High-dimensional quantum key distribution performance estimation method for orbital angular momentum coding

Abstract

The invention provides a high-dimensional quantum key distribution performance estimation method for orbital angular momentum coding, and the method comprises the steps: obtaining OAM light beam characteristic parameters through employing a Laguerre-Gaussian mode as an orthogonal mode base set, and carrying out the estimation of the high-dimensional quantum key distribution performance of orbital angular momentum coding based on the OAM light beam characteristic parameters; respectively calculating the probability of receiving the correct OAM state under the influence of the pointing error and the probability of receiving the correct OAM state under the influence of the atmospheric turbulence; and according to the probability of receiving the correct OAM state under the influence of the pointing error and the atmospheric turbulence, carrying out decoy state OAM coding HD-QKD key estimation, and finally carrying out decoy state OAM coding HD-QKD finite length security key estimation to obtain the finite length security key rate of the HD-QKD system. According to the method, the high-dimensional quantum key distribution performance can be accurately estimated, the effective long security key rate of the HD-QKD system is obtained, and detection efficiency mismatch and system security vulnerabilities are avoided.

Description

Technical Field

[0001] The present invention relates to the technical field of quantum cryptography, and in particular to a method for estimating the performance of high-dimensional quantum key distribution of orbital angular momentum encoding. Background Art

[0002] Driven by the grand vision of the global quantum internet, achieving long-distance, high-code-rate, all-weather quantum communication has become a research hotspot in the field of quantum information, leading and promoting the rapid development of quantum information technology. Quantum Key Distribution (QKD) relies on the quantum properties of single photons to encode, decode, transmit, receive, and post-process information, and can generate theoretically untapped and tamper-proof secure keys between the two communicating parties. At present, most research on quantum communication is based on two-dimensional quantum systems implemented by polarization or phase encoding of photons. Although these systems have significant advantages in quantum bit manipulation, due to the boundedness of Hilbert space, the information capacity is always limited to 1 bit / photon. In order to break the information entropy threshold of the system, high-dimensional quantum key distribution (HD-QKD) provides a feasible idea and has developed rapidly in theory and experiment. Among them, the use of the transverse degree of freedom of photons - orbital angular momentum (OAM) for high-dimensional quantum state encoding provides advantages for improving the transmission performance of free-space quantum communication systems:

[0003] 1) Utilize the coding advantages of infinite-dimensional Hilbert space in a single quantum system to improve the transmission efficiency and fidelity of coded photon states, thereby enhancing the anti-noise capability and communication bandwidth per unit time of the quantum communication system;

[0004] 2) Combining the rotational invariance characteristics of the vortex light field, the problem of aligning the reference frame of the ground-to-air scene is effectively avoided, providing a new coding scheme for building a high-speed and stable mobile quantum communication network based on a high-speed mobile aerial platform.

[0005] Based on the characteristics of typical aviation channel environments and the results of research on optical quantum manipulation technology, researchers have conducted theoretical and experimental explorations on free-space HD-QKD from different perspectives. Research shows that this work is still in its infancy. Under the conditions of optical diffraction limit, communication distance limit, and photoelectric detection limit, atmospheric turbulence causes nonlinear superposition of the degree of degradation of OAM quantum states. Pointing errors and turbulence effects in the atmospheric environment amplify the differences in the reception efficiency of OAM quantum states, leading to detection efficiency mismatch and system security vulnerabilities, which ultimately affect the secure key generation of OAM-QKD.

[0006] Based on the above problems, there is an urgent need for a method for estimating the high-dimensional quantum key distribution performance of OAM coding under mode-dependent effects, which can calculate the differences in the receiving efficiency of OAM quantum states under pointing errors and turbulence effects, thereby realizing accurate estimation of the high-dimensional quantum key distribution performance. Summary of the invention

[0007] Based on this, it is necessary to provide a method for estimating the performance of high-dimensional quantum key distribution encoded with orbital angular momentum to address the above technical problems.

[0008] A method for estimating the performance of orbital angular momentum encoded high-dimensional quantum key distribution comprises the following steps: using Laguerre-Gaussian modes as orthogonal modal basis sets to obtain OAM beam characteristic parameters; calculating the probability of receiving a correct OAM state under the influence of pointing errors according to the OAM beam characteristic parameters; calculating the probability of receiving a correct OAM state under the influence of atmospheric turbulence according to the OAM beam characteristic parameters; performing key estimation of decoyed state OAM encoded HD-QKD based on the probability of receiving a correct OAM state under the influence of pointing errors and atmospheric turbulence; and performing finite-length security key estimation of the decoyed state OAM encoded HD-QKD according to the key estimation result to obtain the finite-length security key rate of the HD-QKD system.

