Abstract
The launch of a satellite capable of distributing entanglement through long distances and the first loopholefree violation of Bell inequalities are milestones indicating a clear path for the establishment of quantum networks. However, nonlocality in networks with independent entanglement sources has only been experimentally verified in simple tripartite networks, via the violation of bilocality inequalities. Here, by using a scalable photonic platform, we implement starshaped quantum networks consisting of up to five distant nodes and four independent entanglement sources. We exploit this platform to violate the chained nlocality inequality and thus witness, in a deviceindependent way, the emergence of nonlocal correlations among the nodes of the implemented networks. These results open new perspectives for quantum information processing applications in the relevant regime where the observed correlations are compatible with standard local hidden variable models but are nonclassical if the independence of the sources is taken into account.
Introduction
Bell’s theorem^{1}, i.e., the incompatibility of quantum predictions with local hidden variable (LHV) models, is among the most influential results in quantum foundations. Despite being almost 60 years old, only recently the phenomenon of Bell nonlocality has been proven in a loopholefree manner in a series of independent experiments^{2,3,4,5,6}. Apart from its fundamental importance, witnessing Bell nonlocality is at core of many applications in quantum information processing, ranging from quantum communication^{7} and distributed computing^{8}, to quantum cryptography^{9}, quantum key distribution^{10,11}, and randomness generation^{12,13}.
Indeed, the violation of a Bell inequality allows to bound the secure key rate that can be exchanged among distant nodes in a deviceindependent (DI) manner^{14,15,16}, that is, with minimal assumptions on measurement apparatuses and relying exclusively on observed data. Most of these findings, however, only hold for the paradigmatic bipartite Bell scenario. Generalizations to the multipartite case have been proposed, for instance, based on multipartite entangled states, such as the GreenbergerHorneZeilinger states (GHZ)^{17}. Notwithstanding, generation of such states on a photonic platform is still highly demanding^{18}, affecting the nearterm experimental relevance of deviceindependent communication protocols based on it. Accordingly, quantum networks of growing size and complexity aiming at the socalled quantum internet^{19,20} are much more likely to be composed of independent sources, each one generating small size entangled states but at much higher quality and rate.
The independence of the sources in a quantum network gives rise to a much richer set of correlations^{21,22,23,24,25,26,27,28,29,30} as compared with the standard Bell nonlocality. Not only the network scenario allows for nonlocality activation^{31} and less stringent detection efficiencies^{32}, but also the emergence of truly new kinds of nonclassical behaviors^{30}. In spite of much recent theoretical advance, experimental implementations^{6,33,34,35} have been so far limited to the simplest possible quantum network, the bilocality model^{21,22}. Such scenario is akin to the entanglement swapping experiment, involving two independent sources of entangled states, and only very recently was implemented while closing the locality and independence loopholes at the same time^{36}.
In this work, we provide a proof of principle demonstration of the scalability of such networks, moving beyond the bilocality scenario, by considering up to four independent sources distributing entanglement among five nodes in a starnetwork topology. To ensure the independence of the sources, we employ four photonic setups located in four different laboratories, each one consisting of a source of entangled photons, pumped by a different laser, and a measurement station. Via this platform, we report the violation of polynomial chained Bell inequalities with an increasing number of parties and measurement settings, hence detecting the presence of nonlocal correlations among the nodes of multipartite networks. Our scalable approach could be useful both for demonstrating the nonlocality of topologically different scenarios^{30}, as well as a testbed for deviceindependent protocols of information processing in quantum networks^{37}.
Results
Causal modeling approach
Topologically complex networks, composed of several sources and parties, are notoriously hard to characterize, especially if the independence of the sources is taken into consideration. A new approach has been introduced in the last few years to properly analyze these networks, which leverages on the theory of causal inference^{38}. Indeed, the causal modeling approach^{39} has become a customary and powerful tool to represent classical and quantum networks^{27,40,41,42,43}. In this framework, directed acyclic graphs (DAGs) provide both a graphical and a mathematical way to visualize causal structures (see Fig. 1). Each of the nodes stands for a random variable involved in the process, whose cause and effect relations are encoded by directed edges (arrows).
Bilocality scenario
The simplest quantum network beyond the paradigmatic Bell scenario (Fig. 1a) is the socalled bilocality scenario^{21,22} depicted in Fig. 1b, that is, a network with two independent sources Λ_{1} and Λ_{2}. The three nodes, Alice, Bob, and Charlie, can perform measurements of different observables denoted by x_{1}, x_{2}, and y, chosen independently, with outcomes labeled as a_{1}, a_{2}, and b, respectively. Note that such independence in the choice of the measurements is also crucial in standard Bell tests^{6}, in particular for the derivation of Bell inequalities^{22,27,41}.
