ARTICLE
Received 1 Aug 2016 | Accepted 1 Feb 2017 | Published 16 Mar 2017 DOI: 10.1038/ncomms14775 OPEN
Experimental violation of local causality in a
quantum network
Gonzalo Carvacho1, Francesco Andreoli1, Luca Santodonato1, Marco Bentivegna1,
Rafael Chaves2,3 & Fabio Sciarrino1
Bell’s theorem plays a crucial role in quantum information processing and thus several
experimental investigations of Bell inequalities violations have been carried out over the
years. Despite their fundamental relevance, however, previous experiments did not consider
an ingredient of relevance for quantum networks: the fact that correlations between distant
parties are mediated by several, typically independent sources. Here, using a photonic setup,
we investigate a quantum network consisting of three spatially separated nodes whose
correlations are mediated by two distinct sources. This scenario allows for the emergence of
the so-called non-bilocal correlations, incompatible with any local model involving two
independent hidden variables. We experimentally witness the emergence of this kind of
quantum correlations by violating a Bell-like inequality under the fair-sampling assumption.
Our results provide a proof-of-principle experiment of generalizations of Bell’s theorem for
networks, which could represent a potential resource for quantum communication protocols.
1 Dipartimento di Fisica - Sapienza Università di Roma, P.le Aldo Moro 5, I-00185 Roma, Italy. 2 International Institute of Physics, Federal University of Rio
Grande do Norte, 59070-405 Natal, Brazil. 3 Institute for Theoretical Physics, University of Cologne, 50937 Cologne, Germany. Correspondence and requests
for materials should be addressed to F.S. (email: fabio.sciarrino@uniroma1.it).
NATURE COMMUNICATIONS | 8:14775 |DOI: 10.1038/ncomms14775 | www.nature.com/naturecommunications 1
ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14775
s demonstrated by the celebrated Bell’s theorem1,
Acorrelations arising from experiments with distant a bquantum mechanical systems are at odds with one of
our most intuitive scientific notions, that of local realism. The Y
assumption of realism formalizes the idea that physical quantities
have well-defined values independently of whether they are X Y X Z
measured or not. In turn, local causality posits that correlations B
between distant particles can only originate from causal
influences in their common past. These two rather natural
assumptions together imply strict constraints on the empirical A B A C
correlations that are compatible with them. These are the famous
Bell inequalities, which have been recently violated in a series of
loophole-free experiments2–4 and thus conclusively established c d
the phenomenon known as Bell non-locality. Apart from their
profound implications in our understanding of nature, such
experiments provide a proof-of-principle for practical Y X Y Z W
applications of non-local correlations, most notably in the
context of quantum networks5–7. X Z
B A B C D
In a quantum network, short-distance nodes are connected by
sources of entangled systems which can, via an entanglement
swapping protocol8,9, establish entanglement across long A C
distances as well. Importantly, such long-distance entanglement 1 2 1 2 3
can in principle also be used to violate a Bell inequality and thus
establish a secure communication channel10–12. Clearly, for these Figure 1 | Representation of the causal structures underlying the
and many other potential applications13–16, the certification of networks45. Directed acyclic graphs45 can represent different causal
non-local correlations across the network will be crucial. The structures, for instance the nodes in the graph represent the relevant
problem, however, resides on the fact that experimental random variables with arrows accounting for their causal relations. There
imperfections accumulate very rapidly as the size of the are three different kinds of nodes: hidden variables (orange boxes),
network and the number of sources of states increase, making measurement settings (green boxes) and measurement outcomes (blue
the detection of Bell non-locality very difficult or even impossible boxes). (a) Bipartite LHV model. (b) Tripartite LHV model. (c) Tripartite
by usual means17,18. One of the difficulties stems from the scenario with two independent LHVs, that is, bilocal hidden variable (BLHV)
derivation of Bell inequalities themselves, where it is implicitly model. (d) Possible extension of the bilocal model to a linear chain of four
assumed that all the correlations originate at a single common stations with three independent LHVs.