[0009] In one of the embodiments, the method of using the Laguerre-Gaussian mode as the orthogonal mode basis set to obtain the OAM beam characteristic parameters includes: using the Laguerre-Gaussian mode as the orthogonal mode basis set, and the complex amplitude of the OAM beam in the Laguerre-Gaussian mode is:

[0010]

[0011] In the formula, l and p represent the angular distribution and radial distribution respectively, is the beam radius, which indicates the spot size of the LG beam transmitted to the propagation distance z, w0 is the beam waist radius, represents the Rayleigh distance, k = 2π / λ, λ is the wavelength of light, is the generalized Laguerre polynomial, (r,θ,z) is the polar coordinates; when p = 0, the LG mode intensity distribution of the lowest order radial mode is:

[0012]

[0013] In the formula, A l,p (r, θ, z) represents the amplitude of the OAM beam. Normalizing equation (2) yields ∫∫ D I l (r,θ,z)rdrdθ=1, where D represents the transmission distance.

[0014] In one embodiment, the calculation of the probability of receiving the correct OAM state under the influence of the pointing error according to the characteristic parameters of the OAM beam includes: under the lateral deviation, according to the central limit theorem, assuming that the random displacement deviations x and y caused by the jitter have a mean of 0 and a variance of σ ρ 2 is a Gaussian random variable, ρ represents the jitter offset of the beam center, and the jitter offset ρ is represented by Rayleigh distribution:

[0015]

[0016] The LG mode intensity distribution of OAM after transmission under the influence of pointing error is re-expressed as:

[0017]

[0018] Combined with the finite receiving aperture effect, the signal transmission efficiency caused by the lateral displacement is:

[0019]

[0020] Where P R Represents the received optical power, P T represents the transmitted optical power; the point coordinate method is used to calculate the surface integral:

[0021] L pe (ρ) = L pe (△x,△y)=∫ S I l (x+△x,y+△y,z)dxdy, (6)

[0022] Where, (△x) 2 +(△y) 2 =ρ 2 ; Approximate the telescope area S to a square aperture π(D r / 2) 2 , then the update formula (6) is:

[0023]

[0024] In the formula, Based on the Rayleigh distribution, the transmission efficiency under pointing error is expressed as:

[0025]

[0026] The geometric variance of the pointing error at the receiver is approximately:

[0027] σ ρ 2 (z) = πtan 2 (δ / 2)z 2, (9)

[0028] In the formula, δ represents the angular jitter error of the transmitter, and δ smaller than the preset threshold is approximately expressed as Assuming that the radial deviation and angular deviation of the OAM beam under the action of pointing error are ρ and α, the complex amplitude expression of the misaligned l-state OAM beam is expanded into the helical harmonic function:

[0029]

[0030] In the formula, represents the amplitude component of the m-state OAM beam, E l is the amplitude, represents the binomial coefficient, J m-n (·) represents the first kind of modified Bessel function; the energy of the received beam is expressed as:

[0031]

[0032] In the formula, represents the probability density of the OAM beam in the m-state, ε0 is the vacuum dielectric constant; the proportion of each spiral harmonic energy in different OAM states to the total beam energy is used as the weight factor of each harmonic component, and the probability of receiving the correct OAM state under the influence of pointing error is:

[0033]

[0034] In the formula, C q Represents the spiral harmonic energy of different OAM states under the influence of pointing error.

[0035] In one embodiment, the probability of receiving the correct OAM state under the influence of atmospheric turbulence is calculated according to the characteristic parameters of the OAM beam, including: using Kolmogorov turbulence theory to evaluate the random statistics of atmospheric turbulence, and expressing the probability of receiving the m-state OAM light after the l-state OAM light is transmitted as:

[0036]

[0037] In the formula, <·> represents the function of the statistical mean, represents the atmospheric coherence coefficient, The refractive index structure constant representing the atmospheric turbulence intensity, λ is the wavelength of light, △=|lm|; link transmission efficiency η atm It represents the light intensity loss at the receiving end caused by the atmospheric turbulence effect during the transmission of the normalized OAM initial light, which is used to describe the link attenuation and is:

[0038]

