## Abstract

Quantum mechanics admits correlations that cannot be explained by local realistic models. The most studied models are the standard local hidden variable models, which satisfy the well-known Bell inequalities. To date, most works have focused on bipartite entangled systems. We consider correlations between three parties connected via two independent entangled states. We investigate the new type of so-called “bilocal” models, which correspondingly involve two independent hidden variables. These models describe scenarios that naturally arise in quantum networks, where several independent entanglement sources are used. Using photonic qubits, we build such a linear three-node quantum network and demonstrate nonbilocal correlations by violating a Bell-like inequality tailored for bilocal models. Furthermore, we show that the demonstration of nonbilocality is more noise-tolerant than that of standard Bell nonlocality in our three-party quantum network.

- quantum networks
- quantum foundations
- Entanglement
- Bilocality
- Bell nonlocality
- quantum information
- Photonic Quantum Information
- nonlocality

## INTRODUCTION

Bell’s theorem (*1*) resolved the long-standing Einstein-Podolsky-Rosen debate (*2*) by demonstrating that no local realistic theory can reproduce the correlations observed when performing appropriate measurements on some entangled quantum states—so-called (Bell) nonlocal correlations (*3*). Entanglement now finds applications as a resource in many quantum information and communication protocols [for example, see the studies by Ekert (*4*) and Bennett *et al*. (*5*)]. In most fundamental or applied experiments to date, the entangled systems come directly from a single source. However, sometimes, more than one source of entanglement is used, such as in protocols that rely on entanglement swapping (*6*) to generate entanglement between two parties at the ends of a chain (although they share no common history). Because the entanglement swapping results in a bipartite entangled state, one may examine this “network” scenario by considering only the nonlocality of the correlations between the measurement outcomes at the terminal nodes. An “event-ready” Bell test (*6*), heralded on success signals from all intermediate nodes, would then aim to disprove a local theory that is based on a single local hidden variable (LHV) model. However, such a test ignores properties of the intermediate channel, such as the fact that the multiple sources of entanglement may be independent of each other. This raises an important fundamental question: How does source independence affect the notion of nonlocality?

To address this question, a new type of LHV model was recently considered, where the independence properties of the different sources in an experimental setup are also imposed at the level of the hidden variables (*7*, *8*). The simplest nontrival quantum network to analyze this new type of model is a three-node linear network, as depicted in Fig. 1. In such a network, two independent entanglement sources connect the three nodes, Alice, Bob, and Charlie; the corresponding model, which involves two independent LHVs, is termed “bilocal.” Just like standard LHV models satisfy Bell inequalities, it was shown that bilocal models impose constraints on the corresponding correlations in the form of (nonlinear) Bell-like inequalities—so-called “bilocal inequalities”—which can be violated quantum mechanically (*7*, *8*). One advantage of considering bilocal models is that one may demonstrate nonbilocality in situations where no nonlocality could be obtained. For example, in an entanglement swapping experiment that generates a two-qubit Werner state between Alice and Charlie of the form (where |ψ〉 is a maximally entangled state and is the maximally mixed state), a visibility is required to violate the commonly used Clauser-Horne-Shimony-Holt (CHSH) Bell inequality (*9*), whereas bilocal inequalities can detect nonbilocality for any *v* > 1/2 (*7*, *8*); thus, one can certify the absence of a bilocal LHV model under more noise compared to a Bell local model.

The aim of the present work is to experimentally investigate quantum nonbilocal correlations. We implement the scenarios of Fig. 1 in a photonic setup. In our experiment, the entangled photon pairs originate from two nonlinear crystals pumped separately, although by the same laser beam. To enhance the independence of the two sources, we actively destroy any coherence in the pump beam between the two crystals. We test two different bilocal inequalities and find violations that allow us to disprove bilocal models for the quantum correlations observed.

