## Abstract

A high-dimensional quantum key distribution (QKD), which adopts degrees of freedom of the orbital angular momentum (OAM) states, is beneficial to realize secure and high-speed QKD. However, the helical phase of a vortex beam that carries OAM is sensitive to the atmospheric turbulence and easily distorted. In this paper, an adaptive compensation method using deep learning technology is developed to improve the performance of OAM-encoded QKD schemes. A convolutional neural network model is first trained to learn the mapping relationship of intensity profiles of inputs and the turbulent phase, and such mapping is used as feedback to control a spatial light modulator to generate a phase screen to correct the distorted vortex beam. Then an OAM-encoded QKD scheme with the capability of real-time phase correction is designed, in which the compensation module only needs to extract the intensity distributions of the Gaussian probe beam and thus ensures that the information encoded on OAM states would not be eavesdropped. The results show that our method can efficiently improve the mode purity of the encoded OAM states and extend the secure distance for the involved QKD protocols in the free-space channel, which is not limited to any specific QKD protocol.

© 2021 Chinese Laser Press

## 1. INTRODUCTION

Quantum key distribution (QKD) can provide information theoretic security to share keys between two distant parties [1–3]. Currently, free-space quantum communication has progressed out of laboratories into real-world scenarios [4–8], which paves the way towards global-scale and highly secure quantum communication networks. However, most experiments mainly rely on either the polarization [9,10] or phase [11] of faint laser pulses as information carriers, where the typical two-dimensional encoding scheme limits the capacity of QKD systems due to an intrinsically bounded Hilbert space.

Unlike the limited degree of freedom on polarization or phase states, the orbital angular momentum (OAM) is a high-dimensional encoding scheme for free-space QKD owing to an infinite number of available OAM eigenstates in principle. Recently, Refs. [12–15] implemented OAM-encoded QKD experiments, and Ref. [16] realized OAM-based entanglement distribution over a free-space optical (FSO) channel of more than 143 km. Furthermore, OAM has been experimentally demonstrated to be rotational invariant in the propagation direction, which can remove the error caused by reference frame misalignment [17]. These outstanding properties make OAM states useful in both classical communication [18–22] and high-dimensional quantum cryptography [23–31].

Nonetheless, it has been shown that the helical phase of a vortex beam that carries OAM is sensitive to the transmission environment and easily distorted [32–34]. Particularly in a free-space optical channel, the atmospheric turbulence resulting from the inhomogeneity of temperature and pressure in the atmosphere will lead to severe wavefront distortion and coherence destruction of the beam, which will directly increase the crosstalk among the adjacent OAM modes [35,36] and further influence the performance of the key rate in OAM-encoded QKD schemes. Some meaningful methods, such as a post-selection of the data or an increase in the mode spacing, which are usually used in OAM-encoded QKD systems to improve the stability of vortex beams against atmospheric turbulence, will lead to a reduction of single-photon gain and coding dimension. In addition, an adaptive optics (AO) system is often applied in the classical OAM-encoded communications to implement the phase compensation [37–39]. In such scenarios, the compensation accuracy depends on the convergence of algorithms used in AO systems, which requires relatively more processing time. Especially for the satellite-based QKD system with time-varying channel conditions, the AO systems may not be fully adequate for an efficient turbulence correction. Therefore, methods for quickly compensating for the turbulence-distorted vortex beams are still urgently needed for practically applying OAM-encoded QKD systems, especially for satellite-based QKD applications with the limited link durations.

In this paper, we propose a fast and efficient method to adaptively compensate for the atmospheric turbulence effects, which borrows the ideas of deep learning technology. Deep learning technology is one of the most popular artificial intelligence (AI) tools, and it shows extraordinary ability in image recognition [40], optical fields [41], and even parameter optimization for QKD [42,43]. In our proposed method, a convolutional neural network (CNN) model is first designed to predict phase distortion information of atmospheric turbulence. Then CNN combined with common setups is integrated as an accurate and stable phase feedback module to generate a phase screen [44–46] and correct the distorted vortex beam in our free-space OAM-encoded QKD scheme. Our results show that the compensation ability of our method is independent of the turbulence intensity and only relies on the structure of the CNN. In addition, our method can efficiently eliminate the turbulence distortion and improve the mode purity of OAM states compared to the OAM-encoded QKD scheme without compensation, which extends the secure distance for the involved QKD protocols in the free-space channel.

The organization of the article is as follows. In Section 2, a CNN is presented for estimating phase distortion induced by atmospheric turbulence. Then, in Section 3, we describe the components and phase compensation operation of our free-space OAM-encoded QKD scheme and show the improvement of mode purity of OAM states at different situations to verify the effectiveness of our method. In Section 4, we analyze the performance of the involved OAM-encoded QKD scheme with the decoy-state method. The article is ended in Section 5 with concluding remarks.

