Experimental nonlocality-based network diagnostics of multipartite entangled states

We introduce a novel diagnostic scheme for multipartite networks of entangled particles, aimed at assessing the quality of the gates used for the engineering of their state. Using the information gathered from a set of suitably chosen multiparticle Bell tests, we identify conditions bounding the quality of the entangled bonds among the elements of a register. We illustrate the effectiveness of our proposal by characterizing a quantum resource engineered combining two-photon hyperentanglement and photonic-chip technology. Our approach opens up future studies on medium-sized networks due to the intrinsically modular nature of cluster states, and paves the way to section-by-section analysis of larger photonics resources.


Theoretical background
Consider N observers each with the possibility to choose between two dichotomic observables A j (n 1 ) and A j (n 2 ), determined by some local parameters denoted by n 1 and n 2 , which means that each local observer can choose independently two arbitrary directions. The correlation function is then the average over many runs of the experiment: Note that in general it is true that: With this choice of the function S, one recovers the Mermin-Ardehali-Belinskii-Klyshko (MABK) inequalities 12 , which are a particular instance of a Multipartite Nonlocality Inequality (MNLI). Let us now derive the correlation function in the specific case of N = 4, i.e. the expectation value of the tensor product of four general rotations, over our state, the cluster state. We firstly define the Pauli matrices: and σ = (σ x , σ y , σ z ).
From the general form of S shown above, the correlator has the following expression: The Werner, Wolf, Zukowski and Brukner , (WWZB) parameter is composed by sixteen correlators, each for a two-setting, four-party configuration 12 : Notice that all the θ 's angles are π 2 . This means that the all the observables used in the WWZB inequality for the cluster state are off-diagonal: Each term in the WWZB parameter, with the angles chosen beforehand, contributes to the violation of the local realistic constraint by a certain amount, that we report in the column labeled c teo of Table S.I, after renaming the correlators as in the following: c 9 = C(θ α2 , φ α2 , θ β 1 , φ β 1 , θ γ1 , φ γ1 , θ δ 1 , φ δ 1 ) c 11 = C(θ α2 , φ α2 , θ β 1 , φ β 1 , θ γ2 , φ γ2 , θ δ 1 , φ δ 1 ) Therefore, in order to obtain all the sixteen correlators given above, we need to perform the 4-qubit operators shown in Table  S.I.
Each observable for the single qubit is in the form: 0 e iφ e −iφ 0 with a suitable choice of the phase. Therefore its measurement amounts to projecting its eigenvectors (−e iφ , 1) and (e iφ , 1). For instance, taking φ = π 4 , the eigenvectors of J are: . Performing a projection on these eigenvectors means finding the unitary transformation that rotates the computational basis into the rotated basis of these eigenvectors, and then invert it. This operation can be implemented by a combination of a HWP and a QWP: The inverse is then easily derived by noting that, if we use φ = π 4 : In this way we can perform the projection on the rotated basis of the eigenvectors of J and K, while for X and Y we can use the standard tomographic settings. The same applies for the observables in the path degree of freedom: a phase shifter and a beamsplitter, allowing to measure all observables in the form we need.

Experimental realization
The experimental setup The experimental setup consists of an hyperentanglement (HE) photon state source, a manipulation stage (integrated photonic circuit) and a detection apparatus.
In this work we use both path and polarization degrees of freedom to build a 2-photon 4-qubit linear cluster state from an hyperentangled photon state source. This source generates two photons, entangled both in path and polarization degrees of freedom: where A and B label the two photons, r A , A , r B , B label the path modes and H, V label the two possible polarizations for each photon. Each of the four spatial modes can be manipulated independently by custom-made half mirrors, half lenses and a single mode fiber array, with four GRIN lenses at the input side. The output side of the fiber array is coupled to a glass chip, manufactured with the femtosecond laser writing tecnique: it consists of two directional couplers, mainly two beam splitters (BSs). r A and A are coupled on the first BS, while r B and B on the second. This integrated quantum circuit preserves polarization and, when thermically insulated, grants phase stability thus allowing to perform arbitrary basic operations on path-encoded qubits, without compromising the polarization encoding. Besides, polarization compensation has been performed individually for each of the four modes with a set of two QWPs and a HWP, by which it is possible to compensate any arbitrary rotation of the polarization due to fibers and chip. The output of the chip is collected by a 10x objective and then coupled to the detectors through multi mode fibers, in order to measure coincidences between the output modes r A and B (two of the outputs of the two BSs inside the chip). To ensure the photon indistinguishability two interference filters centred at 710 nm select a 5 nm bandwidth. Passing through a set of HWP, QWP and PBSs the photons are analyzed in polarization. In this way we are able to characterize the HE state by performing a complete two-qubit tomography on each of the couples | A r B and |r A B . The tomography of the states are reported in Fig.S1. Fidelity and concurrence of such tomographies are reported in the table S.II.

