State-dependent misalignment and turbulence effects on high-dimensional quantum key distribution with orbital angular momentum

High-dimensional quantum key distribution (HD-QKD) is a topic of growing interest in the quantum communication community, not only for its inherent properties but also for its possible applications. As the typical freedom in HD-QKD, orbital angular momentum (OAM) has made significant advancements in experiments recently. However, in the airborne scenario, different states suffer different amounts of misalignment and turbulence. A complete theoretical analysis model for the transmission characteristics of OAM in atmospheric channels is lacking. In this paper, we systematically analyze the extent to which degeneration including channel power loss and mode crosstalk are influenced by misalignment and turbulence effects. Furthermore, the performance of OAM-encoded HD-QKD system in different dimensions is evaluated while incorporating finite-key effects. We demonstrate that the performance of OAM-encoded HD-QKD will be better at short range, which provide a reference to implement QKD based on task requirements. Since OAM is desired to increase the capacity of QKD system and experiments have already been carried out, our work can not only bridge the gap between theory and practice, but also optimize experimental parameters and improve system performance.


Introduction
Quantum key distribution (QKD), as a potentially revolutionary cryptographic technique, can establish a bit string of common true random bits between two legitimate parties.Based on quantum mechanics, QKD could provide information-theoretic approach against an eavesdropper during key distribution tasks [1][2][3].Currently, along with the breakthrough of fiber-based [4][5][6] and satellite-based [7,8] quantum communication, building and operating a high-efficient and flexible quantum internet has become a grand vision attracting global attention [9,10].However, most of these achievements are based on encoding information with two-dimensional quantum systems such as polarization and phase of photons.Although these systems are advantageous in qubit manipulation and control, the information capacity is limited to 1 bit per photon due to the intrinsically bounded Hilbert space.To break through the system information entropy threshold, high-dimensional (HD) QKD offers an exceptional means and has developed rapidly in both theoretical [11][12][13][14][15] and experimental [16][17][18][19][20] aspects over past two decades.By utilizing larger Hilbert space, HD-QKD has following benefits [21]:

Challenges in an OAM-based channel
Based on the advantages of high-dimensional coding, OAM modes could be potentially utilized in free space optical (FSO) communication.However, for complex atmospheric environment, there are different key challenges, which might induce power loss and mode crosstalk.Several critical factors cannot be ignored including atmospheric turbulence, divergency and misalignment.Some details about potential challenges will be discussed in this section.The concept of OAM beam is introduced firstly.

OAM of light beam
Due to the phase singularity in a helical phase profile, an OAM beam with a nonzero order usually has a donut-shaped intensity profile, as shown in figure 1.In general, an OAM beam could refer to any helically  phased light beam, irrespective of its radial distribution.Laguerre-Gaussian (LG) modes are exploited as the orthogonal modal basis set of OAMs.The complex amplitude expression of OAM beam in LG mode is: where normalizing factor A = 2p!/π (|l| + p)!, the indices l and p correspond to azimuthal and radial distribution, respectively.w(z) = w 0 1 + z 2 /z R 2 represents the spot size of the LG beam transmitted to the distance z. w 0 is the beam waist, z R = 1 2 kw 0 2 is the Rayleigh range, k = 2π/λ and λ is wavelength.L |l| p () are the generalized Laguerre polynomials and (r, θ, z) is the cylindrical coordinate.
The intensity distribution of a LG mode for the lowest-order radial mode (p = 0) is: where A l,p (r, θ, z) represents the amplitude of the OAM beam.For ease of calculation, the expression is normalized such that ˜Il (r, θ, z)rdrdθ = 1.

State-dependent divergence
Limited by the size of optical components like limited-size receiver aperture, the transmission of OAM beam in free space is affected by SDD.In such a scenario, different states (azimuthal mode) ℓ experience different amounts of diffraction, leading to state-dependent loss.The reason is that the mode transmission efficiency induced by SDD is a function of Fresnel product N f and OAM quantum number ℓ.According to the research of [44], the radial position of the maximum of intensity is: which confirms the existence of SDD.Link structure of OAM under SDD effect is shown in the figure 2. The beam divergence could increase the difficulty of capturing the beam profiles, which contributes to signal power loss.Since the distribution of light intensity is normalized, the geometric attenuation L geo caused by diffraction could be obtained by calculating the optical power at the receiving end: where, A represents the aperture area of the receiving telescope, Γ(s, t) represents an incomplete gamma function.
Because of the SDD, OAM states with higher ℓ will have larger far-field sizes, and acquire more propagation phase.Thus, in practical free-space communication links, different OAM states will suffer different amounts of loss for a given collection aperture of finite size, leading to ℓ-dependent detection efficiency.