The aforementioned experiment is described by a probability distribution that, under the bilocality assumption, should be compatible with a bilocal LHV model, given by refs. ^{21,22}
Assuming that all these variables are dichotomous (with values 0 or 1), any correlation compatible with the bilocal model (1) should fulfill the following nonlinear Bell inequality:
with
and \(\langle {A}_{1}^{{x}_{1}}{A}_{2}^{{x}_{2}}{B}^{y}\rangle\) being the expectation value of the measurements outcomes of the three nodes:
In the last 2 years, inequality (2) has been experimentally violated in a number of photonic setups^{6,33,34,35,36}. The first violation^{33} exploited polarizationentangled photons and the measurements performed in the central node B relied on a complete Bellstate measurement. Notably, bilocality violation can also be achieved with separable measurements^{22,23}, as experimentally realized in refs. ^{6,35}. The violation of bilocality is relevant, in particular, in the intermediate situation where local variable models can reproduce the observed correlations but bilocal models cannot^{33}, namely one can violate a bilocal causality inequality even if the data admits a LHV model, where the independence of the sources is not taken into account. Accordingly, quantum states generating classical correlations in conventional scenarios can become powerful resources in a network, thus enlarging the capabilities to process information in a nonclassical way.
nlocality in a starnetwork
The bilocality scenario can be extended to more complex networks, considering an increasing number of independent sources and involved nodes, as well as different topologies. In particular, a nlocality scenario involves n independent sources distributing correlations among the nodes. A topology that has received much theoretical attention is the starnetwork^{21,22,23,24,25}, with n + 1 nodes interconnected by n independent sources, which contains the bilocality scenario as a particular case (see Figs. 1b–d). Each of the n peripheral nodes (called A_{i}) is connected through the source Λ_{i} to the central node of the network (called B). Labeling the measurements for B and external nodes A_{i} by y and x_{i} and their outcomes by b and a_{i}, respectively, a classical nlocal hidden variable model implies that any classical observed distribution should be decomposable as
Considering that each of the nodes can perform k measurements described by dichotomous observables B^{y} (central node) and \({A}_{i}^{{x}_{i}}\) (external nodes), the nlocal model implies the following nonlinear Bell inequality^{23}:
with \({A}_{i}^{k}={A}_{i}^{0}\). This is a generalization both of bilocality inequality (2), as well as of the socalled chained Bell inequality^{44}, which are recovered in the case of n = 2 and n = 1, respectively. For this reason, we call inequality (7) the chained nlocality inequality. A larger number of measurement settings k for the chained form considered here provides advantages in deviceindependent protocols, either by reducing the experimental constraints for their violation^{45} or by leading to better security tests^{46}. The set of classical correlations allowed by nlocal models is shown in Fig. 3.
Let us now focus on the quantum violation of these inequalities, by highlighting a few important points. First, inequality (7) makes the explicit assumption that the sources of correlations are independent, thus any implementation of the corresponding causal structure should take that into account. For instance, the first experimental implementations^{33,34} adopted a single laser in order to generate the two pairs of entangled states, opening a loophole in the quantum violation of inequality (2). To avoid this loophole, as detailed below, in our experimental implementation, we used independent entanglement sources in separated laboratories. The second important point is that, in the starnetwork, Bob (the central node) has access to n independent physical systems, thus the most general measurement he can perform is a measurement in an entangled basis. In fact, theoretical results showing the activation of nonlocality in such networks^{31} rely on measurements on a GHZ basis. From the experimental perspective, however, such entangled measurements represent an extremely demanding task, as even measurements on a complete Bell basis cannot be implemented using linear optics, without resorting to hybrid or nonlinear approaches^{47,48}. Nicely, however, it has been shown that the optimal quantum violation of the nlocality inequality can already be achieved if all measurements, including those of the central node in the networks, are made in separable bases^{22,23,24}, thus avoiding the request to synchronize photons from different pairs as needed for projection on Bell basis. In summary, not only one can rely on independent sources but also make the simplest possible measurements to detect nonlocality in complex networks, paving the way of an experimentally scalable approach.
Let us now quantify the quantum violation of the nlocality inequality. By generating singlet entangled states and by performing the measurements reported in the Methods section, the upper bound of inequality (7) is the following:
that can be shown to be the optimal quantum violation^{24}. We note that such bound does not depend on the number of nodes, n, but only on the number of measurements settings, k.