Building the cluster state
Starting from the HE state, we engineered the four-qubit linear cluster state, encoded in path and polarization degrees of freedom, as shown above in Eq.S8: The desired cluster state can be obtained from the hyperentangled state by placing a zero-order HWP at zero degrees on mode A , which performs the transformation: We verify that the HWP implements the correct transformation by performing polarization tomography on projecting on each of the path pairs | A r B and |r A B . The results are reported in Fig.S2, along with the values of the fidelities and concurrences shown in Table S.III.

The WWZB parameter
As we mentioned above, operations on polarization-encoded qubits can be easily performed by rotating the analysis waveplates, while the two beam splitters inside the chip and the tilting of two additional phase shifters (one for photon A and one for photon B) perform transformations on path. The experimental values of each correlator c i is reported in Tab. S.I. The correlators,

5/13
Figure S1. Reconstruction of density matrices obtained with θ = 0: tomographies of | A r B and |r A B respectively (on the left side the real part, on the right side the imaginary part). As expected, they both show |φ + behaviour. summed as in Eq. S12, give the experimental WWZB parameter for our resource:

State
where the error has been calculated by propagating the errors of the coincidences counts, assuming them to be Poissonian distributed. The value obtained is 11 standard deviations away from the bound of local realistic theories.

Non Locality based assessments of multipartite quantum resources
In this section we demonstrate that it is possible to infer the quality of the cluster resource by considerations on its non-locality. Given a four-qubit linear cluster state it is always possible to measure a σ x operator on one of the four qubits, projecting its state on one of his eigenvalues ( |+ or |− ). The resulting three-qubit state is again a cluster state, which violates a general Mermin inequality (i.e. a WWZB parameter) that involves now three qubits instead of four. This three-qubit state is generally different depending on which of the four qubits has been crossed out by the measurement, implying that the measures that characterize the WWZB parameters to calculate can be different. In this way it is possible to study non locality properties of different groupings of the qubits: 1-2-3, 1-2-4, 1-3-4, 2-3-4, having used always the same convention for the order of the qubits (π A , π B , k A , k B ), that means the first two are polarization encoded and the last two path encoded.
This process can be iterated performing a second measurement on one of the three qubits remained. In this case we obtain a two-qubit entangled state, that can be tested with a two-qubit Mermin inequality that reduces to a simple Bell test. The state is Figure S2. Examples of density matrices obtained: tomographies of | A r B and |r A B respectively (on the left side the real part, on the right side the imaginary part). As expected, they show |φ + and |φ − behaviors respectively.
decided by the eigenstates considered in measuring the two crossed out qubits. Given the violations of all Mermin inequalities for all the eleven possible four-, three-and two-qubit subgroupings, we will show that it is possible to quantify the quality of the entangelment that characterizes our resource.
The bound imposed by local realistic theories is 2 3 = 8, then |WW ZB 3 | < 8 is the inequality to beat. To construct the WWZB 3 parameter, we have to find the angles that maximize its value, which in this case can attain the value of 16. However, we can exploit the same measurement of J and K performed in the four-qubit case at the cost of reaching a maximum violation of 11.31 instead of 16. The angles that allow this violation are: This means that the observables used in the WWZB 3 inequality for the GHZ state are: Each term in the WWZB parameter, with the angles identified by Eq. S21, is a 3-qubit observable and contributes to the violation of the local realistic constraint according to the following definition: We have measured the values for each corrrelator as we did for the four-qubit case. Results are reported in Table S.IV. The WWZB 124 parameter is: where the error has been calculated by propagating the errors of the coincidences counts, assuming them to be Poissonian distributed. The value obtained is 7 standard deviations away from the bound of local realistic theories. This procedure can be easily repeated for all the possible groupings of three qubits out of four, crossing out one qubit at a time, projecting it on a σ x eigenvalue. It must be underlined that each of the resulting three-qubit GHZ state can always be reduced in the |ψ 124 form as in Eq.S19 by operating a simple one qubit rotation. All the results, which are reported in the main text, are inconsistent with the expectation of local realistic theories.

