Machine-learning-based device-independent certification of quantum networks

Witnessing nonclassical behavior is a crucial ingredient in quantum information processing. For that, one has to optimize the quantum features a given physical setup can give rise to, which is a hard computational task currently tackled with semidefinite programming, a method limited to linear objective functions and that becomes prohibitive as the complexity of the system grows. Here, we propose an alternative strategy, which exploits a feedforward artificial neural network to optimize the correlations compatible with arbitrary quantum networks. A remarkable step forward with respect to existing methods is that it deals with nonlinear optimization constraints and objective functions, being applicable to scenarios featuring independent sources and nonlinear entanglement witnesses. Furthermore, it offers a significant speedup in comparison with other approaches, thus allowing to explore previously inaccessible regimes. We also extend the use of the neural network to the experimental realm, a situation in which the statistics are unavoidably affected by imperfections, retrieving device-independent uncertainty estimates on Bell-like violations obtained with independent sources of entangled photon states. In this way, this work paves the way for the certification of quantum resources in networks of growing size and complexity.

Figure 1

Causal structures and admitted correlations set. (a) Causal structure of the standard Bell's scenario with a common source $\rho$, whose generated bipartite state is shared between the nodes $A$ and $B$, performing measurements labeled by $X$ and $Y$. (b) Sketch of the correlation (convex) sets in a standard Bell bipartite scenario [61]: local $L$, quantum $Q$, and no-signaling $NS$, where $L\subset Q\subset NS$. (c) Instrumental causal structure, where node $A$ is influenced by a classical variable $X$ (the instrument) and directly influences $B$. Also in this case there is only one common cause $\rho$. (d) Sketch of the nonconvex sets of correlations produced in the bilocality scenario. (e) DAG of the bilocality scenario, where two independent sources share correlations among three nodes: ${\rho }_1$ between $A$ (with measurement choice $X_1$) and $B$ (with measurement choice $z$) and ${\rho }_2$ between $B$ and $C$ (with measurement choice $X_2$). (f) The triangle scenario, where three independent sources interconnect three nodes $A$, $B$, and $C$, each of which performs a single measurement. This implies that, in contrast with previous scenarios, the distant parties have no measurement input.

Figure 2

ANN-based approach to the optimization of nonlinear functions over a superset of the quantum set. Our ANN optimizer works in two different configurations. In one case, it takes as input a generic nonlinear function $g\left(p\right)$ to be optimized over the superset of quantum correlations compatible with the causal scenario under analysis at the chosen level of the NPA hierarchy (see Secs. 3 and 4). Let us recall that, for a given causal structure, we indicate with $p(a,b,c,...|x,y,z...)$ or, shortly, $p$, the corresponding input-output probability distribution. In the second case, the ANN gets additional constraints because it is required to optimize $g\left(p\right)$ also compatibly with experimental observations, within an arbitrary confidence level $1\epsilon$ (see Sec. 5). In this instance, the ANN takes as an additional input the numerical value of the observable terms of $p$, which can be simulated or directly amount to experimental frequencies $f^{\text{expt}}(a,b,c,...|x,y,z,...)$. In both cases, the objective function $g\left(p\right)$ is set as a part of the loss function, see Eq. (3). Furthermore, the loss function embodies other constraints on the correlations terms, such as causal conditions (including no-signaling) and the confidence interval on the experimental frequencies. In the end, the ANN outputs the set of probabilities that optimize the objective function and that are compatible with the set constraints.

Figure 3

ANN optimization of the CHSH inequality. In these plots, we show how the ANN-based optimizer finds the maximum violation of the CHSH inequality. (a) The blue bars indicate the correlations maximizing the CHSH inequality violation and belonging to the superset of the quantum set corresponding to the first level of the NPA hierarchy. The orange ones, instead, are the analytical values, saturating the Tsirelson's bound, i.e., $2\sqrt{2}$. On the $x$ axis the probabilities are ordered in blocks that correspond to different choices of measurement operators. Every block contains four pairs of columns corresponding to different sets of outcomes: $(a,b)=\left\{\right(0,0),(0,1),(1,0),(1,1\left)\right\}$. (b) The red triangles represent the CHSH inequality maximal values that the ANN finds over the quantum superset corresponding to the first level of the NPA hierarchy, by solving the primal optimization problem. The green ones, instead, represent the values obtained by the ANN performing the dual problem. On the $x$ axis, we show the number of iterations of the training. The horizontal blue line represents the analytical value corresponding to the maximum violation, i.e., the Tsirelson's bound. The fact that primal and dual solutions coincide, benchmarks the good operation of the ANN [60].

Figure 4

ANN optimization of the bilocal inequality. Bilocal inequality maximal value predicted by the ANN as a function of the number of iterations of the training. The horizontal blue line represents the theoretical upper bound equal to $\sqrt{2}\approx 1.41421$.