To experimentally implement the starnetwork causal structure depicted in Fig. 1d, we exploited the photonic platform of Fig. 2 with four independent polarizationentangled photon pair sources (a detailed description of the sources can be found in Supplementary Note 1). As shown in Fig. 2, three of the sources, that are pumped by different pulsed lasers located in the laboratories 1, 3, and 4, rely on spontaneous parametric downconversion (SPDC) of type II, achieved through BetaBarium Borate crystals, which emits degenerate photon pairs at 785 nm. Instead, the fourth source, located in laboratory 2, is pumped in the continuouswave regime and generates photon pairs at 808 nm. This source scheme exploits a SPDC process of type II in a periodically poled KTP nonlinear crystal, placed into a Sagnac interferometer^{49}. Thus, in our experimental implementation of the starnetwork (Fig. 1d), we strongly enforce the independence of the sources for the implemented causal structure.
When a pair of photons is generated in each crystal, it is split: one photon is sent to the central node B, while the other remains within the laboratory A_{i} where it was generated, to be measured locally. The distribution of the shared photons among the nodes was accomplished by exploiting different optical fiber links. The maximum length of the fibers used was 25 m. At the output of each fiber, the polarization of the photon was properly compensated, to counteract unavoidable rotations along the path. In each laboratory, there is a measurement station, where polarization analysis can be performed by rotating a halfwave plate (HWP) placed right before a polarizing beam splitter (PBS). This apparatus allows each party to perform a number k of different projective measurements on their subsystem, since they are all represented by linear combinations of the Pauli matrices σ_{x} and σ_{z}. Accordingly, to perform the k measurement settings required to test inequality (7), each HWP was rotated to switch up to four different settings, in order to perform all the possible measurement combinations, which, in our starshaped network, amount to 1024 combinations.
Photons arrival events were registered by single photons detectors (SPD) and a different timetagger in each laboratory, and immediately sent to a central station where the synchronization and the coincidence detection took place. In particular, two detected photons were considered as simultaneous, provided that they were registered within a time, shorter than a given window. The results presented here were obtained using a coincidence time window of 80 μs, but a quantum violation was also observed for narrower values, up to 0.49 μs (see Supplementary Note 3).
We carried out the experiments for causal structures of star networks scenarios with n = 2, 3, 4 sources, whose independence is enforced by exploiting different independent pump lasers for SPDC processes occurring in dislocated laboratories, as mentioned above. Let us note that our setup relies, inevitably, on the measurement independence assumption, i.e., free choice of inputs measurements, as well as on the lack of correlations among the sources. Both such assumptions, indeed, cannot be enforced by any physical principle, but only be made less plausible as possible. For this reason, as already mentioned, we enforced the independence of the sources by adopting different laser devices, which, furthermore, were not relying on the same electric power source. Moreover, it requires the fairsampling assumption, analogously to other most relevant experiments dealing with quantum networks^{6,33,34,35,36}. In the end, given that the measurements were not performed providing spacelike separation, we are assuming the lack of uncontrolled communication channels among the measurements stages, located in the separated laboratories.
In this framework, we managed to distinguish synchronous events by designing a sophisticated software, keeping track of the generation times of all of the four sources, as detailed in the Supplementary Note 3. On the other hand, by not requiring photons of different sources to interfere, we offered the possibility of using different types of sources within the same network and to have significantly higher generation rates.
Violations of nlocality with k = 2 measurement settings
The scheme previously described allows for the violation of nlocality with k different measurement settings. When considering the particular case of k = 2, the optimal measurement settings to perform at the external nodes and summarized in Eq. (10), are performed by the HWPs at measurement stages before PBS and with angles \({\theta }_{0}^{{\rm{A}}}={0}^{\circ }\) for choice x_{i} = 0 and \({\theta }_{1}^{{\rm{A}}}=22.{5}^{\circ }\) for choice x_{i} = 1. For the central node B, the HWP’s angles that project along states in Eq. (13) are \({\theta }_{0}^{{\rm{B}}}=11.2{5}^{\circ }\) for choice y = 0 and \({\theta }_{1}^{{\rm{B}}}=33.7{5}^{\circ }\) for choice y = 1. The corresponding results for this case are presented in Fig. 3 and can be extended by increasing the number of measurements k. The data in Fig. 3 show that the observed values are incompatible with the classical models for nlocality. Nonetheless, the obtained correlations are still compatible with a LHV model with a single shared source among all parties. To witness the nonclassicality of the network correlations, the source independence has to be taken into account. In particular, for n = 2, we measured such parameters exploiting all of the six combinations of pairs among the four sources. The maximum observed value is \({S}_{\max }^{{\rm{obs}}}=1.218\pm 0.002\), violating the classical bound by 109 standard deviations. For n = 3, our setup allows for four triples of sources, achieving \({S}_{\max }^{{\rm{obs}}}=1.199\pm 0.004\), a violation of 50 standard deviations with respect to the classical bound. Finally, for n = 4 we obtained a value \({S}_{\max }^{{\rm{obs}}}=1.192\pm 0.005\), corresponding to a violation of 38 standard deviations. All the results for k = 2 are reported in Table 1.
