifically, let us assume there are no unwanted rotations, and that the reduction of fidelity comes from the presence of white noise. Then, one can insert the measured value of the purity $P$ in Eq. (29) into Eq. (28) which effectively yields an upper bound to the measured value of the fidelity,

[TABLE]

Finally, if one assumes that $|\psi\rangle=|\mathcal{G}\rangle$, i.e., $\rho=p|\mathcal{G}\rangle\langle\mathcal{G}|+\frac{1-p}{8}{\rm 1\kern-2.20001ptl}$, then we can easily determine the tripartite negativity $N_{\textrm{tri}}(\rho)=[N_{A|BC}(\rho)N_{B|AB}(\rho)N_{C|AB}(\rho)]^{1/3}$. Here, $N_{i|jk}(\rho)=\frac{||\rho^{\Gamma_{i}}||-1}{2}$ indicates the bipartition negativity of qubit $i$ and joined qubits $jk$ and $\rho^{\Gamma_{i}}$ is the partial transpose of $\rho$ with respect to subsystem $i$. After some straightforward calculations one has

[TABLE]

where $\theta$ is defined in Eq. (7) and we use the relation $p=\sqrt{\frac{8P-1}{7}}$ given in Eq. (29).