Figure 5

Triangle scenario and cut inflation. The triangle structure is an instance of a causal structure whose compatible correlations are extremely hard to characterize since no independence conditions can be enforced. To circumvent this issue, we adopt the cut inflation causal structure, to get results that can be then translated to the triangle scenario: (a) triangle causal structure, (b) cut inflation causal structure, where the source ${\rho }_2$ is split (which then constitutes the bilocal scenario), (c) cut inflation with four observable nodes.

Figure 6

Experimental apparatus. We employ a platform already introduced in Refs. [20, 26] to implement the bilocality scenario. In detail, we employ three different laboratories. Those held respectively by Bob and Charlie (Laboratories 1 and 3) are equipped with one quantum state source and one measurement station, while the one held by Alice (Laboratory 2) only includes two measurement stations. The source in Laboratory 1, ${\mathrm{\Lambda }}_1$, sends one photon to Bob's measurement station, while source ${\mathrm{\Lambda }}_2$, in Laboratory 3, sends one to Charlie. Moreover, both sources send the other photon to Alice, namely to Laboratory 2, through a $\approx 30\text{m}$ long single-mode fiber. The two employed sources are spontaneous parametric down-conversion sources implemented through a beta barium borate type II crystal (Laboratory 1) and through a periodically poled potassium tytanil phosphate in a Sagnac loop (Laboratory 3). The measurement stations are composed by a half-wave plate, followed by a polarizing beam splitter, which allows performing any projective measurement, in the polarization degree of freedom, of the form $cos\left(\alpha \right){\sigma }_x+sin\left(\alpha \right){\sigma }_z$, where ${\sigma }_x$ and ${\sigma }_z$ are Pauli matrices. Then, to implement two instances of the Bell's scenario, we separately used two of the three laboratories at a time, i.e., only one state source and two measurement stations.

Figure 7

Experimental results in the bilocality scenario. Comparison between experimental statistics $f^{\text{expt}}(a,b,c|x,y,z)$ (a) and values predicted by the ANN using a confidence interval $CI=1-{10}^{-6}$. (b) The columns and rows represent, respectively, $(a,b,c)$ and $(x,y,z)$ triples. (c) Lower bounds on the bilocal inequality violation predicted by the ANN tested on the experimental values as a function of the complementary of the confidence interval $\epsilon =1-Cl$. The green line represents the minimum value needed for the network to certify nonlocality. The blue line indicates the experimental violation value. Due to the training with numerically simulated experimental data, the ANN can then be employed to obtain lower bounds for any experimental violation obtained in the bilocal scenario, i.e. with no additional time requirements. At the same time, this implies that ${\mathrm{\Delta }}_{\mathcal{NS}}$ [see Eq. (3)] will be of the same order of magnitude of the Hoeffding's uncertainties.

Figure 8

CHSH experimental violation confidence (Hoeffding inequality). In this plot, we report the lower bound on the experimental violation of the CHSH inequality, obtained considering different confidence levels in the result, through Hoeffding's inequality. This inequality requires the iid assumption, i.e., that the runs are independent and identically distributed. The green dashed line stands for the theoretical upper limit, admitted by quantum mechanics. The orange and purple solid lines represent the experimental violations obtained through the apparatus depicted in Fig. 6, considering the two sources separately (i.e., two instances of the Bell scenario).

Figure 9

Bonet experimental violation confidence (Hoeffding inequality). In this plot, we report the lower bound on the experimental violation of the Bonet inequality, obtained considering different confidence levels in the result, through Hoeffding's inequality. This inequality requires the iid assumption, i.e., that the runs are independent and identically distributed. The green dashed line stands for the theoretical upper limit, admitted by quantum mechanics. The orange and purple solid lines represent the experimental violations obtained through the apparatus depicted in Fig. 6, considering the two sources separately (i.e., two instances of Bell scenario, lifted to obtain instrumental correlations [62]).

Figure 10

CHSH experimental violation confidence (Azuma inequality). In this plot, we report the lower bound on the experimental violation of the CHSH inequality, obtained considering different confidence levels in the result, through Azuma's inequality, and different levels of white noise [see Eq. (A10)]. This inequality drops the iid assumption, i.e., that the runs are independent and identically distributed, and hence these results are fully device-independent. The green dashed line stands for the theoretical upper limit, admitted by quantum mechanics. The black one, instead, stands for the theoretical upper limit, admitted by classical physics. The data used in this case are numerically simulated.

Figure 11

Bonet experimental violation confidence (Azuma inequality.) In this plot, we report the lower bound on the experimental violation of the Bonet inequality, obtained considering different confidence levels in the result, through Azuma's inequality, and different levels of white noise [see Eq. (A10)]. This inequality drops the iid assumption, i.e., that the runs are independent and identically distributed, and hence these results are fully device-independent. The green dashed line stands for the theoretical upper limit, admitted by quantum mechanics. The black one, instead, stands for the theoretical upper limit, admitted by classical physics. The data used in this case are numerically simulated.