Violations of nlocality with k > 2 measurement settings
An increasing number of settings provides advantages in DI protocols based on the violation of Bell inequalities^{37}. For that reason, we also consider the violation nlocality chained inequality (7) with an increasing number k of settings. In particular, we violated the 4locality (n = 4) with up to four different measurement settings corresponding to a total 1024 combinations. Using this apparatus, we were able to violate the nlocality chained inequality (7) for the 2, 3, and 4starshaped networks scenarios, counting the 4, 6, and 8fold coincidences events, in the case of k = 2, 3, 4 measurement settings. For the case of n = 2 and n = 3, each of these measurements was performed on every pair and triplet of the different four parties, thus allowing for six and four different combinations, respectively. The results for the different cases are presented in Fig. 4 as well as Table 2. For the particular case of n = 4 with k = 4, our results yields S = 3.157 ± 0.002, surpassing the classical limit by 71 standard deviations. The experimental values are fully compatible with the theoretical predictions where the noise of the whole system is taken into account (see Supplementary Note 2). Albeit with our apparatus we cannot close the locality loophole, because we would require an increase of the spatial distance among the parties, we approached such condition as much as possible through a reduction of the time window for the detection of coincidence events. Indeed, we present results also for narrower coincidence windows, up to 0.49 μs, which still allow for significant quantum violations (see Supplementary Note 3). Such reduction of the coincidence window lowers the probability of mutual causal influences among the parties in realistic experimental conditions.
Discussion
Bell’s theorem is a conceptual cornerstone in our understanding of quantum theory that, with the rise of quantum information science, has also been turned into a novel tool for information processing. In spite of the key developments over the years in tasks such as quantum cryptography, selftesting, and entanglement certification, generalizations of Bell’s theorem beyond the simple single source scenario, in particular from the experimental point of view, are still an almost uncharted territory. That is precisely the aim of this work, to provide the first experimental implementation of a scalable quantum network consisting of an increasing number of independent entanglement sources, separated laboratories and measurement settings.
Here, we chose to investigate a topology that has received considerable theoretical attention lately, the starnetwork^{21,22,23,24,25}, with a central node sharing an entangled state with n other peripheral nodes. In particular, we focused on the crucial task of certifying, in a deviceindependent manner, the presence of nonclassical correlations among the nodes of the network. To this aim, we violated the chained nlocality inequality, an extension of the famous chained inequalities^{44} to the network case. The main strengths of this inequality are its adaptability to consider an increasing numbers of nodes in the network and of measurement settings for each party and also the possibility of achieving maximum quantum violation with separable measurements, a significant experimental advantage. We violated the chained nlocality inequality, thus ensuring the presence of truly nonclassical correlations, by considering networks up to five nodes (four independent sources) and that each of the nodes could measure up to four different measurement settings.
In particular, networks of this kind might prove relevant, in the future, for the execution of multipartite protocols such as secret sharing^{50,51,52}. Such task consists in a central node aiming to securely share a message with all the other parties, even if some of the receivers are untrusted. The crucial point, in this work, lies in the experimental realization of a quantum network of five stations, within which, the simultaneous presence of nonlocal correlations is certified. In this context, even though the ideal condition of spacelike separated parties is not reached, we approximate such condition, by taking coincidence counts within a small time interval. Let us note that the limits on such interval are given by the rates of the sources. This approximate simultaneity, together with the independent sources, can certainly be considered as a very versatile tool for the future implementation of quantum networks and makes our platform easily adaptable to scenarios like the ones proposed by Lee and Hoban^{37}, that could be applied to cryptographic tasks. Furthermore, for practical purposes and applications, the short time interval, that we adopt to detect simultaneous events, makes eventual communications among the parties less plausible in a plethora of realistic scenarios. To conclude, the novelty of our work stems from the presentation of the prototype of a scalable quantum network, going beyond the bilocality scenario, which allowed, for the first time, the violation of the chained nlocality inequalities. Moreover, the versatility of our platform can be exploited to study networks that are attracting growing attention, like the triangle network^{30} and the linear chain topology underlying quantum repeaters^{53}.