source (see Fig. 1b), the so-called local hidden variable (LHV)
models. Notwithstanding, in a network a precise description must Results
take into account that there are several and independent sources Local and bilocal correlations in a tripartite scenario. We start
of states (see Fig. 1c), which introduce additional structure to the describing the typical scenario of interest in the study of Bell non-
set of classically allowed correlations. In fact, there are quantum locality shown in Fig. 1b for the case of three distant parties. A
correlations that can emerge in networks that, while admitting a source distributes a physical system to each of the parties that at
LHV description, are incompatible with any classical description each run of the experiment can perform the measurement of
where the independence of the sources is considered19–25. For different observables (labelled by x, y and z), thus obtaining the
instance, a network with two independent sources allow for the corresponding measurement outcomes (labelled by a, b and c). In
emergence of a different kind of non-local correlations violating a classical description of such experiment, no restrictions other
the so-called bilocal causality assumption19,20. than local realism are imposed, meaning that the measurement
The aim of this study is to experimentally observe this different devices are treated as black boxes that take random (and inde-
type of Bell non-locality. We experimentally implemented, using pendently generated) classical bits as inputs and produce classical
pairs of polarization-entangled photons, the simplest possible bits as outputs as well. After a sufficient number of experimental
quantum network akin to a three-partite entanglement swapping runs is performed, the probability distribution of their measure-
scheme (see Fig. 1c). Two distant parties, Alice and Charlie, ments can be estimated, that according to the assumption of local
perform analysis measurements over two photons (1 and 4, see realism can be decomposed as a LHV model of the form
Fig. 2), which were independently generated in two different X
sources, whereas a third station, Bob, performs a Bell-state pða; b; cjx; y; zÞ¼ pðlÞpðajx; lÞpðbjy; lÞpðcjz; lÞ: ð1Þ
measurement over the two other photons (2 and 3), one l
entangled with Alice’s photon and the other entangled with The hidden variable l subsumes all the relevant information in
Charlie’s one. This scheme allows us to observe Bell non-bilocal the physical process and thus includes the full description of the
correlations by violating the Bell-like inequality proposed in source producing the particles as well as any other relevant
refs 19,20. Further, showing that our experimental data is information for the measurement outcomes.
nevertheless compatible with usual LHV models where the In the description of the LHV model (equation (1)), no
independence of the sources is not taken into account, we can mention is made about how the physical systems have been
conclude that the quantum correlations we generate across the produced at the source. For the network we consider here (see
network are truly of a different kind. Moreover, we Fig. 1c), the two sources produce states independently, thus the
experimentally show that beyond a certain noise threshold one set of classically allowed correlations
can enter a region where no standard local causality violation can P
be extracted from the shared state between Alice and Charlie after pða; b; cjx; y; zÞ ¼ pðl1Þpðl2Þ
entanglement swapping and, nevertheless, the correlations of the l1;l2 ð2Þ
entire network can still violate the bilocal causality assumption. pðajx; l1Þpðbjy; l1; l2Þpðcjz; l2Þ;
2 NATURE COMMUNICATIONS | 8:14775 | DOI: 10.1038/ncomms14775 | www.nature.com/naturecommunications
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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14775 ARTICLE
Second harmonic A C
generation crystal
AT
SPDC crystal CT
EPR1 EPR2
AR CR
PBS 1 2 3 4
Alice Charlie
Mororized HWP
B Coincidence
50-50 BS p box
Coupler
BIT B2T
Motorized delay line
BIR B2R
Detector Bob
Figure 2 | Experimental apparatus for the violation of bilocal causality. Two polarization-entangled photon pairs are generated via Spontaneous
Parametric Down-Conversion (SPDC) in two separated nonlinear crystals. Photon 1 (4) of the first (second) pair is directed to Alice’s (Charlie’s) station,
where one of the local observables A0, A1 (C0, C1) is measured via a motorized half-wave plate (HWP) (angles yA and yC) followed by a polarizing BS (PBS).
Photons 2 and 3 are sent to Bob’s station, where a complete Bell-state measurement is performed. A 50/50 in-fibre BS followed by two PBSs allows to
discriminate jC i and jCþ i when the HWP angle yB is set to 0 and discriminate jF i and jFþ i when yB¼45. A motorized delay line is adopted to
control the amount of noise p in the Bell measurement, by changing the photons wavepacket temporal overlap in the BS.