### Local versus bilocal models

The differences between testing locality and bilocality on a three-node quantum network are highlighted in Fig. 1. Let us first introduce a standard LHV three-party model: Consider a tripartite probability distribution of the form(1)
where Alice, Bob, and Charlie have measurement inputs *x*, *y*, *z* and measurement outputs *a*, *b*, *c*, respectively, and the LHV λ with the distribution ρ(λ) can be understood as describing the joint state of the three systems. *P*(*a*|*x*, λ), *P*(*b*|*y*, λ), and *P*(*c*|*z*, λ) are the local probabilities for each separate outcome, given λ. A probability distribution *P*(*a*, *b*, *c*|*x*, *y*, *z*) of the form of Eq. 1 is said to be (Bell) local; one that cannot be expressed in that form is called (Bell) nonlocal (*3*).

In a practical experiment, where the abovementioned tripartite probability distribution is obtained by measuring some physical systems—for example, particles—it is natural to assume that the LHV λ originates from the source that prepares and sends those systems. However, for our three-node quantum network of Fig. 1, there are two independent sources of entangled particles—*S*_{1} and *S*_{2}. It is then natural to consider two LHVs, λ_{1} and λ_{2}, one attached to each source, and write(2)Here, the local probabilities of each party are conditioned only on the LHV(s) attached to the source(s) from which they receive the particles: λ_{1} for Alice, λ_{2} for Charlie, and both λ_{1} and λ_{2} for Bob, at the intermediate node. So far, the correlations producible by the local decompositions in Eqs. 1 and 2 are equivalent. For example, the joint distribution of the two LHVs ρ(λ_{1}, λ_{2}) could be nonzero only when λ_{1} = λ_{2} = λ (*7*). However, we shall now introduce the critical bilocality assumption, based on the physical arrangement of our quantum network: The independence of the two sources *S*_{1} and *S*_{2} carries over to the LHVs λ_{1} and λ_{2}. That is, their joint distribution ρ(λ_{1}, λ_{2}) must factorize(3)

Probability distributions *P*(*a*, *b*, *c*|*x*, *y*, *z*) that can be expressed as in Eq. 2, with ρ(λ_{1}, λ_{2}) satisfying Eq. 3, are said to be “bilocal”; those that cannot be expressed as such are termed “nonbilocal” (*7*, *8*).

### Demonstrating nonbilocality

The decomposition of Eq. 2, together with Eq. 3, imposes certain restrictions on the correlations that can be produced by bilocal models. Note that any bilocal model is in particular Bell local, so that it must satisfy all Bell inequalities; any violation of a Bell inequality is already a demonstration of nonbilocality. However, it is also possible to derive stronger constraints for bilocal models, which specifically make use of the independence condition of Eq. 3. In the study by Branciard *et al*. (*8*), different bilocal inequalities were obtained, of the general form(4)
where *I* and *J* are linear combinations of the observed probabilities *P*(*a*, *b*, *c*|*x*, *y*, *z*) (see Methods for details). A violation of such an inequality, that is, a value greater than 1, is a proof of nonbilocality, as it rules out any possible bilocal model—in a similar way that a CHSH value, greater than 2 disproves any Bell local model (see the Supplementary Materials) (*9*).

The bilocal inequalities described above apply to scenarios where Alice and Charlie have binary inputs and outputs. As for Bob, we consider two cases that are of particular experimental relevance. In the first case, he has a single fixed input (measurement setting) and four possible outputs (measurement results); following the notations of Branciard *et al*. (*8*), we shall label this case “*14*” and write the corresponding inequality as ≤ 1. In the second case, Bob still has a fixed input, but he now has three possible outputs; we shall label this case “*13*” and write ≤ 1. As discussed below, these two cases will correspond in the experiment to a full and a partial BSM implemented by Bob, respectively.

## RESULTS

To test the two bilocal inequalities , ≤ 1, we realized a photonic implementation of an entanglement swapping type of experiment [for example, see the study by Pan *et al*. (*10*)] that implements the three-node quantum network of Fig. 1. Two “sandwich” type 1 spontaneous parametric downconversion (SPDC) sources (*11*) supplied the entangled photonic links between the nodes (see Fig. 2). To justify that the bilocality assumption is reasonable, one should ideally have truly independent sources. In our case, we used two separate nonlinear crystals to realize the parametric downconversion; however, the two crystals were pumped by a strong beam originating from the same laser. To increase the degree of independence between the two sources, we installed a TVPS in the pump beam before the source *S*_{2}. The TVPS comprised a rotatable optical flat connected to an automated stage and a remote quantum random number generator (*12*), adding a genuinely random phase offset between sources *S*_{1} and *S*_{2} on each trial of the experiment and thus destroying any quantum coherence (see the Supplementary Materials for details).