[0039] In the formula, <s0>represents the probability of receiving l-state OAM light after the normalized OAM initial light is transmitted, β represents the link attenuation coefficient, and the probability of transmitting an erroneous signal is Spiral harmonics C under atmospheric turbulence l for:

[0040]

[0041] Then, under the influence of atmospheric turbulence, the probability of receiving the correct OAM state is:

[0042]

[0043] In one of the embodiments, the probability of receiving the correct OAM state under the influence of pointing error and atmospheric turbulence is used to estimate the key of the decoyed state OAM-coded HD-QKD, including: in the OAM-coded HD-QKD system, two mutually unbiased bases SUP base and OAM base are included, wherein the OAM base includes |-l> and |l>, and the SUP base includes According to the GLLP formula, the security key expression of d dimension is:

[0044]

[0045] In the formula, q m represents the probability of selecting a basis, and They represent the gain and quantum bit error rate of the signal state, △1 represents the proportion of emitted single-photon signals, represents the error correction efficiency of the signal state, represents the bit error of the single photon state, H d (x) = -(1-x)log2(1-x)-xlog2[x / (d-1)] represents the Shannon entropy in d dimension. For the decoy state OAM coded HD-QKD system, the single photon signal transmission efficiency η includes the pointing error and the loss of atmospheric turbulence as well as the internal loss at the receiving end, which is expressed as:

[0046] η=η pe ·η atm ·η Bob ·η D , (18)

[0047] Where η Bob represents the internal optical loss at the receiving end, η D represents the detection efficiency of SPD, then for:

[0048]

[0049] Where, Y0 = 2p d represents the yield when the transmitter does not emit photons, p d represents the dark count of the detector, μ represents the average number of photons in the signal state; based on the influence of the receiving bit error, the pointing error and the mode crosstalk, It is expressed as:

[0050]

[0051] Where, e0 = (d-1) / d represents the dark count bit error rate of HD-QKD, and t = 1-P at ·P Pe represents the crosstalk probability, e d represents the misalignment error; using the weak and vacuum state deception methods, we estimate:

[0052]

[0053] Where, Y1 OAM represents the yield of single-photon states, Y1 L,v,0 represents the lower bound of the yield of single-photon trapped states, μ represents the average photon intensity of the signal state, v represents the average photon intensity of the trapped state, and e v Indicates the bit error in the decoy state, e μ Indicates the bit error in the signal state, represents the gain of the decoy state under the OAM basis, Indicates the bit errors in the decoy state under the SUP basis.

[0054] In one embodiment, the step of performing a finite-length security key estimation of a decoyed state OAM-coded HD-QKD system according to the key estimation result to obtain a finite-length security key rate of the HD-QKD system includes:

[0055] For observed values ​​ζ>0, the Chernoff inequality is used to obtain the confidence interval of the expected value of the observed parameter E[ζ]:

[0056]

[0057] In the formula, E U [ζ] and E L [ζ] represents the upper and lower limits of the confidence interval, δ L and δ U for:

[0058]

[0059] Where ε represents the failure probability used to estimate the expected value of the confidence interval;

[0060] According to equations (24) and (25), the upper and lower bounds of the global gain and bit error rate of the signal state and the decoy state under statistical fluctuations are obtained. Combining equations (21)-(23), the upper and lower bounds of the total gain and bit error rate of the single-photon pulse are obtained. The security key length of the OAM basis of the HD-QKD system under the finite length effect is:

[0061]

[0062] In the formula, K OAM,L is the security key length, H d represents the Shannon entropy in d dimension, is the lower limit of the number of single-photon pulses in the key after OAM sieving, is the total amount of OAM-based key after screening, is the upper limit of the phase error rate of the key after the single-photon signal state SUP basis sieve, is the upper limit of the quantum bit error rate of the key after the signal state OAM basis screening; let the number of transmitted quantum signal pulses be N, then the finite length security key rate of the HD-QKD system is:

[0063]