is now mediated via two independent hidden variables l1 and l2 bosonic bunching, ending up in the same BS output (see
(ref. 19), thus defining a bilocal hidden variable model. Supplementary Note 1). A twofold coincidence corresponding to
In our scheme, Bob always performs the same measurement different polarizations in a single BS output branch corresponds
(no measurement choice) obtaining four possible outcomes that to jCþ i detection. A half-wave plate placed before one of the
can be parameterized by two bits b0 and b1. Alice and Charlie can arms of the BS allows, by setting yB¼ 45, to change the
choose each time one of two possible dichotomic measurements. incoming state from jFþ i to jC i and from jF i to jCþ i. In
Thus, in this case the observable distribution containing the full this way, depending on the setting yB, we are able to detect either
information of the experiment is given by p(a, b , b , c|x, z). This jCþ i and jC0 1 i or jFþ i and jF i states. This detection can be
allows us to violate the bilocal causality inequality proposed in interpreted as a probabilistic Bell-state measurement, where, for
ref. 19 and further developed in refs 20,22–25: each pair of incoming photons, only two out of four outcomes
pffiffiffiffiffi pffiffiffiffiffi
B¼ j jþ j j  ð Þ can be unambiguously identified.I J 1 3 In the ideal case, Bob should be able to distinguish between all
The Pterms I and J are sums oPf expectation values, given by of the four Bell states, but this cannot be done by means of
I¼ 1 hAxB0Czi and J¼ 1 ð 1Þxþ zhAxB1Czi, where linear optics
26. By this approach, however, we are able to measure
 4 x;z P 4 x;z
¼ ð Þaþ by þ c ð j Þ all the combinations (A0, C0),(pAffi0ffi , C1),(A1, C0),(A1, C1) opf ffiffitheAxByCz a;b ;b ;c 1 p a; b0; b1; c x; z and x, z, a,0 1 observables A0¼C0¼ðsz þ sxÞ= 2 and A1¼C¼ 1
¼ðsz sxÞ= 2 of
b0, b1, c 0, 1. Inequality (equation (3)) is valid for any classical Alice and Charlie, for the two possible yB configurations. The
model of the form (equation (2)) and its violation demonstrates fair-sampling assumption allows us to reconstruct from these data
the non-local character of the correlations we produce among the the probability p(a, b0, b1, c|x, z) and then to compute the
network. quantities I and J, which appear in equation (3).
The maximum value reached in our experimental setup was
Violation of the bilocal causality inequality. We generate B¼1:268 0:014, corresponding to a violation of inequality
entangled photon pairs via type-II spontaneous parametric down- (equation (3)) of almost 20 sigmas. This value is fully compatible
conversion process occurring in two separated nonlinear crystals with a theoretical model that considers both colored and white
(Einstein-Podolsky-Rosen (EPR) 1 and EPR 2) injected by a noise in the state generated by the spontaneous parametric down-
pulsed pump laser (see Fig. 2). When a pair of photons is gen- conversion process sources and takes into account the partial
erated in each of the crystals, one photon from source EPR 1 distinguishability of the generated photons (see Supplementary
(EPR 2) is sent to Alice’s (Charlie’s) measurement station, where Note 2).
polarization analysis in a basis that can be rotated of an arbitrary Next, we address the robustness of the bilocal causality
angle yA (yC) is performed (see Supplementary Note 1). The inequality violation with respect to experimental noise. To this
other two photons (2 and 3) are sent to Bob’s station, which aim, we tuned the noise in the Bell-state measurement by
consists of an in-fibre 50/50 beam splitter (BS) followed by two modifying the temporal overlap between photons 2 and 3. This
polarizing BSs for the polarization analysis of each of the outputs. can be achieved by using a delay line before one of the two inputs
In the ideal case (which relies on perfect photons’ indis- of the BS, thus controlling the temporal delay between these
tinguishability), an incoming jC i (singlet) state will feature photons (see Fig. 2). We can therefore define a noise parameter p,
antibunching, giving rise to coincidence counts at different out- which is equal to 1 in the ideal case of a perfect Bell-state
puts of the BS. All the other cases (triplet states) will experience measurement and is equal to 0 when the probability of having a
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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14775
a b J
1.4 1.0
1.3
0.8 B
A C
1.2
B
B B
1.1 0.6
A C A C A C1 2
1 2
1.0 B
0.4 A C1 2
B
0.9
A C
1 2
0.8 0.2 B
A C
0.7
B
0.0 I
0.6 A C1 2
0.0 0.2 0.4 0.6 0.8 1.0
p –0.2 0.0 0.2 0.4 0.6 0.8 1.0
Figure 3 | Experimental violation of bilocal causality. (a) Measured quantity B as a function of the noise parameter p, with fixed (blue circles) and
optimized (orange squares) measurements settings. Theoretical predictions are shown by blue- and orange-shaded regions compatible with our state
preparation and varying the other noise parameter p. The regions are obtained considering the propagation of the uncertainty in the experimental
estimation of noises and are bounded by 1 s.d. upper (dashed) and lower (line) curves. The dotted horizontal line indicates the bound of the inequality
(equation (3)), whereas error bars indicates 1 s.d. of uncertainty, due to Poissonian statistics. (b) Measured values in the I–J plane. Error bars show 1 s.d. for
both I and J values. The dashed line bounds the bilocal region as prescribed by inequality (equation (3)). Green lines define the local set and the white area
represents correlations, which are compatible with local models but incompatible with bilocal causality assumption. The grey area shows the set of
correlations, which are incompatible with both local and bilocal models.