At the central node, Bob implements an entangling BSM (*13*) to essentially fuse the two sources of entanglement *S*_{1} and *S*_{2} via entanglement swapping (*6*). Using linear optics only, it is impossible to construct an ideal BSM device that reliably discriminates between all four Bell states, which would be necessary for deterministic entanglement swapping (*14*). However, it is possible to experimentally simulate the statistics of an ideal BSM. We construct such a BSM device that projects onto one of the four Bell states. We then implement local unitaries to project separately, in different experimental runs, onto the three remaining states and combine the statistics at the end of the experiment to mimic a universal BSM device. In this case, Bob’s implemented measurement device has four input settings (one for each of the canonical Bell states, |Φ^{+}〉, |Φ^{−}〉, |Ψ^{+}〉, and |Ψ^{−}〉) and one bit of output (indicating successful projection onto the relevant state)—such that on each run of the experiment, we only project onto a single Bell state. After recombining the statistics at the end of the experiment, Bob has simulated an ideal BSM device with a single input setting (corresponding to precisely performing a BSM) and four possible measurement results (outputs b), one corresponding to each of the four Bell states. It is precisely in this one-input/four-output scenario that one can test the ≤ 1 bilocal inequality introduced previously. Conveniently, it is also possible using linear optics to construct a partial BSM device that projectively resolves two of the four Bell states (for example, |Φ^{+}〉 and |Φ^{−}〉), accompanied by a third projection that groups the remaining two Bell states (for example, |Ψ^{±}〉) into a single outcome (*15*)—a single-input, three-output measurement that allows one to test the ≤ 1 bilocal inequality. As for Alice and Charlie, as mentioned above, they should have binary inputs (measurement settings) and outputs (measurement results) to test these two inequalities. We implemented projective measurements of the observables and (depending on the inputs *x*, *z* = 0, 1) defined as (using the corresponding half–wave plate setting θ_{0} = 11.25°) and (θ_{1} = − 11.25°) in the *14* case (where are the standard Pauli matrices) and as (θ_{0} ≈ 8.82°) and (θ_{1} ≈ − 8.82°) in the *13* case, which, in principle, provide the optimal violations of the two inequalities (*8*).

Each entanglement source *S*_{i} (*i* = 1, 2) ideally produces a pure Bell state. However, because of minor experimental imperfections, the produced states were close to Werner states (as described in the Introduction) with visibility *v*_{i} ≳ 0.94 [determined via quantum state tomography (*16*)] for both sources for all implementations of Bob’s BSM. The fidelity of the BSM was maximized using single-mode fibers, narrowband frequency filters (~3-nm full width at half maximum), and a high-precision translation stage, affording subcoherence length timing resolution and ensuring high-quality Hong-Ou-Mandel (HOM) interference. We measured a resultant HOM visibility of when Bob implemented the *14*-BSM and for Bob’s *13*-BSM. The visibility of the closest Werner state to the resultant entangled state at Alice’s and Charlie’s terminal nodes (conditioned on Bob’s BSM result) was estimated using quantum state tomography, yielding *v*_{14} ≈ 0.78 and *v*_{13} ≈ 0.85, respectively—in agreement with the product of the visibility of each entangled source and the BSM visibility, as expected.