State-dependent misalignment
In FSO links, mechanical errors in the acquiring, pointing and tracking (APT) system and vibrations of the transmitter/receiver platform will cause the random beam jitters, contributing to the fluctuating losses.Assuming a Gaussian beam, the misalignment could lead to power loss, as the limited-size receiver aperture might fail to fully capture the beam profile.However, in an OAM-based communication link, such misalignment could induce not only power loss but also inter-channel crosstalk, which would affect the SKR and QBER.
Due to the special beam profile of OAM, lateral displacement is analyzed emphatically to ensure the receiving efficiency.Figure 3 shows the link structure of OAM when the transmitter and receiver are misaligned with lateral displacement.
In scenario of lateral displacement, based on central limit theorem, the random position deviations and caused by jitter can be assumed to be Gaussian random variables with a mean of zero and a variance σ ρ 2 .Jitter variance σ ρ 2 represents stability of the transmitter.ρ denotes the radial vector representing random beam displacements.Due to the symmetry of the beam shape and receiver area, the characterization of transmittance efficiency caused by misalignment η pe depends only on the radial jitter distance ρ = ∥ρ∥.So ρ can be described by the Rayleigh distribution: Given the factor of misalignment, the expression (2) of intensity distribution could be calculated renewedly: Under the limited receiving aperture effect, transmittance efficiency caused by lateral displacement can be expressed: In the case of misalignment, the power of OAM beams received on each circle of the receiving telescope varies, so the method of calculating the received optical power by using polar coordinates is no longer applicable.Here, the point coordinate method is applied to obtain the surface integral.
where ξ = √ π D r /2 .Because ρ obeys the Rayleigh distribution, received power (also called transmittance efficiency) caused by misalignment is given as: In order to get closer to the practical application, we analyze the displacement errors at different distances.The geometric variance of the pointing error at the receiver can be approximated by [62]: where δ is the angular jitter error at the transmitter.For small amounts of δ one can write σ ρ 2 (z) ≃ (δz) 2 .Furthermore, misalignment will also cause bit error, that is, lateral displacement will lead to energy transfer between receiving OAM beam modes.The mode purity of an OAM state will decrease accordingly, which contributes to incorrect receiving bits.Assuming the radial and angular deviation of OAM beam under the action of misalignment is ρ and α, respectively.The complex amplitude expression of deflected l-state OAM beam could be expanded in form of spiral harmonic function: where represents amplitude component of m-state OAM beam.E l is amplitude, and J m (x) is the modified Bessel function of the first kind of integer m.
According to the formula (12), the orientation error causes mode expansion of pure l-state OAM beam, and the power spectrum of the mode is symmetrically distributed with order m = l as the center.Therefore, when the OAM beam is transmitted along different axes, it is the result of the superposition of different OAM modes on the measurement axis.Below, we represent the proportion distribution of different modes according to the energy weight distribution.The energy of the received beam can be expressed as: where C m = ´Dr/2 0 |A ml (r, ρ)| 2 rdr represents the probability density of OAM beams revealing in m -state.ε 0 is permittivity of vacuum.The weight factor of each harmonic component represents the ratio of the energy of each spiral harmonic with different OAM states to the total energy of the beam.Therefore, influenced by pointing error, the probability of received correct OAM state can be given as: According to formulas ( 9) and ( 12), both the detection efficiency and probability caused by misalignment are related to OAM index l, which theoretically proves the existence of state-dependent misalignment.

