Experimental Simultaneous Learning of Multiple Nonclassical Correlations

Nonclassical correlations can be regarded as resources for quantum information processing. However, the classification problem of nonclassical correlations for quantum states remains a challenge, even for finite-size systems. Although there exists a set of criteria for determining individual nonclassical correlations, a unified framework that is capable of simultaneously classifying multiple correlations is still missing. In this Letter, we experimentally explored the possibility of applying machine-learning methods for simultaneously identifying nonclassical correlations. Specifically, by using partial information, we applied an artificial neural network, support vector machine, and decision tree for learning entanglement, quantum steering, and nonlocality. Overall, we found that, for a family of quantum states, all three approaches can achieve high accuracy for the classification problem. Moreover, the run time of the machine-learning methods to output the state label is experimentally found to be significantly less than that of state tomography.

Figure 1

Quantum correlation is divided into four categories by three typical ML models. (a) The structures of artificial neural network (ANN). (b) The support vector machine (SVM). (c) The decision tree (DT). (d) The Venn diagram of these four categories, in which I, II, III, and IV represent the separable state, the entangled state, the one-way steerable state, and the Bell nonlocal state.

Figure 2

Experimental setup. A pair of polarization-entangled photons are generated in the preparation stage by pumping a type-II PPKTP crystal located in a Sagnac interferometer with an ultraviolet laser at 404 nm. The polarization state of the pump light is rotated by a half-wave plate (${\mathrm{HWP}}_1$). The dual-wavelength ${\mathrm{HWP}}_2$ set to be 45° in the Sagnac interferometer is used to exchange the polarizations of the pump light and down-converted photons. The ${\mathrm{HWP}}_3$ set to be 45° is used to change the form of the entangled state. One of the photons passes through an unbalanced interferometer and is sent to Alice. ${\mathrm{HWP}}_4$ is set at $22.5°$ and two sufficiently long calcite crystals (CCs) with the length of the last one being two times larger than the first one to completely destroy the coherence in different components. Two variable filters (VF, in the red dotted line circle) are used to control the relative amplitudes between these two arms. The other photon is sent directly to Bob. Quarter-wave plates (QWPs), HWPs, and polarization beam splitters (PBSs) on both sides of Alice and Bob are used for observables and tomography measurement. The photons are detected by single-photon detectors (SPDs), and the signals are sent for coincidence. The training and testing procedures are marked with black and red colors, respectively. In the training case, the criteria of quantum correlations are determined from tomographic data and the results are sent to the ML models together with the two characteristic features obtained from partial information measurement. In the testing case, only the two measured characteristic features are used to predict the labels.

Figure 3

(a) The training data (blue dots) and test data (red dots). (b) The accuracies of ANN, SVM, and DT distinguish whether a state is entangled, one-way steerable, or Bell nonlocal. The test results for multilabel classification with ANN, SVM, and DT are shown in (c), (d), and (e) with the accuracies of 90.11%, 90.11%, and 85.17%, respectively. The parameter $\theta$ varies from 0 to $2\pi$ while the visibility $p$ changes from 0 to 1. The space is divided into four parts by the gray theoretical boundary, in which states in I, II, III, and IV are theoretical predicted to be separable, entangled, one-way steerable, and Bell nonlocal, respectively. “$\times$” represents the mistakenly labeled states via ML models compared with those labeled by traditional criteria. The error bars are smaller than the size of the dots. The inset shows the enlargement of the corresponding region in the black pane. The error bars are due to the Poissonian counting statistics.

Figure 4

The comparison between three ML methods. (a) The performance of ANN vs the Epochs. The black curves represent the value of the loss function and the red dotted line represents the training accuracy. Red triangles show the tested accuracy in different Epochs. (b) The performance of DT and SVM vs max tree’s depth and error penalty factor with the data marked in black and red colors, respectively. Circles and triangles represent the training and testing results respectively. (c) The relationship among labeling time, error, and iterations (N2) of SDP (black); the relationship among labeling time, error, and hidden units (N1) of ANN (red). (d) The variation of error along with the number of hidden units for distinguishing entangled and separable states.