[0064] Compared with the prior art, the advantages and beneficial effects of the present invention are as follows: using Laguerre-Gaussian modes as orthogonal modal basis sets, obtaining OAM beam characteristic parameters, and based on the OAM beam characteristic parameters, respectively calculating the probability of receiving the correct OAM state under the influence of pointing error and the probability of receiving the correct OAM state under the influence of atmospheric turbulence; according to the probability of receiving the correct OAM state under the influence of pointing error and atmospheric turbulence, performing key estimation of decoyed state OAM coded HD-QKD, and finally performing finite-length security key estimation of decoyed state OAM coded HD-QKD, obtaining the finite-length security key rate of the HD-QKD system, and realizing atmospheric turbulence. By combining the flow and OAM, high-dimensional coding is incorporated into the security key rate analysis through the calculation of the influence of OAM quantum state degradation caused by pointing errors and atmospheric turbulence, and the influence of reception errors on system performance under the finite key effect is fully considered. The accurate estimation of the performance of high-dimensional quantum key distribution is achieved, and the effective long security key rate of the HD-QKD system is obtained, avoiding detection efficiency mismatch and system security vulnerabilities, thereby smoothly generating the security key of OAM-QKD. HD-QKD based on OAM coding provides ideas for effective information transmission between local hot spots between ground and air links, which is conducive to promoting the practical application of quantum communication in free space transmission. BRIEF DESCRIPTION OF THE DRAWINGS

[0065] Figure 1 A schematic diagram of a flow chart of a method for estimating performance of high-dimensional quantum key distribution encoded with orbital angular momentum in one embodiment;

[0066] Figure 2 Schematic diagram of wavefront, intensity distribution, and phase distribution of different OAM quantum states in one embodiment;

[0067] Figure 3 A schematic diagram of a link structure in which a transmitting end and a receiving end are misaligned due to a lateral displacement deviation in an embodiment;

[0068] Figure 4 A schematic diagram of OAM phase distortion caused by atmospheric turbulence in one embodiment;

[0069] FIG5( a ) is a graph showing the relationship between the transmission efficiency and transmission distance of an OAM signal caused by different jitter errors under pointing error conditions in one embodiment;

[0070] FIG5( b ) is a graph showing the relationship between OAM signal transmission efficiency and transmission distance under the condition of mode-dependent pointing error in one embodiment;

[0071] FIG6( a ) is a diagram showing the spiral spectrum distribution of an OAM beam under pointing error conditions in one embodiment;

[0072] FIG6( b ) is a graph showing the relationship between the probability of receiving the correct OAM quantum state and the transmission distance under the condition of mode-dependent pointing error in one embodiment;

[0073] Figure 7 is a graph showing the relationship between signal transmission efficiency and transmission distance under different atmospheric turbulence intensities in one embodiment;

[0074] FIG8( a ) is a diagram showing the spiral spectrum distribution of an OAM beam under atmospheric turbulence conditions in one embodiment;

[0075] FIG8( b ) is a graph showing the relationship between the probability of receiving the correct OAM quantum state and the transmission distance after different OAM quantum states are transmitted through atmospheric turbulence in one embodiment;

[0076] Fig. 9 Graph showing the relationship between the HD-QKD key rate and the transmission distance in one embodiment. DETAILED DESCRIPTION

[0077] Before describing the specific embodiments of the present invention, the overall concept of the present invention is described as follows:

[0078] The present invention is mainly developed based on the quantum cryptography distribution process. At present, when a vortex light beam propagates in free space, a mode-dependent effect occurs (the mode-dependent effect in this embodiment includes pointing error and turbulence effect), causing the light spot to be distorted with the OAM quantum state, resulting in differences in the receiving efficiency of high-dimensional quantum signals encoded under different OAM quantum states, leading to security vulnerabilities such as detection efficiency mismatch, which in turn affects the estimation of security keys.

[0079] Therefore, the present invention proposes a method for estimating the performance of high-dimensional quantum key distribution encoded in orbital angular momentum, using Laguerre-Gaussian modes as orthogonal modal basis sets to obtain OAM beam characteristic parameters, and based on the OAM beam characteristic parameters, respectively calculate the probability of receiving the correct OAM state under the influence of pointing error and the probability of receiving the correct OAM state under the influence of atmospheric turbulence; according to the probability of receiving the correct OAM state under the influence of pointing error and atmospheric turbulence, the key of the decoyed state OAM encoded HD-QKD is estimated, and finally the finite-length security key of the decoyed state OAM encoded HD-QKD is estimated to obtain the finite-length security key rate of the HD-QKD system, By combining atmospheric turbulence with OAM, the effects of pointing errors and OAM quantum state degradation caused by atmospheric turbulence are calculated, high-dimensional coding is incorporated into the security key rate analysis, and the impact of reception errors on system performance under the finite key effect is fully considered. Accurate estimation of the performance of high-dimensional quantum key distribution is achieved, and the effective long security key rate of the HD-QKD system is obtained, avoiding detection efficiency mismatch and system security vulnerabilities, thereby smoothly generating OAM-QKD security keys. HD-QKD based on OAM coding provides ideas for effective information transmission between local hot spots between ground and air links, which is conducive to promoting the practical application of quantum communication in free space transmission.