8/13
Two-qubit groupings The strategy shown in the previous section can be iterated performing a projection on two qubits instead of only one, obtaining a two-qubit entangled state. This can be tested with a Mermin inequality that reduces to a simple Bell test, for all the six possible two-qubit groupings out of four: 1-2, 1-3, 1-4, 2-3, 2-4, 3-4. Generally the kind of state in which the system collapses depends on the eigenstates considered in measuring the σ x operator on the two crossed out qubits. Besides we can always rotate the final state with a single-qubit operation, being able to use the same measures taken in the four-qubit case to maximise the violation also in a two-qubit system.
If we measure the σ x operator on the second and third qubit (first path encoded) and consider the projection on |+ 2 and |+ 3 eigenstates, the state of the remaining 1-4 qubits is therefore: With this kind of state we can build a two-qubit Mermin inequality as it has been done in the four-and three-qubit case.
The WW ZB 2 parameter is now composed by four correlators, each for a two-setting, two-party configuration, as in the Bell inequality: The bound imposed by local realistic theories is 2 2 = 4, then |WW ZB 2 | < 4 is the inequality to beat. We proceed as above to find the angles that maximize its value, in this case 4 √ 2. The angles that allow this violation are: The observables used in the WWZB 2 inequality are: Each term in the WWZB parameter, with the angles defined in S26, is a two-qubit observable and contributes to the violation as indicated by c teo , while experimental results are indicated by c exp in Table S.V. The WWZB 14 parameter is: The value obtained is 4 standard deviations away from the bound of local realistic theories. We have followed the same procedure for the remaining cases of two-qubit subgroupings. The results are summarised in the main text.

Assessment of correlations quality
Given the violations of all the Mermin inequalities for all the eleven possible four-, three-and two-qubit subgroupings, we can reconstruct the strength of the corelations between the cluster components, i.e. the quality of the entangelment shared among the four qubits. This can be done by comparing the results of the WWZB parameters obtained, with the ones of a theoretical specular resource corrupted by noise.

9/13
Correlators Operation  Table S.V. Contribution of each correlator c i to the WW ZB 2 parameter defined in Eq.S25. In the second column, the corresponding operator to perform to the cluster state. c teo is the theoretical expected value, c exp is the experimental result.

Noisy entangling gates
As shown in Ref. 4 and Ref. 5 , a for-qubit linear cluster state can be obtained applying a chain of three C-phase gates on an initial | + + + + 1234 state. If it is prepared in states |0000 1234 or |1111 1234 , no entanglement will build up. The three C-phase gates actually build the entanglement links between the cluster's qubits, according to the scheme in Fig. S3: Figure S3. Building of entanglement on an initial separable | + + + + 1234 state. A linear cluster state can be easily obtained by chaining three C-phase gates on the four initial qubits: C-PHASE 1,2 , C-PHASE 2,3 , C-PHASE 3,4 .
The application of a chain of the three operators on the prepared state produces the following linear cluster state: that can be reduced to our linear cluster state by simply applying the unitary rotation: Here H is the single qubit Hadamard gate. To account for imperfections, we apply to the initial state a probabilistic C-phase gates, i.e. gates that acts with a probability p as a C-phase and fail completely with a probability (1 − p). We can write the effect of these gates as a Kraus maps: When these three three maps act on the initialρ in = (| + + + + 1234 )( 1234 + + + + |) state in a chain, we obtain the state: In this way, after applying the unitary U, we have expressed the linear cluster state, Eq. S8, as a function of the probabilities 10/13 that describe the action of the probabilistc C-phase gates. From the analitic expression of the state we can calculate the expected values of the WWZB parameters.
Now we can find the values of p 1 , p 2 , p 3 that reproduce the WWZBs obtained in the experiment, by a maximum likelihood fitting: having imposed as constraints that each of the WWZB parameters is Gaussian distributed assuming its error as standard deviation.

