The Bell state is a crucial resource for the realization of quantum information tasks, and when combined with orbital angular momentum (OAM), it enables a high-dimensional Hilbert space, which is essential for high-capacity quantum communication. In this study, we demonstrate the recognition of OAM Bell states using interference patterns generated by a classical light source and a single-photon source from a Sagnac interferometer-based OAM Bell state evolution device. The interference patterns exhibit a one-to-one correspondence with the input Bell states, providing conclusive evidence for the full recognition of OAM Bell states. Furthermore, we introduce machine learning to the field of Bell state recognition by proposing a neural network model capable of accurately recognizing higher order single-photon OAM Bell states, even in the undersampling case. In particular, the model’s training set includes interference patterns of OAM Bell states generated by classical light sources, yet it is able to recognize single-photon OAM Bell states with high accuracy, without relying on quantum resources during training. Our innovative application of neural networks to the recognition of single-photon OAM Bell states not only circumvents the resource consumption and experimental difficulties associated with quantum light sources but also facilitates the study of OAM-based quantum information.

## I. INTRODUCTION

Quantum entanglement^{1} is not only a unique property of quantum mechanics but also a core resource of quantum information.^{2} The Bell state is the largest entangled state in a two-qubit system. Among them, the two-qubit Bell states based on two photons are the most common standard Bell states and are widely studied.^{3–5} In recent years, multidimensional entanglement theory has provided an alternative approach to extend the two-particle state,^{6} such as the single-photon polarization-path Bell state.^{7–10}

The orbital angular momentum (OAM)^{11,12} of a photon has extensively been studied as an additional degree of freedom. The high-dimensional nature of the OAM of a photon enhances the ability of single photon to carry information.^{13} In previous experiments,^{14,15} single-photon OAM Bell states are already prepared. In addition, OAM Bell states are measured by converting the OAM modes to the fundamental Gaussian mode. Because OAM states are theoretically infinite dimensional, measurements that convert to the fundamental Gaussian mode are not able to obtain information about all OAM modes, resulting in a loss of information. In contrast, using imaging methods to measure the spatial distribution of single-photon OAM Bell states can obtain complete information.^{16,17} However, as the order of photonic OAM increases, the interference pattern of the OAM Bell state becomes more complex, resulting in a distinction that is no longer possible for the human eye. For this reason, machine learning is naturally considered for application to this demanding task. Machine learning,^{18} as a branch of artificial intelligence, learns from past experience in order to optimize performance. In recent years, interest in applying it to quantum information tasks has been growing, and it has achieved applications in classification tasks,^{19–21} Hamiltonian learning,^{22–24} quantum autoencoders,^{25} quantum state reconstruction,^{26,27} and quantum state learning.^{28}

In this study, we use a Sagnac interferometer based OAM Bell state evolution device to generate interference patterns from a classical light source and a single photon source. These patterns correspond to the input OAM Bell states, enabling full recognition of the OAM Bell state. To improve the accuracy and speed of single-photon OAM Bell state recognition, we introduce machine learning into Bell state recognition by constructing a convolutional neural network. We demonstrate that a neural network model can be trained on interference patterns generated by classical light sources and then used to accurately recognize OAM Bell states generated by quantum light sources, even at low sampling rates. Given that quantum properties such as entanglement are susceptible to environmental perturbations, our use of data generated by classical light sources for training provides an alternative approach to preserving quantum resources. This not only reduces the consumption of quantum light sources but also improves the efficiency of Bell state recognition compared to previous experiments. Further exploration of this approach for other types of quantum states and investigating additional ways in which machine learning can enhance quantum experiments would be of great interest.

## II. THE THEORY OF THE OAM BELL STATE

*H*⟩ represents the horizontal polarization, |

*V*⟩ represents the vertical polarization, and

*m*represents the topological charge of OAM. We design a Sagnac interferometer-based OAM Bell state evolution device as shown in Fig. 2(a). Taking $\Psi +=1/2|H,m\u232a+1/2|V,\u2212m\u232a$ as an example, after it passes through a polarization beam splitter (PBS3) and is split into $1/2|H,m\u232a$, which propagates clockwise, and $1/2|V,m\u232a$, which propagates counterclockwise, reflections occur on the counterclockwise path, so the sign of the topological charge changes. $1/2|H,m\u232a$ passes through mirrors (M3 and M2), the liquid crystal phase controller (LC), and M1, while $1/2|V,m\u232a$ passes through M1, the LC, M2, and M1 in turn. Here, the phase

*e*

^{iθ}is generated with the LC. These two beams finally meet and output at PBS3, where the state is $1/(2)ei\theta |H,\u2212m\u232a+1/(2)|V,m\u232a$. A 22.5° half-wave plate (HWP) and PBS are used to measure this state. Ultimately, the state $\Psi +$ corresponds to the output state and is $\Psi +\u2032=1/2(ei\theta |\u2212m\u232a+|m\u232a)$. The corresponding output states for the four OAM Bell states are

It can be seen that the interference patterns of the OAM Bell state are phase-dependent. Simulations of the interference pattern for any of the OAM Bell states in Eq. (1) at different phase differences are shown in Fig. 1.