To further verify that our network was producing Werner-like states, we compared the measured CHSH inequality with the inferred entanglement visibilities. We tested the CHSH inequality (see the Supplementary Materials) (*9*) on the resultant state of Alice and Charlie after successful entanglement swapping, a standard [event-ready (*6*)] test of Bell locality. We recorded and , agreeing with the measured *v*’s above, and both with clear violations of the local bound. Next, the bilocal inequalities of Eq. 4, for both the full (case “*14*”) and partial (case “*13*”) BSMs, were tested in our network, with clear violations in both cases: = 1.25 ± 0.04 > 1 and = 1.17 ± 0.02 > 1 (see the Supplementary Materials for further details). To explore the noise robustness of our locality and bilocality tests, we added various amounts of white noise to our experimental data (see Fig. 3). We implemented this by swapping the labels on Alice’s measurement outcomes on selected experimental runs, mimicking the effect of white noise by washing out the correlations (see Methods). This experimentally verified the prediction that in the presence of noise, there exists a region where nonbilocal correlations can be observed but nonlocal correlations cannot (*7*, *8*).

## DISCUSSION

We have thus experimentally demonstrated the violation of two Bell-like inequalities tailored for quantum networks with independent entanglement sources and verified that those inequalities can be violated at added noise levels for which a CHSH inequality cannot. As with quantum (EPR-) steering (*17*), for example, the addition of an extra assumption (here, source independence) relaxes the stringent intolerance to noise of nonlocality demonstrations.

Our violation of bilocal inequalities shows, in principle, that no bilocal model can explain the correlations we observed. However, we acknowledge that, like most Bell tests until very recently (*18*–*20*), our experiment is subject to some loopholes. In addition to the issue of space-like separation and the detection loophole (*21*), the specificity of the bilocality assumption opens a new “source independence loophole” when the entanglement sources are not guaranteed to be fully independent. In our experiment, we enhanced the source independence by erasing the quantum coherence between the pump beams of our two separate SPDC sources. Nevertheless, the bilocality violations we observed could still, in principle, be explained by some hidden mechanism that would correlate the two sources (and the two LHVs λ_{1}, λ_{2} attached to them in a bilocal model), for instance, via the shared pump beam. To be able to draw more satisfying conclusions with regard to nonbilocality, the next step will be to realize a similar experiment with “truly independent” sources [following in the footsteps of Kaltenbaek *et al*. (*22*) and Erven *et al*. (*23*)]—but keeping in mind that just like a Bell test can never rule out a superdeterministic explanation (*24*), it is impossible to guarantee that two separate sources are genuinely independent, as they could have been correlated at the birth of the universe.

The bilocality assumption, as well as its extension to “*N*-locality” in more complex scenarios involving *N*-independent sources, provides a natural framework to explore and characterize quantum correlations in multisource, multiparty networks (*7*, *8*). “*N*-local inequalities” have been derived in the line of Bell and bilocal inequalities (*25*–*33*), which could be tested in possible extensions of the present experiment and in future larger quantum networks. An interesting question is whether the violation of these inequalities could be directly exploited and could allow for useful applications in quantum information processing—similar to the demonstration of Bell nonlocality or quantum steering which can, for example, be used to certify the security of quantum key distribution, or the privacy of randomness generation, in a device-independent way (*34*–*37*). We note that, contrary to the event-ready Bell test, the violation of the bilocal inequalities tested here does not by itself certify that Bob must have performed an entangling measurement and that Alice and Charlie end up sharing an entangled state (a counterexample is presented in the Supplementary Materials); thus, it is not sufficient for information processing protocols that require such a certification. However, we expect other possible applications to be discovered, which will fully harness the non–*N* locality of quantum correlations, for instance, in cases where nonlocality cannot be demonstrated. The problem of characterizing and demonstrating non–*N*-local correlations will become more and more crucial as future quantum networks continue to grow in size and complexity.

*Note Added.* During the preparation of the paper, we became aware of an independent experimental study of nonbilocality (*38*).

## METHODS

### Bilocal inequalities

The quantities *I* and *J* in the bilocal inequalities (Eq. 4) that we tested in our experiment were defined from the observed probabilities *P*(*a*, *b*, *c*|*x*, *z*) as follows. [Because we consider cases where Bob has a single fixed measurement setting *y*, we can ignore it when writing *P*(*a*, *b*, *c*|*x*, *z*).]

