Experimental violation of local causality in a quantum network

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.


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2, corresponds to θ A 1 = 78.75 • . Analogously, C 0 and C 1 can be measured at Charlie's station using the same angles θ C 0 = θ A 0 and θ C 1 = θ A 1 . By measuring fourfold coincidences for all the possible combinations (θ A i , θ C j ), with i, j = 0 or 1, and performing the measurements with θ B = 0 • and with θ B = 45 • , we are able to reconstruct the probability table to extrapolate the quantities I and J which appear in Eq. (3) of the main text. The noise parameter p was defined as p = 2P B success − 1, where P B success is the probability that the Bell-state measurement returns the correct outcome given an input Bell state. Since a Bell-state measurement via BS can only mix |ψ − and |ψ + (or |φ − and |φ + ) outcomes, depending on photons indistinguishability, the minimum P B success will be 1/2 and it will occur in case of totally distinguishable photons. The degree of indistinguishability was computed via a Hong-Ou-Mandel dip in the fourfold coincidence counts with H (V) polarized photons in B and V (H) polarized photons in stations A and C. Indeed P B success = [2C(∞) − C(∆s)]/2C(∞) where C(∆s) is the count value at distance ∆s from dip center and C(∞) is the count value at the plateau. This leads to the estimation p = [C(∞) − C(∆s)]/C(∞) whose maximum experimental value was p max = 0.846 ± 0.007.

Noise Modeling
In an entanglement-swapping scenario there are different sources of noise that must be considered. The main ones affect state preparation and Bell-state measurement. Let us recall the definition of the correlation functions used in the main text given by This mean value can be evaluated with two different approaches. The first one exploits the evaluation of the probability p Q (a, b 0 , b 1 , c|x, z), which reads where P i denotes the projection operator on the eigenstate |i and AB ⊗ BC represents the quantum state density matrix.
The second approach consists in the modeling of the quantum operators associated with equation (1), followed by a direct evaluation of the mean value through the equation: Supplementary Figure 1: Table showing the expected values of the operator B y , varying y, b 0 and b 1 .
This second approach, which is computationally easier, allows to directly evaluate the mean values after a simple modelling of the measurement operators. It is thus possible to define the following operators where A x and C z are general single qubit projective measurements with eigenvalues 1 and −1.
The possible values of the operator B y are shown in Fig. 1.
The operator B y is then defined as which relates each value of y = 0, 1 with its correct set of outcomes.

Imperfect Bell-state measurement
Let us now derive a generalization of equation 5 for an imperfect Bell-state measurement, which we model with a well defined POVM.

Single qubit POVM
In order to introduce this model it is convenient to discuss the simpler case of a single qubit projective measurement, e.g. σ z = |0 0| − |1 1|. Imperfections in the measurement can be seen as the probability 1 − p of no success, i.e. to read |0 when the real state is |1 and viceversa. It is thus necessary to find two positive operatorsF 1 andF −1 which satisfy the following constraints This can be obtained by defining the following operatorŝ Further, one can see that for an arbitrary state |ψ = α|0 + β|1 the following relation holds thus showing that these operators represent a well defined POVM.
This leads us to the definition of the following measurement operatorÔ which models an imperfect measurement.

POVM generalization of B y
A Bell-state measurement in a Beam Splitter (BS) strongly relies on the indistinguishability of the photons. It has been shown [1] that partial distinguishability in Bell-state measurement acts mixing the |ψ + results with |ψ − ones and viceversa (and similarly for |φ + and |φ − states), but it doesn't mix elements belonging to these two different categories. In this section we will develop the single qubit case shown before, extending it to the two-qubit case, and taking into account that an imperfect Bell-state measurement can only mix |φ + with |φ − or |ψ + with |ψ − .
Following the previous conclusions we can rewrite equation (5) as And then apply the substitution where p was chosen as a parameter of imperfection such that the probability of success is never below 50% for 0 ≤ p ≤ 1.
Equation (11) gives the correct probabilities and at the same time it satisfies that for an arbitrary state These constraints guarantee that F i operators shown in equation (11) represent a well defined POVM. The imperfect Bell-state measurement is then modeled as Presence of noise in the quantum state preparation In our entanglement swapping scenario two pairs of entangled photons in the singlet state are generated by two independent SPDC sources. It has been shown [2] that these sources suffer from two different kinds of noise: 1) White noise (i.e. isotropic depolarization): where I represents the identity matrix.
2) Colored noise (i.e. depolarization on a preferred direction): The state given by this mixture of noises can be modeled as: where v X stands for the total noise while λ X is the fraction of colored noise and where the label X can represent either the measurement station A or the measurement station C.