In Fig. 1, $\Psi \xb1$ and $\Phi \xb1$ represent the four Bell states, while *θ* values on the left represent the phase difference between $m$ and $\u2212m$.

## III. EXPERIMENT AND RESULTS

A diagram of the experimental setup is shown in Fig. 2. In Fig. 2(a), a continuous wave laser with a wavelength of 810 nm is modulated by H1 and PBS2. The combination of half-wave plates (H2 and H3), quarter-wave plates (Q1, Q2, and Q3), and a vortex wave plate (VR) is used to prepare the OAM Bell state. The mirrors (M1, M2, and M3) and PBS3 make up the Sagnac interferometer. In the interferometer, PBS3 acts as the input and output ports, which divides the incident light into horizontally polarized light propagating in a clockwise direction and vertically polarized light propagating in a counterclockwise direction. The two beams propagate and then regroup at PBS3 and exit from the non-incident end. A programmable liquid crystal waveplate in the Sagnac interferometer controls the phase difference between the horizontally and vertically polarized light. Finally, the OAM Bell state is received by a CCD after passing through an H4 at 22.5° and PBS4. To generate single-photon OAM Bell states, we prepare a single-photon source using the same method as in Refs. 8, 17, 29, and 30, as shown in Fig. 2(b). The pump light is a fundamental mode Gaussian continuous wave laser with a power of 20 mW and a wavelength of 405 nm. It pumps a type-II phase-matched periodically poled potassium titanyl phosphate (PPKTP) crystal and generates an entangled photon pair with a wavelength of 810 nm according to spontaneous parametric downconversion (SPDC). The photon pairs are then split by PBS1 according to horizontal and vertical polarization, where the vertically polarized photons are received by a single photon avalanche diode (SPAD0, Excelitas Technologies SPCM-800-14-FC), while the horizontally polarized photons are fed into the setup shown in Fig. 2(a) as a light source. Since CCDs are not sensitive enough to detect signals at the single-photon level, we use a single-pixel camera composed of digital micromirror device (DMD), SPAD1, and Time-Correlated Single Photon Counting (TCSPC, Siminics FT1040) to replace CCD. The single-pixel camera is loaded with sampling matrices, which modulates the interference pattern of the OAM Bell state and reflects it back to SPAD1. The combination of TCSPC and SPAD1 is used to record the number of photons corresponding to each sampling matrix. Finally, a computer algorithm is used to reconstruct the interferogram of the single photon OAM Bell state.

As shown in Fig. 1, the differences in the interference patterns of the OAM Bell states are characterized by the position of these corners of the petals. Therefore, we can recognize the OAM Bell states entirely (i.e., with 100% efficiency) based on the differences in the positions of the corners of the petals in the interference patterns of the OAM Bell states. In previous experiments,^{17} the interference patterns were observed directly by the human eye and the OAM Bell states were inferred. However, when the OAM order increases, the images become complex and the human eye is unable to accurately detect these differences. Therefore, we construct a neural network model to accurately recognize OAM Bell states of arbitrary order. First, we collected 360 laser interference patterns from the experimental setup as a training set for the neural network model. The interference patterns of the four laser OAM Bell states are shown in Fig. 3. Before using the training set, we crop the patterns to squares and reduce their size proportionally using bi-triple interpolation. We do this to ensure that the patterns match the size of the model, which results in the best performance of the network. In addition, each pattern in the training set is independent of each other, so we train the network using multiple patterns simultaneously to reduce training time. Our training uses asynchronous stochastic gradient descent with a fixed learning rate schedule of 0.01 and 0.5 momentum. With the operation of the multi-pattern training, the training time of the central processing unit (CPU)-based network [Intel (R) Core (TM) i7-9750H CPU, RAM 16G] can be on the order of minutes. The variation of the loss during training with the number of training epochs is shown in Fig. 4(a). From the variation of the loss, it can be noticed that the accuracy of the model starts to improve significantly around the 250th epoch and tends to stabilize by around the 300th epoch. For testing, we prepare OAM Bell states using the single-photon source and collect interference patterns of single-photon OAM Bell states with a single-pixel camera, which are shown in Fig. 3. The resolution of the images in Fig. 3 is 256 × 256 for both the laser source and the single-photon source. Because the intensity of the single-photon source is of the single photon order of magnitude, a longer sampling time is required to obtain an image of comparable quality to the laser source. A sampling time of 183 h is required to fully sample an image of 256 × 256 resolution with the DMD operating at 0.2 Hz. Single-pixel cameras combined with compressed sensing technology can reduce the sampling rate (SR: the number of samples/the number of samples for full sampling) while maintaining image quality. In Fig. 3, images from the single-photon source are shown for the SR of 10%, 1%, and 0.5%, which require 18, 2, and 1 h of sampling time, respectively. At 0.5% SR, the image quality deteriorates and the petals of the interference pattern are already linked together, making it difficult for the human eye to recognize the OAM Bell states. Figure 4(b) shows how the accuracy of the proposed model in recognizing single-photon OAM Bell states with different SRs varies with the number of training epochs. It can be seen that after 200 training epochs, the model is able to achieve more than 99% recognition of single-photon OAM Bell states at all SRs. This shows that the model trained with data generated by classical light sources is still valid for data generated by quantum light sources. On the one hand, it is not necessary to spend a lot of time collecting interference patterns of single-photon OAM Bell states as the training set, which reduces the consumption of quantum light sources, and on the other hand, the neural network model can still recognize single-photon OAM Bell states well at low SRs, which improves the efficiency of Bell state recognition compared to previous experiments.