Let us start with the full BSM, with four possible outcomes—the case labeled *14*. Here, Bob’s output consists of two bits, *b* = *b*^{0}*b*^{1}. Using some of the notations and forms introduced by Branciard *et al*. (*8*), we first define, for *j* = 0 and 1, the tripartite correlators (expectation values)(5)

where the sum is over all outputs *a*, *b*^{0}, *b*^{1}, *c* = 0, 1 of the three parties. These correlators, for the various values of *x*, *z* = 0, 1, then sum together in the following way to define *I*_{14} and *J*_{14} as(6)

The case of a partial three-outcome BSM, labeled *13*, is slightly complicated by the asymmetry in the partial BSM. Here, we denote Bob’s three possible outcomes as *b* = *b*^{0}*b*^{1} = 00, 01, {10 or 11}. The tripartite correlators are defined as(7) and, restricting to the case where Bob gets one of the first two outcomes (that is, *b*^{0} = 0)(8)Similarly as before, these correlators then sum together to now define(9)

We provide in the Supplementary Materials all the probabilities *P*(*a*, *b*, *c*| *x*, *y*) measured in both our *14* and *13* tests, which allowed us to compute our experimental values for

### Adding noise to our network

To investigate the noise tolerance properties of testing different local and bilocal models in our quantum network, we added white noise to our measured correlations. Ideally, the joint state shared between Alice and Charlie (outer nodes) after entanglement swapping would be one of the four Bell states. However, because of experimental noise, the states produced can instead be approximated by a Werner state of the form(10)
where |ψ〉 is the resulting shared Bell state after the swapping operation and is the maximally mixed two-qubit state. In our experiment, the state *W* had visibilities of *v*_{14} ≈ 0.78 and *v*_{13} ≈ 0.85 before introducing further white noise; the values of *v* in the *14* and *13* cases mainly differ because of the differences in Bob’s BSM visibility between runs. This agrees with the source visibilities determined using quantum state tomography and with the visibility of the BSM determined by measuring a heralded HOM dip visibility, by scanning the automated delay stage in Bob’s BSM apparatus. This also agrees with the measured CHSH parameter values for our network.

To add further noise, our procedure was inspired by the effect of white noise on the observed statistics. We flipped Alice’s measurements with probability *p* = (1 − *v*_{added}/2) to simulate adding further noise with visibility *v*_{added}, ending up with a global visibility *v* = *v*_{14/13} *v*_{added} for the state of Eq. 10. This mirrors the effect of a depolarizing channel for our polarization-encoded qubits, allowing us to vary the value of *v* for the Werner states produced in our network, up to the limit of *v* ≤ *v*_{14/13}.

## SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/4/e1602743/DC1

Characterization of the TVPS

Photon counting and experimental details

Experimental violation of the bilocal inequalities

Experimental violation of the CHSH inequality

Our bilocal inequalities violations are not device-independent certifications of A-C entanglement: Counterexample

table S1. Measured probabilities *P*_{14}(*a*, *b*, *c*|*x*, *z*) and correlators in our test of the ≤ 1 bilocal inequality.

table S2. Measured probabilities *P*_{13}(*a*, *b*, *c*|*x*, *z*) and correlators in our test of the ≤ 1 bilocal inequality.

table S3. Observed violations of the bilocal inequalities .

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is **not** for commercial advantage and provided the original work is properly cited.

## REFERENCES AND NOTES

**Acknowledgments:**We thank A. Abbott, N. Brunner, N. Gisin, S. Pironio, and D. Rosset for helpful discussions.

**Funding:**This research was conducted by the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (project number CE110001027). D.J.S. and C.B. acknowledge European Union Marie Curie Fellowships PIIF-GA-2013-629229 and PIIF-GA-2013-623456, respectively. C.B. acknowledges a “Retour Post-Doctorants” grant from the French National Research Agency (ANR-13-PDOC-0026). D.J.S. acknowledges support from an ERC advance grant (MOQUACINO).

**Competing interests:**The authors declare that they have no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.

**Author contributions:**D.J.S., C.B., and G.J.P. conceived the experiment. D.J.S. and A.J.B. constructed and performed the experiment, with guidance from G.J.P., D.J.S., and C.B. adapted the theory to the experiment. DJS performed the data analysis with help from C.B., G.J.P., and A.J.B. All authors contributed to writing the manuscript.

- Copyright © 2017, The Authors