## 2. DEEP LEARNING FOR PHASE DISTORTION PREDICTION IN THE TURBULENCE CHANNEL

When the vortex beam transmits through atmospheric turbulence, the refractive index fluctuation causes the helical phase wavefront distortion. In our work, the phase screen produced by the Kolmogorov model [47], which has been widely used to model the phase distortion information over the propagating beam, is adopted to provide quantitative agreement with analytical results. In parallel, in order to simulate the atmospheric turbulence using the random phase screen located in front of the receiver, the propagation of the optical beam through the atmospheric turbulence channel is usually divided into two processes: vacuum transmission and phase modulation by the turbulence phase screen (details in Appendix A). Consequently, we first need to predict the phase screen containing the distortion information induced by the atmospheric turbulence to perform the phase compensation for correcting the distorted vortex beam.

The CNN is composed of multiple-layered structures built from human-brain-like behavior (i.e., neurons), which can learn data with multiple levels of abstraction and can extract high-level abstract features. Based on the universal approximation theorem of a neural network, it is possible to infinitely approximate any complicated but smooth functions on a defined domain with a CNN. Stated differently, it can be applied to accept new incoming data $x$ and predict the corresponding $\widehat{y}$ after adequate iterations. Here we designed a CNN to learn the phase screen in our atmospheric turbulence prediction work as illustrated in Fig. 1. The CNN structure consists of 15 learned layers, including 12 convolutional layers and 3 deconvolutional layers. The first convolutional layer uses $5\times 5$ convolutional kernels, and the other layers use $3\times 3$ convolutional kernels.

The CNN is trained with a set of training data. This set contains $N$ mappings $({x}_{i}^{T},{y}_{i}^{T}),i=1,2,\dots ,N$. Specifically, ${y}_{i}^{T}$ is a gray image of the random phase screen produced by the Kolmogorov model in different turbulence intensities, and the input ${x}_{i}^{T}$ represents the two gray images, i.e., the intensity distribution of the Gaussian probe beam (GPB) affected by turbulence and that without turbulence. Namely, the network can analyze distortion information in the form of images, which can automatically find the characterization of turbulence to generate a predicted phase screen for the subsequent phase compensation.

In our CNN, a loss function mean squared error (MSE) is defined on the output layer between the network’s calculated outputs ${\widehat{y}}_{i}=f({x}_{i}^{T},w)$ on a set of input data ${x}_{i}^{T}$, versus the set of desired output ${y}_{i}^{T}$. The loss function MSE takes the form

Here we take a linear combination of inputs ${x}_{i}^{T}$, with weight $w$ and constant factor $b$, and the activation in each neuron can be expressed asTo achieve the trained CNN with a better generalization ability to prevent overfitting, we use an amount of training data to decrease the training error and the model complexity. Here we use 84,000 GPB intensity images with specified turbulence (i.e., the atmospheric refractive index structure constant ${C}_{n}^{2}$) ranging from $[{10}^{-15},{10}^{-13}]$, where the size of each image is fixed at $256\times 256$ and the gray values are normalized into $[0,1]$. In detail, 80,000 of them are used as the training data, 2000 are for the test data, and the remaining 2000 are for the validation data. Figure 2 shows the loss function, where the yellow and blue curves represent the loss value of the validation data set and training data set, respectively. The loss value of the validation data set is very close to that of training data set, and both converge to be stable. After 400 iterations, the loss value of the validation data set dropped to 0.03, which demonstrated that the CNN at this time has a good performance of prediction on turbulence and no overfitting occurred.

To reflect the trend of the loss curve more intuitively, Fig. 3 presents the phase images predicted by our CNN. From the images, we can see that when the number of iterations is sufficient, the predicted phase images are almost the same as the real phase screens. In detail, the time required for the iterations and processing is about 0.1 ms per image by using an i9-8950 CPU, which is much lower than the turbulent freezing time (10 ms) [51], and thus it brings advantages to optimize the performance of the OAM-encoded QKD scheme.

## 3. DEEP-LEARNING-BASED PHASE COMPENSATION IN THE FREE-SPACE OAM-ENCODED QKD SCHEME

We use the trained CNN for phase compensation in our OAM-encoded QKD scheme. Figure 4 shows the schematic diagram of an OAM-encoded QKD system with decoy-state method, which contains the real-time atmospheric turbulence compensation feedback at Bob’s side. In detail, at Alice’s side, a Gaussian beam from a continuous-wave (CW) laser (operating at 1550 nm) is modulated into a pulse train with a 1 MHz repetition rate. A polarization beam splitter (PBS) is utilized to separate the pulses into two subpulses, where the signal pulses are linearly polarized in the $x$ direction and the Gaussian probe pulses are linearly polarized in the $y$ direction. In practice, it is usually difficult to create a perfect vacuum state in the decoy-state QKD experiments [52]. To realize the preparation of decoy states, an intensity modulator is used to modulate the signal pulses into different intensities. When the gain of a state is modulated to be close (a finite extinction ratio below 30 dB) to the dark count rate of our single-photon detector, the state will be regarded as the “vacuum state.” After that, the signal pulses are converted by spatial light modulator (SLMs) to carry OAM-encoded qubit signal and then attenuated to a single-photon level signal by an attenuator. By using a polarizing beam splitter (PBS), the quantum signals together with the Gaussian probe pulses are sent to Bob through a simulated free-space channel.