Next, we conducted higher-order state recognition experiments in numerical simulations to demonstrate the reliability of the proposed neural network. We generated interference patterns of 3, 6, 8, 10, and 12 orders as training sets through numerical simulations and then undersampled and reconstructed the interference patterns of these five orders with a 0.5% sampling rate as the test set. After the training process, we presented the variations of losses under different orders in Fig. 5(a) and the changes in accuracy in Fig. 5(b). It can be observed that within a certain range of orders, the proposed neural network can accurately recognize the OAM Bell states. However, with the increase in order, the required number of training iterations also increases. It should be noted that we preprocessed the collected images into 28 × 28 resolution before inputting them into the network. When the order is higher than 12, the interference pattern petals become too dense. Moreover, we reconstructed the images with a sampling rate of 0.5%, so the recognition accuracy decreases at this point. Expanding the neural network’s structure will overcome this issue. Finally, we visualized the outputs of the first and second inception layers, and the visualization results are shown in Fig. 6. It can be observed that our neural network model does learn the circular petal feature of the OAM Bell state, and the features learned become more and more abstract as the convolutional layer deepens. Nevertheless, similar to other neural networks, it still cannot directly explain the complete picture of how the network operates internally. The working principle of neural networks remains a complex and active research area.

## IV. DETAILS OF NEURAL NETWORKS

We refer to the proposed convolutional neural network as the “OAM-LeNet.” The structure of the proposed neural network is described below. The network is seven layers deep, and the total number of layers used to construct the entire network is 23. Among these, the used layers are the convolutional layer, maximum pooling layer, and ReLU layer. The structure of the network is shown in Fig. 7. The inception layer is a combination of convolutional layers of our own design, which contains 1 × 1, 3 × 3, and 5 × 5 convolutional layers and a 3 × 3 maximum pooling layer. A diagram of the inception layer is shown in Fig. 8.

The inception layer has four parallel paths such that the layer helps to find the optimal local sparse structure in the network. For example, in the training process, if a certain path plays a dominant role in the discrimination of Bell states, its weight will gradually be greater than that of other paths, which completes the search for local sparse structure. In addition, the parallel paths also help speed up the training process and optimize the performance of the network.

## V. CONCLUSION

We developed a Sagnac interferometer to generate OAM Bell states. We obtained interference patterns from both a classical light source and a single-photon source. To recognize higher-order single-photon OAM Bell states, we introduce machine learning and construct a convolutional neural network model with inception layers. We find that the model could be trained with interference patterns from a classical light source and still accurately recognize single-photon OAM Bell states, suggesting that quantum resources are not a necessary component of the training process. Furthermore, the trained neural network model could fully recognize single-photon OAM Bell states in the undersampling case, which is a significant advance in the recognition of higher-order single-photon OAM Bell states. Our single photon source is based on SPDC generation and has an intensity limited by the pump laser power and crystal efficiency to thousands of photon levels. The interference pattern of the higher-order single-photon OAM Bell state is complex and requires a large number of photons to be clearly demonstrated, making it a challenging task to detect and recognize these states quickly and efficiently. To address this, the proposed scheme uses neural networks and compressed sensing methods to reduce the detection and recognition time from 18 to 1 h, laying the foundation for the identification of high-order single-photon OAM Bell states. Overall, our study has important implications for overcoming quantum resource constraints and expanding the scope of OAM-based quantum information technology.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

Qing-Yuan Wu and Zhe Meng contributed equally to this work.

**Qing-Yuan Wu**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal); Software (equal); Writing – original draft (equal); Writing – review & editing (equal). **Zhe Meng**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Methodology (equal). **Xiao-Xiao Chen**: Conceptualization (equal); Data curation (equal); Formal analysis (equal). **Jian Li**: Conceptualization (equal); Data curation (equal); Methodology (equal); Software (equal). **Jia-Zhi Yang**: Conceptualization (equal); Data curation (equal); Formal analysis (equal); Software (equal); Writing – original draft (equal). **An-Ning Zhang**: Supervision (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are openly available in GitHub at https://github.com/qingyuanwu/OAM_Lenet.^{31}