State-dependent turbulence
Atmospheric turbulence refers to irregular motion in the atmosphere.Specifically, temperature fluctuations and air currents in a turbulent atmosphere cause the appearance of eddies.These eddies lead to fluctuations of the local index of refraction at any position in the atmosphere.This deterioration becomes particularly evident in OAM communication.The reason lies in the phase distortion caused by atmospheric turbulence.The phase distortion will cause the wavefront distortion of OAM beams, so that the peak position of OAM beams will be shifted and the information transmission will be interfered.The phase distortion caused by atmospheric turbulence is shown in figure 4.
During practical implementation, a beacon light can be used to eliminate the low order aberrations from turbulence.Therefore, this scenario focuses on wavefront distortion caused by state-dependent turbulence.
From literature [51], the Kolmogorov turbulence theory is used to evaluate the random statistics of atmospheric turbulence.The probability that OAM beam with quantum number l receives OAM beam with quantum number m after transmission is: where r 0 = 0.1853( λ 2 C 2 n z ) 3/5 represents Fried parameter, C 2 n is the refractive-index structure parameter, which is the strength of atmosphere turbulence.λ is the wavelength.∆ = |l − m|.
The transmission probability η atm , which illustrates the probability of the received photon with the same OAM state as the input, is used to describe the aberration about atmospheric turbulence and link attenuation.η atm is expressed as: where ⟨s 0 ⟩ represents the mode purity of the receiving initial OAM state.Without loss of generality, we are only interested in the ensemble average of this quantity.β is the link attenuation coefficient.Correspondingly, the transmission error probability is η atm = 1 − η atm .
It is similar to the analysis of the pointing error on the receiving probability of the helical spectrum.The same formula C l = ´Dr/2 2 rdr applies to atmospheric turbulence.
According to the analysis in section 2.3, it can be concluded that in atmospheric turbulence, the helical harmonic C l with OAM index l is: It is worth mentioning that the above analysis of state-dependent turbulence includes two folds: (1) high order OAMs have larger beam cross section which leads to a worse turbulence; (2) high order OAMs have multi-fold phase structures which are more sensitive to phase distortions.Therefore, influenced by atmospheric turbulence, the probability of received correct OAM state can be given as: Formula ( 18) represents the normalized energy weight for each of the spiral harmonics of OAM beams in the paraxial region.Formulas ( 15) and (17) show that the influence of OAM in turbulence is also related to state.

Analysis on the key generation rate performance
In OAM-encoded HD-QKD, the SUP basis and the OAM basis form two mutually unbiased bases, which guarantees the unconditional security of QKD.OAM basis is composed of different OAM states, and the Fourier conjugate SUP basis consists of an equal superposition of OAM states with fixed relative phase between adjacent OAM components.The OAM basis and SUP basis can be represented as: where d is the dimension of the Hilbert space and L is the maximum OAM quantum number in use, which satisfies the relation 2L + 1 = d.Generally, Alice independently modulates quantum states by spatial light modulator.Then, the weak coherent pulses are attenuated to the single-photon level.Another attenuator is used to realize the 'vacuum + weak decoy state' method [63].The OAM beams are transmitted in free-space link.In measurement parts, photons carrying OAM are detected by OAM sorters and single-photon detector (SPD).Bob applies two different mode sorters to fully discriminate quantum states in a specific dimension, one is used to measure the OAM basis, the other constructs the regulation required by the mapping to obtain the SUP basis according to the mathematical transformation method.Subsequently, Bob uses the results with OAM basis as the key bits and the results with SUP basis as the testing bits, and then testify the security of the key distribution.After post-processing procedure containing basis sifting, error estimation, key reconciliation, error verification, and privacy amplification, QKD implements secure key distribution.
The deterioration on OAM-encoded HD-QKD in free space from divergence, misalignment and turbulence concentrate on SKR and QBER.According to the Gottesman-Lo-Lütkenhaus-Preskill formula [64] for HD-QKD, the secret key rate of the d-dimensional BB84-QKD can be estimated by: (20) where q m represents the probability of choosing basis and depends on the QKD protocol.Q OAM For decoy-state OAM-based QKD system, the transmittance η of single photon signal consists of the transmittance efficiency caused by misalignment and turbulence, as well as the internal loss at the receiver, which is expressed as: where η Bob is the internal optical loss at Bob. η D is the detection efficiency of SPD.Thus, Q µ can be calculated by: where Y 0 = 2p d is the yield when Alice does not send a photon.p d is the dark count of the detector.µ is the mean photon number of signal state.Meanwhile, considering the errors are induced by misalignment and crosstalk between states, E OAM µ can be given as [53]: Different from the general two-dimensional QKD system, e 0 = (d − 1)/d is the error rate of the dark count in HD-QKD.Take into account the bit errors caused by misalignment and turbulence, ε t = 1 − P at P pe is the crosstalk probability.
By using 'vacuum + weak decoy state' method, the yield, the fraction and the error rate of the single-photon states Y OAM 1 , ∆ 1 and e SUP 1 can be estimated by:

Finite-key security analysis for decoy-state OAM-encoded HD-QKD
Due to the differences between HD-QKD and 2D-QKD in key analysis, the security of decoy-state OAM-encoded QKD has been obtained in the asymptotic regime, especially, in the limit of infinite-key effects.In the asymptotic case, the yield and the error rate of the single-photon state required by equation ( 19) can be estimated accurately from the measured values of OAM-encoded QKD, which will converge to the approximated true values.Then, the final key length of OAM-encoded QKD can be obtained safely.However, in the practical HD-QKD system, the number of quantum signal pulses emitted is limited, so there are statistical fluctuations between the observed and true values of the system parameters.When analyzing the security of HD-QKD system, it is necessary to consider the influence of finite-key effects response final security key rate caused by statistical fluctuation.Here, before giving a finite-key security analysis for OAM-encoded HD-QKD, we first review the improved Chernoff bound method used for fluctuation analysis in [65,66].For observation ζ > 0, using the Chernoff inequality, the confidence interval for the expected value E [ζ] of the observed parameter can be obtained: where are the upper and lower limits of the confidence interval, δ L and δ U can be obtained from the following equations: where ε is the failure probability to estimate the confidence interval of the expected value.
Based on formulas ( 26) and ( 27), the upper and lower bounds of statistical fluctuation of overall gain and bit error rate of signal state and decoy state can be obtained.Furthermore, the upper and lower limits of the single-photon pulse overall gain and bit error rate are obtained by combining the formulas ( 23)- (25).Based on this, the security key length of OAM-basis of HD-QKD system under finite-key effects is obtained: where M OAM,L 1µ is the lower limit number of OAM-basis sifted key from the single-photon signal state.M OAM µ is the number of OAM-basis sifted key from the overall signal state.e SUP,U 1µ is the upper limit of the phase error rate.E OAM,U µ is the upper limit of the QBER in the OAM-basis sifted key.Therefore, if the total number of quantum signal pulses emitted by Alice is N, then the final secret key rate of the practical HD-QKD system is:

Numerical simulations
In view of the rotation invariance of the vortex light field, it effectively avoids the problem of the reference frame alignment, and provides a new coding scheme for the construction of high-speed and stable mobile quantum communication networks.The most likely application in the future is, for example, the scenario of drone-based quantum communication, with a typical distance range of kilometers.According to the above theoretical analysis, targeting 10 km free space link scenarios, the simulation method is used to analyze the influence of different factors on OAM-encoded HD-QKD system.The main parameters for simulation are as follows in table 1 [65,67]:

Divergence of OAM beams propagation in free space
In order to evaluate the transmission characteristic of OAM under SDD, the relationship between distance and efficiency is shown in figure 5.As we mentioned above, different OAM states will suffer different amounts of diffraction, as do the OAM components of a SUP state.As shown in figure 5, for a long distance (z ≫ 1 km ), there are huge efficiency differences between lower-order and higher-order OAM states.This difference results in a nonuniform probability of detecting the OAM states, which deteriorates the system performance.

Misalignment of OAM beams propagation in free space
In view of the selection of jitter error at the transmitter, the variance of divergence angle is analyzed numerically.While holding w 0 of the LG mode constant, a convenient form for expressing the state-dependence of the beam divergence is given by [44]: When l = 4, We get the beam divergence angle θ ≈ 78.1 µ rad.Therefore, the jitter error δ is determined according to this divergence angle, and the demand for APT system capability is evaluated.Three values of 10 −6 , 10 −5 and 10 −4 were selected for simulation to analyze the signal transmission efficiency caused by misalignment under different variances.At the same time, similar to SDD analysis, the transmission efficiency of different OAM states under the misalignment is also analyzed.
As shown in the figure 6(a), a small angular jitter error has a greater advantage in short-distance transmission (z ≪ 3 km), which is higher for the transmission efficiency of the signal, and the impact of misalignment jitter in long-distance transmission can be negligible.Considering the cost of the system and the capability requirements for the APT system, δ = 10 −5 rad is selected as the angle jitter error for further analysis.Figure 6(b) demonstrates the state-dependent misalignment effect on transmission efficiency.The curve trend in figure 6(b) is similar to that in figure 5, indicating that misalignment has subtle impact on the state-dependent effects.original OAM, which means that the probability of receiving other states is symmetrical distribution centered on the receiving correct state.Figure 9(b) simulates the receiving probability of correct OAM states at different transmission distances, confirming the existence of state-dependent turbulence effect, and providing data support for subsequent high-dimensional QKD key analysis.