In the free-space channel, the equivalent phase screens are produced by SLM2 under the Kolmogorov model. At Bob’s side, the compensation feedback is used to perform phase compensation, among which SLM3 is only sensitive to $x$ polarization and a CCD is utilized to extract the intensity distribution of Gaussian probe pulses, which is used as the input of a laptop PC with our CNN to predict a real-time atmospheric turbulence phase screen. Thus, by getting a reversed phase of the predicted phase screen on SLM3, the phase distortion of the only vortex pulses induced by atmospheric turbulence can be restored. After these operations, the OAM states are differentiated by employing the OAM sorter [12], which consists of two static optical transformations [unwrapper (R1) and phase corrector (R2) shown in Fig. 4] and a lens. Finally, two single-photon detectors are adopted to detect the results.

Note that the measurement probabilities of the desired OAM state relate directly to the probabilities of obtaining correct or incorrect measurements of a transmitted “symbol.” Therefore, for quantitative analysis of the correction effect in terms of the measurement probability, we now need to compare the mode purity of the OAM states before and after compensation. The mode purity, which can be regarded as the normalized power, is an important parameter describing the crosstalk between different spatial modes. Specifically, we assume that the receiver may measure a photon with the adjacent OAM mode ${l}^{\prime}={l}_{0}+\mathrm{\Delta}l,\mathrm{\Delta}l=0,1,2,\dots $. Thus, the mode purity of OAM states can be estimated by taking ratio ${s}_{\mathrm{\Delta}}={P}_{\mathrm{\Delta}}/P$ between the power contained in each OAM mode and the total power collected by the receiver. Without loss of generality, we are only interested in the ensemble average of this quantity (i.e., $\u27e8{s}_{\mathrm{\Delta}}\u27e9$). Meanwhile, the OAM topological charge is set as ${l}_{0}=1$. For Kolmogorov turbulence theory, the mode purity of an OAM state propagating through turbulence can be written as [34]

To show the applicability of our method in various environments, we update the training data set for our CNN, where the ranges of $\mathrm{\Delta}z$ and ${C}_{n}^{2}$ are set as $[1\times {10}^{3},1\times {10}^{5}]\text{\hspace{0.17em}}\mathrm{m}$ and [${10}^{-14}$, ${10}^{-13}$] m^{−2/3}, respectively. In particular, we generate 70,000 GPB images every 10 km to form the final data set, i.e., a total of $10\times 70\text{,}000$ pieces of data set to train the CNN, where the entire process from generating the data set to training the CNN took about 6 h. After retraining, we show the mode purity of OAM state ${l}_{0}=1$ at different situations. As shown in Fig. 6, the mode purity can always be well recovered under a wide range of parameter combinations of the intensity of turbulence and the distance. Although the prediction effect under strong turbulence or long distance is reduced (some pixels of the phase screen are out of the gray value range, which leads to the reduction of effective information during the normalization process), the mode purity values in different situations are still improved to above 0.82. This is because the most key working mechanism of the proposed CNN is to find the mapping relationship between inputs and outputs. Next, we will discuss the key rate performance optimization of the OAM-encoded QKD system.

## 4. PERFORMANCE OPTIMIZATION OF THE FREE-SPACE OAM-ENCODED QKD

To further evaluate the performance of the deep-learning-based phase compensation in improving the performance of the OAM-encoded QKD scheme with the decoy-state method, we discuss the secret key rate of the involved system. For simplicity and generality, a two-dimensional OAM-QKD protocol is discussed, where the OAM basis and superposition basis (SUP) [12] are both used in the OAM-QKD protocol. In such a decoy-state QKD, Alice randomly chooses an OAM basis $(|-{l}_{0}\u27e9|{l}_{0}\u27e9)$ or the SUP basis consisting of $\frac{1}{\sqrt{2}}(|{l}_{0}\u27e9+{e}^{in\pi}|-{l}_{0}\u27e9)$, where $n=0,1$ is the encoding bit. Subsequently, Bob uses the results with OAM basis as the key bits and the results with SUP basis as the testing bits, and then they testify the security of the key distribution. After error correction and privacy amplification, they can distill a secure key.