Violation of bilocal causality depending on noise
The presence of noise in the experimental setup tends to destroy quantum correlations, reducing QM's Bell nonlocal behaviour. It is thus essential to estimate the dependence of the parameter B (defining the bilocality inequality) from experimental noise in our model. This can be achieved using equations (14) and (17) and applying equation (3). Let Alice and Charlie perform the following measurements (A 0 , A 1 and C 0 , C 1 respectively) : After several algebraic manipulations we obtain where we defined Hereafter we will restrict to the case of interest given by Thus, equation (19) leads to In the following to analyze the maximal value of B as a function of noise. This can be done by considering α and β as parameters and then maximizing the function B(θ, φ). This leads to the equations These system of equations admits only solutions constrained by The maximization of equation (22) is thus equivalent to maximize the simplified expression that leads to Importantly, these analytical results allow for the evaluation of the best experimental settings (i.e. θ(α, β)) given particular experimental conditions (i.e. α and β).
It is now interesting to study how equation (26) behaves in different regimes of noise. When dealing with colored noise only (i.e. λ = 1), so that β(v, λ = 1) = 1, it is possible to perform a full optimization, allowing for bilocality violation for all possible values of v and p.
In the opposite case (i.e. λ = 0, white noise only) we have β(v, λ = 0) = v 2 . The noise parameter v can be factorized while the dependence of θ(α, β) from v vanishes. This leads to the conclusion that no settings optimization can counteract the effects of white noise in the state preparation. However, performing the optimized measurements settings prescribed in equation (26) allows also to maximize B with respect to the imperfection in the Bell-state measurement due to partial distinguishability in the BS (i.e. parameter p), which means that an advantage can always be obtained by such optimization.

Violation of bilocal causality for a wider class of measurements
We now extend the conclusions of the previous section to the case in which Alice and Charlie can perform a wider class of measurements. First, let us define Suppose A and C perform the following measurements: This choice leads to Further, it is possible to show that In order to maximizeB(θ 0 , θ 1 , φ 0 , φ 1 ), it is necessary to find its stationary points, solving the equations: where we have defined Excluding singular points, the solutions of the system (31) are constrained by: The problem of finding stationary points forB(θ 0 , θ 1 , φ 0 , φ 1 ) can thus be simplified, leading to This demonstrates that equation (22) maximizes equation (29). We conclude then that the measurements defined in equation (18) represent the subset of the class given in equation (28) which provides a full optimization of the parameter B. Numerical maximization of equation (29) was also performed and numerical results perfectly agree with these analytical conclusions.

Local and bilocal correlations
We focus now on the relation between QM and both the local and the bilocal set. As discussed in the main text, in terms of the observables I and J the following inequalities hold Local set : L ≡ |I| + |J| ≤ 1, Bilocal set : B ≡ |I| + |J| ≤ 1.
QM mean values I and J depend on several factors, such as the measurements performed in A and C, the quantum state preparation an the noise level under consideration in the Bell-state measurement. Considering our noise modeling and performing the optimized measurements described in equation (26) it is possible to obtain , where the sign ± depends on the choice of φ(α, β) = nπ ± θ(α, β) , n ∈ Z. For the case v = 1 (regardless of λ) or λ = 1 (regardless of v) equation (37) leads to the QM values shown in Fig. 2-1 (red line). It is interesting to compare this result with the orange line, which shows QM behaviour in case of fixed measurement angles (θ = π/4 and φ = nπ ± π/4 , n ∈ Z). It is clear that the maximization protocol keeps QM out of the bilocal set. In the presence of white noise, after a certain threshold value, measurement optimization is not able to keep QM completely out of the bilocal set (see Fig. 2-2). It is worth pointing out that equation (37) never allows QM to exit the local region, regardless of the value of v, p, and λ. As a matter of fact, no violation of the inequality |I| + |J| ≤ 1 is possible within quantum mechanics thought some non-signaling (post-quantum) correlations can do so [3].
If AC is written in the form given by equation (41), the computation of T gives us that T T T eigenvalues : It is easy to show that (47) and (48) are equivalent. In fact and thus m 2 1 and m 2 2 will be given by β(v, λ) 2 and α(v, p) 2 .

Comparison between the protocols
The comparison between the bilocality and CHSH (for the swapped state) inequality violations is based on equations (26) and (47) which provide the maximum violation that can be obtained given a certain quantum state, for the two different protocols. This analysis is summarized in Fig. 3. It can be seen that if white noise is present (even partially), there is a wide region of v and p where only bilocality inequalities can be violated. In the case in which both inequalities can be violated, bilocality assumption provides a greater violation. In fact, since α(v, p) ≤ 1 and β(v, λ) ≤ 1 are satisfied, equations (26) and (47)

Excursus on the two parties Bell scenario
It is interesting to investigate how CHSH maximization works when dealing with the Bell scenario depicted in Fig.  1-a of the main text. This is not directly related to our main argument, which concerns a quantum network of three parties, but could be very useful in order to characterize the best measurements settings in case of a simpler Bell scenario (for different noises involved). Suppose that A and C directly shares a state analogous to the one defined in equation (17), with parameters v and λ.