successful measurement is 1/2. This parameter can be tuned from constraining p(a, b0, b1, c|x, z) compatible with LHV models.
pmax to zero by changing the delay from zero to a value larger Apart from trivial ones, there are 61 of these inequalities and we
than the coherence time of the photons. have checked for all the collected data with different noise
The measured values of B versus p are shown in Fig. 3a. As parameter p whether they are violated. The results are shown in
expected, the violation decreases with increasing noise19,20. This Fig. 4a. It can be seen that none of the points (even those that do
plot shows two sets of different data points: considering a fixed violate the bilocal causality inequality (equation (3)), as shown in
measurement basis (optimal in the absence of the additional Fig. 3) show any significant evidence (taking into account the size
noise) and optimizing the measurement basis at Alice and of the error bars) for the violation of any of the all LHV
Charlie’s stations as a function of p, that is, changing the constraints.
measurement basis in order to counteract noise effects (see Finally, we addressed the question whether, in an entanglement
Supplementary Note 2). In both cases, our setup can tolerate a swapping scenario, bilocal causality violation could represent a
substantial amount of noise before inequality (equation (3)) is stronger test rather than the usual CHSH violation29, in order to
not violated anymore, but it is clear how the optimization certify non-local correlations in presence of experimental noise.
increases both the degree and region of bilocal causality We therefore performed a tomography of the quantum state
violation. shared between Alice and Charlie upon conditioning on Bob’s
Another relevant way to visualize the non-bilocal correlations outcome (that is, entanglement swapped state) followed by an
generated in our experiment and its relation to usual local models experimental test of bilocal causality (see Supplementary Note 4).
is displayed in Fig. 3b. A bilopcalffiffiffimffiffi odpel ffiffi(ffidffiffi efined by equation (2)) This allows us to compare our experimental bilocal causality
must respect the inequality jIj þ jJj  1, while a standard violation with the maximum possible CHSH of the swapped state
LHV model (defined by equation (1)) in turn fulfils jIj þ jJj  1. in different regimes of noise30. Figure 4b clearly shows the
As shown in Fig. 3b, the measured values of I and J are clearly existence of quantum states, which violate bilocal causality (even
incompatible with bilocal causality (apart from the cases with the without any settings’ optimization) but cannot violate the CHSH
highest amount of noise) and behave in good agreement with inequality, thus turning unfeasible any protocol10–12 based on its
the theoretical model. Moreover, it clearly shows how optimizing violation.
the measurement settings improves the robustness of violation
against noise.
Discussion
Characterizing non-bilocal correlations against LHV models. Our results provide an experimental proof-of-principle for
The data in Fig. 3b show that the observed values for I and J do network generalizations of Bell’s theorem. However, similarly to
not violate the corresponding LHV inequality. However, this only any Bell test31, our violation of the bilocal causality inequality is
represents a necessary condition. To definitively check whether subjected to loopholes, in particular the locality and detection
we are really facing a new type of local causality violation beyond efficiency loopholes, as the parties are not space-like separated
the standard LHV model (equation (1)), we also checked that all and we make use of the fair-sampling assumption. Given the
Bell inequalities defining our scenario are not violated in the nature of our experiment, a new loophole—similar to the
experiment. In general, given an observed probability distribu- measurement independence loophole in Bell’s theorem27,32—is
tion, it is a simple linear program to check if it is compatible with also introduced if the sources of states cannot be guaranteed to be
LHV model (see, for example, ref. 27 for further details). truly independent. Regarding usual Bell tests, it was not until
Equivalently, noticing that a LHV model defines a polytope of recently that such loopholes were finally overcome2–4. Thus, from
correlations compatible with it28, one can derive all the Bell the practical perspective it would be highly desirable to design
inequalities constraining that model. As described in the future experiments achieving that also for more complex
Supplementary Note 3, we have derived all the Bell inequalities networks.