We follow the decoy-state QKD theory from Ref. [55]. Using the Gottesman–Lo–Lutkenhaus–Preskill (GLLP) formula [56], the secure key rate of decoy-state OAM-QKD in the asymptotic case is given by

Here the transmittance consists of the probabilities of obtaining correct and incorrect initial OAM states. As discussed in the below section, the probability of obtaining a correct state can be expressed as

where $\beta $ is the link attenuation coefficient. Correspondingly, the other one, the transmission error probability, is ${\overline{\eta}}_{0}=1-{\eta}_{0}$. Thus, ${Q}_{\mu}^{\mathrm{OAM}}$ can be calculated by Similarly, the gain ${Q}_{\mu}^{\mathrm{SUP}}$ in the SUP basis can also be estimated in a similar way. Therefore, ${Q}_{\mu}^{\mathrm{SUP}}={Q}_{\mu}^{\mathrm{OAM}}$. Note that, considering the errors are induced by misalignment and crosstalk between states, ${E}_{\mu}^{\mathrm{OAM}}$ in the OAM basis can be given asIn Fig. 7, we compare the secret key rates versus propagation distance for the situation with perfect phase compensation (i.e., $\chi =1$), the practical situation with phase compensation feedback based on the learned CNN (i.e., the value of $\chi $ is dependent on the transmission distance), and the situation without phase compensation. As shown in Fig. 7, our method offers a higher key rate than that of no compensation. Even if the learned CNN is not perfect, the curves of the practical situation for the secret key rates are quite close to the perfect phase compensation, where the situation of perfect phase compensation can be regarded as having no turbulence effects. This means that our proposed method can be applied in the free-space OAM-encoded QKD system to mitigate the turbulence phase distortion.

## 5. CONCLUSIONS

In this work, considering that the helical phase wavefront induced by the turbulence distortion has significant influence on the performance of a free-space OAM-encoded QKD scheme with the decoy-state method, we integrate the deep learning technology to achieve phase compensation to optimize the involved system. Here a well-trained CNN is designed to predict an equivalent turbulent phase screen. Combined with the commonly used optical devices, fast and accurate phase compensation feedback can be achieved in our experimental scheme. Furthermore, we use the mean-squared error (MSE) between the predicted phase values and original phase values as a performance index for assessing our system. The results show that our scheme can effectively increase the key rate owing to improving the mode purity of OAM states. Our proposed work can provide an efficient solution for a free-space high-dimensional QKD using OAM states.

Note that, considering the time-varied atmospheric turbulence, we use a charge-coupled device (CCD) to extract the real-time turbulence distortions from the intensity distribution of the Gaussian probe beam, which has no coding information. By employing an SLM, a generated compensated phase screen can be directly loaded on the OAM-encoded beams, and thus there is no need to monitor the OAM-encoded signals. In experiments, limited by the response speed of devices such as the SLM, our experimental scheme may degrade the compensation effect, but it can reduce the number of qubits used to estimate channel conditions. Moreover, attacks due to the loopholes of the phase compensation module can also be resisted. In addition, placing the phase compensation module on the measurement side [i.e., measurement-device-independent (MDI)-QKD protocol] can further ensure its safety. Therefore, the experimental and security analysis of the MDI-QKD system integrated with our method will be the subject of our future studies.

## APPENDIX A

When the vortex beam transmits through atmospheric turbulence, the refractive index fluctuation causes helical phase wavefront distortion. To simulate the atmospheric turbulence, we model the atmospheric channel using a phase screen located in front of the receiver. Thus, the propagation of the optical beam through the atmospheric turbulence channel with a length $z+\mathrm{\Delta}z$ can be divided into two processes: a vacuum transmission channel with a length $z$ and a phase modulation process by the turbulence phase screen with a width $\mathrm{\Delta}z$. The phase screen method involves the filtering of a complex Gaussian random field using the phase’s power spectral density (PSD) function of the atmospheric turbulence, where the phase’s PSD function $\mathrm{\Phi}({k}_{x},{k}_{y})$ can be calculated by the Kolmogorov model:

In our work, we choose one of the vortex beam forms, i.e., a Laguerre–Gaussian (LG) beam, making OAM eigenstates correspond to the LG mode set. Here we use the intensity and phase profiles of an LG01 beam after 500 m atmospheric propagation with different ${C}_{n}^{2}$. Here the beam waist radius ${w}_{0}$ is set as 0.002 m. The phase screen is set with a numerical grid of $N\times N=512\times 512$ points and a grid spacing of $\mathrm{\Delta}L=5\text{\hspace{0.17em}}\mathrm{mm}$.

## Funding

National Natural Science Foundation of China (11704412); Key Research and Development Program of Shaanxi (2019ZDLGY09-01); Innovative Talents Promotion Plan in Shaanxi Province (2020KJXX-011); National University of Defense Technology (19-QNCXJ-009).

## Disclosures

The authors declare that there are no conflicts of interest related to this paper.

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