Dephasing channel
An analogous method of evaluating the quality of the cluster resource is comparing it with a resource damaged by a dephasing channel acting on each component of the system. This is an alternative perspective that focalizes on the resource from the point of view of the single qubits quality rather than the quality of a two-qubit link.The dephasing channel is a Kraus operator that acts as follows: where the Kraus operators E k are: so that the dephased state is: If p=1 the state is left unchanged. If p=0, only the diagonal terms of the density matrixρ survive, i.e. the state is completely mixed, having lost all the correlation terms.

11/13
In our case the overall Kraus map, acting on each of the components of cluster state, is the Kronecker product of four dephasing channels as in S34: where ε i (p i ) represents the i th dephasing channel acting on the i th qubit. This overall map is still a Kraus Map, composed by 3 4 = 81 Kraus operators (the product of the sum over three Kraus operators each qubit). Applying this overall map we can therefore obtain the dephased cluster state as a function of (p 1 , p 2 , p 3 , p 4 ) (i.e. the non-dephasing probabilities for each qubit).
Proceeding in an analogous way as in the case of noisy gate, we can determine the degree of dephasing characterizing our resource. Again we can calculate all the eleven WWZB parameters, now functions of (p 1 , p 2 , p 3 , p 4 ), using exactly the same procedure described in the previous section, that means operating the same measures to build the same correlators. (S40) Note that, applying the overall map on the original undisturbed density matrix of the cluster state, with (p 1 , p 2 , p 3 , p 4 ) = (1, 1, 1, 0) we kill not only the correlations between the 4 th qubit and the others, but also the correlations of the 3 rd and the others. In other words we are damaging the correlations between the two couples 1 st − 2 nd and 3 rd − 4 th as we could have done by imposing (p 1 , p 2 , p 3 , p 4 ) = (1, 1, 0, 1) or (1, 1, 0, 0). The same can be observed damaging the qubit 1 st and 2 nd . The loss of all correlations can be therefore observed when at least three qubits are ruined with a p i = 0: (1, 0, 0, 0) , (0, 1, 0, 0) , (0, 0, 1, 0) , (0, 0, 0, 1) or (0, 0, 0, 0). This symmetry can be explained by the fact that the entangling gate that actually builds the cluster state, starting from a HE state, is a C-phase: this relies intrinsically on the action of a phase that, if destroyed, destroys the nature of the cluster itself. It is worth noting that some WWZBs have the same analytic expression. For example WW ZB 124 and WW ZB 123 or WW ZB 14 , WW ZB 13 , WW ZB 24 and WW ZB 23 . Furthermore it must be highlighted that the couple p 1 and p 2 and the couple p 3 and p 4 can never be determined singularly, because they always occur as a product. These two characteristics seem reasonable since the symmetries of the channel combined with the ones of the state discussed above. We can find the values of p 1 , p 2 , p 3 , p 4 that reproduce the WWZBs obtained. The strategy is exactly the same as in the previous case: finding (p * 1 , p * 2 , p * 3 , p * 4 ) that minimize the difference between each of the WWZBs espressed above and the corresponding WWZB experimentally obtained. The results are: p * 1 = 0.954 ± 0.038p * 2 = 0.957 ± 0.038 p * 3 = 0.949 ± 0.045p * 4 = 0.940 ± 0.045.
having imposed as constraints that each of the WWZB parameters is Gaussian distributed assuming its error as standard deviation.