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NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14775 ARTICLE
a LHV Bound
0.00 p
1.0 1.0
–0.05
0.8 0.8
–0.10 0.6 0.6
–0.15 0.4 0.4
0.2 0.2
–0.20
0 0
–0.25
–0.30 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61
b 1.3
B B
1.2 A C A C
A C A C
1 2 1 2
1.1
1.0
0.9
B B
0.8 A C A C
A C A C
1 2 1 2
0.7
0.0 0.5 1.0 1.5 2.0
Smax Maximal CHSH parameter
Figure 4 | Experimental test of LHV models. (a) Experimental violation values for all the 61 Bell inequalities compatible with a LHV model. Each column
corresponds to a different inequality, where a resulting value 40 is not compatible with an LHV model. Each point’s colour represents the estimated
amount of noise p, from dark blue (p¼ 1, that is, absence of noise), to light blue (p¼0, that is, maximum noise). Theoretical predictions are shown in the
background, the red to yellow colour transition representing the dependence from p. Squares (circles) represent those points, which violate (do not violate)
the bilocal causality inequality (equation (3)). (b) Comparison between experimental bilocal causality parameter and maximized CHSH parameter
(multiplied by a factor 1/2 so that the local bound of the CHSH inequality is set to 1) in different regimes of noise. Bilocal causality test is performed with
fixed non-optimized measurement settings, whereas CHSH maximum parameter is computed applying the Horodecki criterion (ref. 30) to a partial
quantum state tomography (red points) or a complete quantum-state tomography (blue points) of the quantum state shared between A and C after the
entanglement swapping protocol (that is, conditioned on singlet state outcome in station B). The purple point was evaluated directly testing both bilocal
causality and CHSH in a particular regime of low noise. Circles (squares) represent entangled (separable) quantum states, where the degree of
entanglement was computed via the partial transpose46. The lower-left region is compatible with both models, the upper-left region denotes
incompatibility with bilocal causality, the upper-right region denotes violation of both the bilocal causality and CHSH inequalities, whereas the lower-right
region is characterized by only CHSH violation. Error bars indicates 1 s.d. of uncertainty, due to Poissonian statistics.
From a fundamental perspective, recent results21,27,33–40 at the even larger quantum networks as the one shown in Fig. 1d. For
interface between quantum theory and causality have shown that sufficiently long networks, the final quantum state swapped
Bell’s theorem represents a very particular case of much richer between the end nodes may be separable and thus irrelevant as a
and broader range of phenomena that emerge in complex quantum resource. Still, the correlations in the entire network
networks and that hopefully will lead to a deeper understanding might be highly non-local25, allowing us to probe a whole new
of the apparent tension between quantum mechanics and our regime in quantum information processing. Finally, we notice
notions of causal relations. Furthermore, given the close that during the review process of this work, an independent
connections between causal inference and machine learning41, experimental investigation of the bilocal causality violation has
it is pressing to consider what advantages the recent progresses in appeared44.
quantum machine learning42,43 can provide in such a causal
context.
From a more applied perspective, such generalizations offer an Methods
almost unexplored territory and it is still unclear how to use this Experimental details. Photon pairs were generated in two equal parametric
new form of non-local correlations in information processing. As down conversion sources, each one composed by a nonlinear crystal beta barium
we showed here, we can still violate a bilocal causality inequality borate (BBO) injected by a pulsed pump field with l¼ 392.5 nm. The data shown
in Figs 3 and 4a and the purple point in Fig. 4b were collected by using 1.5mm -
even if the data admits a LHV model where the independence of thick BBO crystals, whereas for the red and blue points in Fig. 4b we used 2mm-
the sources is not taken into account. That is, quantum states thick crystals to increase the generation rate. After spectral filtering and walkoff
generating classical correlations in conventional scenarios can compensation, photonpaffirffi e sent to the three measurement stations. The observable
become powerful resources in a network, thus hopefully enlarging A0, that is, ðsz þ sxÞ= 2, correpspffiffionds to a half-wave plate rotated by y
A
0 ¼ 11.25,
whereas A1, that is, ðsz  sxÞ= 2, corresponds to yA1 ¼ 78.75. Analogously, Cour current capabilities to process information in a non-classical 0and C1 can be measured at Charlie’s station using the same angles yC0 ¼ yA0 and
way. For instance, a natural next step is to experimentally realize yC1 ¼ y
A
1 .
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Data availability. The data that support the findings of this study are available 33. Oreshkov, O., Costa, F. & Brukner, C. Quantum correlations with no causal
from the corresponding author upon request. order. Nat. Commun. 3, 1092 (2012).
34. Wood, C. J. & Spekkens, R. W. The lesson of causal discovery algorithms for
quantum correlations: causal explanations of bell-inequality violations require
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