Performance of decoy-state OAM-encoded HD-QKD
In the above simulations, the theoretical model and simulation data of OAM transmission in atmospheric channel has been given.In order to further embody the application value of theoretical analysis, the performance of OAM encoded HD-QKD was evaluated.In practical HD-QKD, finite-key effects cannot be ignored.Therefore, we choose the finite size of data N = 10 13 to analyze the relationship between security key rate and transmission distance in different dimensions.It is worth noting that the QBER analysis of HD-QKD takes into account the effect of pointing error and atmospheric turbulence, and each OAM state sends and receives with equal probability.
As can be seen from figure 10, under the background of short-distance transmission, higher-dimensional QKD shows great advantages in the generation of SKR.In the context of long-distance transmission, 2D-QKD should be selected for key transmission, and 2D-QKD can transmit longer distances than higher-dimensional QKD.The above simulation results show that there is no direct relationship between the selection of QKD dimension and the performance of the system.In the actual scenario, we should choose the appropriate dimension according to the task requirements to implement QKD.

Discussion
OAM has been widely used in the field of quantum communication as a way of high-dimensional coding of optical field degrees of freedom.Although recent breakthrough experiments have demonstrated the feasibility of OAM-encoded HD-QKD, the underlying statistical characteristics of quantum atmospheric channels have not been well-understood and experimentally verified in the literature.The research in this paper enhances the theoretical completeness of HD-QKD.The main innovations of this study include the following aspects: (1) Aiming at the problem of SKR constraint caused by high-speed platform maneuvering in airborne scenarios, a complete OAM atmospheric transmission model is established, focusing on the analysis of state-dependent misalignment and turbulence effects, which are characterized by channel loss and mode crosstalk; (2) on the basis of atmospheric channel modeling, security analysis is performed on OAM-encoded HD-QKD, and quantitative evaluation of SKR and QBER is performed in combination with finite-key effects, thus providing theoretical reference for experiments.
This work holds significant reference value for future channel modeling studies.The OAM-encoded HD-QKD provides ideas for efficient information transfer of local hot spots between ground-air links, especially on airborne platforms.As the parameter settings here are temporary and ideal, and should be adjusted with the increase of distance, it is possible to develop a more comprehensive model.Consequently, the help of machine learning should have a better effect, which is the direction of our future research to facilitate the practical implementation of the system.

Figure 1 .
Figure 1.The wavefronts, intensity profiles, and phase profiles of OAM beams with different ℓ values.

Figure 2 .
Figure 2. Link structure of OAM under SDD effect.

Figure 3 .
Figure 3. Link structure of OAM when the transmitter and receiver are misaligned with lateral displacement.

Figure 4 .
Figure 4.The OAM phase distortion caused by atmospheric turbulence.

µ
and E OAM µ are the gain and QBER of the signal states, respectively.∆ 1 is the fraction of single-photon signals transmitted by Alice.f EC (E OAM µ ) is the error correction efficiency of the signal state.H d (x) = −(1 − x)log 2 (1 − x) − xlog 2 [x/(d − 1)] is the d-dimensional Shannon entropy.

Figure 8 .
Figure 8.The transmission efficiency against the distance with different strength of atmospheric turbulence.

Figure 9 .
Figure 9. (a) Helical spectrum distribution of OAM beam under atmospheric turbulence; (b) the probability of receiving the correct OAM state under atmospheric turbulence between different OAM states.

Figure 10 .
Figure 10.The relationships between the secret key rate and transmission distance for HD-QKD.

Table 1 .
Parameters used for the numerical simulation of OAM-encoded HD-QKD.