Methods
Measurements performed
All of the four sources generate polarizationentangled pairs of photons in the Bell state \(\left{\psi }^{}\right\rangle =(\left01\right\rangle \left10\right\rangle )/\sqrt{2}\). To obtain the maximum quantum violation of inequality (7), \(k\cos (\pi /2k)\), all the peripheral parties A_{i} must perform the following projective measurements on their subsystem:
for each setting x_{i} = x. In turn, the central node B measures each of its n subsystems in the local basis:
for each setting y_{i} = y, where the index i refers the system Bob shares with the ith noncentral part. The resulting measurement corresponds to \({B}_{y}={B}_{y}^{1}\otimes \cdots \otimes {B}_{y}^{n}\), where \({B}_{y}^{j}\) represent the measurement performed on each subsystem. Hence, to evaluate the quantum violation in (7), we need to perform k2^{n} combinations of measurement settings, 2^{n} for each term I_{i} appearing in (7).
Experimental details
For the experimental setups of Fig. 2, the three different pump lasers for sources 1, 3, and 4, with λ = 397.5 nm are produced by a second harmonic generation (SHG) process from a Ti:Sapphire modelocked laser with repetition rate of 76 MHz. Photon pairs entangled in the polarization degree of freedom are generated exploiting typeII SPDC in 2mmthick betabarium borate (BBO) crystals. Source 2, instead, employs a continuouswave diode laser with wavelength of λ = 404 nm, which pumps a 20mmthick periodically poled KTP crystal inside a Sagnac interferometer, to generate photon pairs using a typeII degenerate SPDC process. The photons generated in all the sources are filtered in wavelength and spatial mode by using narrow band interference filters and singlemode fibers, respectively.
Coincidence counting
The photon detection events were collected and timed by a different timetagger device for each party, located in the corresponding laboratory (see Fig. 2b). For each 1 s of data acquisition the events were sent to a central server, along with a random clock signal shared between all the timetaggers, which was used to synchronize the timestamps of events relative to different devices. To filter out part of the noise the raw data was first preprocessed by keeping only double coincidence events for each photon source, using a narrow coincidence window of 3.24 ns. Then coincidence events between multiple sources were counted every time one of such double coincidence event was recorded for each source in a window of 80 μs.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Code availability
All the custom code developed for this study is available from the corresponding author upon request.
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Acknowledgements
We thank F. Andreoli for useful discussions. This work was supported by The John Templeton Foundation via the grant QCAUSAL No. 61084, by MIUR (Ministero dell’Istruzione, dell’Università e della Ricerca), via project PRIN 2017 “Taming complexity via QUantum Strategies a Hybrid Integrated Photonic approach” (QUSHIP) Id. 2017SRNBRK, by Sapienza Progetto di Ateneo H2020ERC QUMAC (MAChine learning via hybrid integrated QUANTUM photonics), by the Regione Lazio program “Progetti di Gruppi di ricerca” legge Regionale n. 13/2008 (SINFONIA project, prot. n. 85201715200) via LazioInnova spa, and by the QuantERA ERANET Cofund in Quantum Technologies 2017 project HiPhoP (High dimensional quantum Photonic Platform, projectID 731473). R.C. also acknowledges the Brazilian ministries MEC and MCTIC, CNPq (Grants No. 307172/20171 and No. 406574/20189 and INCTIQ) and the Serrapilheira Institute (grant number Serra170815763). G.C. thanks Becas Chile and Conicyt.
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D.P., I.A., G.C., R.C., and F.S. Conceived the experiment; D.P., I.A., G.M., E.P., T.G., A.S., M.V., G.M.I., N.S., G.C., and F.S. devised and performed the experiment; D.P., G.M., I.A., E.P., N.S., G.C., R.C., and F.S. performed the data analysis; all the authors discussed the results and contributed to the writing of the paper.
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Poderini, D., Agresti, I., Marchese, G. et al. Experimental violation of nlocality in a star quantum network. Nat Commun 11, 2467 (2020). https://doi.org/10.1038/s41467020161896
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DOI: https://doi.org/10.1038/s41467020161896
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Star network quantum steering
Quantum Information Processing (2023)
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