Machine Learning Nonlocal Correlations

The ability to witness nonlocal correlations lies at the core of foundational aspects of quantum mechanics and its application in the processing of information. Commonly, this is achieved via the violation of Bell inequalities. Unfortunately, however, their systematic derivation quickly becomes unfeasible as the scenario of interest grows in complexity. To cope with that, here, we propose a machine learning approach for the detection and quantification of nonlocality. It consists of an ensemble of multilayer perceptrons blended with genetic algorithms achieving a high performance in a number of relevant Bell scenarios. As we show, not only can the machine learn to quantify nonlocality, but discover new kinds of nonlocal correlations inaccessible with other current methods as well. We also apply our framework to distinguish between classical, quantum, and even postquantum correlations. Our results offer a novel method and a proof-of-principle for the relevance of machine learning for understanding nonlocality.

Figure 1

Black-box representation of (a) the bipartite Bell scenario and (b) a tripartite scenario with two independent sources of states. (c) Pictorial illustration of the different sets of correlations: nonsignaling, quantum, local, and bilocal. (d) Blending technique where different machines are combined to improve the overall performance.

Figure 2

The straight line (blue) indicates the exact solution of Eq. (3) or Eq. (6) (see SM [44]) and the circle (red) indicates the ML prediction (considering $10^4$ test set points). In all cases, the machine can predict, with excellent accuracy, the degree of nonlocality without any information about Bell inequalities. (a) Bipartite scenario ($m=2$) with quantum correlations obtained by projective measurements on $|\psi ⟩=\cos \theta |00⟩+\sin \theta |11⟩$, that maximally violate the Clauser-Horne-Shimony-Holt inequality [48]. (b) Bipartite scenario ($m=3$), (c) ($m=5$), and (d) the bilocality scenario employing four features $(I,J,⟨A_0⟩,⟨A_1⟩)$. For (b), (c), and (d), the $x$ axis denotes the point $i$th correlations (${\mathbf{q}}_i$) with an increasing ordering in the value of ${\mathrm{NL}}_{\text{test}}^i$.

Figure 3

Bilocality scenario. (a) Diamonds (blue), indicate the exact value of $\mathrm{NBL}=v^2-1/2$ [49] obtained by measurements on a Werner state maximally violating the inequality (5). The circles (red) indicate the deep learning prediction. (b) The triangles (grey) indicate the results of a numerical optimization for the maximum value of the regression ML function for quantum correlations obtained by measurements on the state $|\psi {⟩}_{AB}=|\psi {⟩}_{BC}=\cos \theta |00⟩+\sin \theta |11⟩$ (not violating inequality 5). The diamonds (blue) indicate the exact value and the circles (red) indicate the prediction made by a neural network trained with NS correlations. Strikingly, the ML approach can discover new quantum correlations without any information about Bell inequalities.