[0080] After introducing the overall concept of the present invention, in order to make the purpose, technical solution and advantages of the present invention more clear, the present invention is further described in detail by specific implementation methods in combination with the accompanying drawings. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0081] In one embodiment, Figure 1 As shown, a method for estimating the performance of high-dimensional quantum key distribution encoded with orbital angular momentum is provided, comprising the following steps:

[0082] Step S110, using the Laguerre-Gaussian mode as an orthogonal mode basis set to obtain OAM beam characteristic parameters.

[0083] Specifically, due to the phase singularity in the spiral phase profile, a non-zero-order OAM beam usually has a donut-shaped intensity profile, such as Figure 2 As shown, in general, the OAM beam can be directed to any helically phased beam regardless of its radial distribution, using the Laguerre-Gaussian (LG) mode as the orthogonal modal basis set, and the desired OAM beam characteristic parameters are calculated.

[0084] Wherein, step S110 includes: using the Laguerre-Gaussian mode as the orthogonal mode basis set, and the complex amplitude of the OAM beam in the Laguerre-Gaussian mode is:

[0085]

[0086] In the formula, l and p represent the angular distribution and radial distribution respectively, is the beam radius, which indicates the spot size of the LG beam transmitted to the propagation distance z, w0 is the beam waist radius, represents the Rayleigh distance, k = 2π / λ, λ is the wavelength of light, is the generalized Laguerre polynomial, (r,θ,z) is the polar coordinates; when p = 0, the LG mode intensity distribution of the lowest order radial mode is:

[0087]

[0088] In the formula, A l,p (r, θ, z) represents the amplitude of the OAM beam. Normalizing equation (2) yields ∫∫ D I l (r,θ,z)rdrdθ=1, where D represents the transmission distance.

[0089] Specifically, the Laguerre-Gaussian mode is used as the orthogonal modal basis set to represent the complex amplitude of the OAM beam, and the LG mode intensity distribution of the lowest-order radial mode is obtained, which is normalized for ease of calculation.

[0090] Step S120, calculating the probability of receiving the correct OAM state under the influence of the pointing error according to the OAM beam characteristic parameters.

[0091] Specifically, in free-space optical links, mechanical errors in the tracking and aiming system and vibrations in the transmitting / receiving platform can cause random beam jitter, which in turn causes wave loss. Due to the limited receiving aperture effect, pointing errors can cause energy loss in the beam. At the same time, for OAM-coded optical communication systems, pointing errors can cause received bit errors, which in turn affect quantum security keys and quantum bit error rates.

[0092] Due to the particularity of the OAM beam profile structure, in order to ensure the signal reception efficiency, the displacement deviation of the beam is analyzed. According to the characteristic parameters of the OAM beam, the probability of receiving the correct OAM state under the influence of the pointing error is calculated, such as Figure 3 , which is a schematic diagram of a link structure in which the transmitter and the receiver are misaligned due to lateral displacement deviation.

[0093] Wherein, step S120 includes: under the lateral deviation, according to the central limit theorem, assuming that the random displacement deviations x and y caused by the jitter have a mean of 0 and a variance of σ ρ 2 is a Gaussian random variable, ρ represents the jitter offset of the beam center, and the jitter offset ρ is represented by Rayleigh distribution:

[0094]

[0095] The LG mode intensity distribution of OAM after transmission under the influence of pointing error is re-expressed as:

[0096]

[0097] Combined with the finite receiving aperture effect, the signal transmission efficiency caused by the lateral displacement is:

[0098]

[0099] Where P R Represents the received optical power, P T represents the transmitted optical power; the point coordinate method is used to calculate the surface integral:

[0100] L pe (ρ) = L pe (△x,△y)=∫ S I l (x+△x,y+△y,z)dxdy, (6)

[0101] Where, (△x) 2 +(△y) 2 =ρ 2 ; Approximate the telescope area S to a square aperture π(D r / 2) 2 , then the update formula (6) is:

[0102]

[0103] In the formula, Based on the Rayleigh distribution, the transmission efficiency under pointing error is expressed as:

[0104] η pe =∫0 ∞ L pe (ρ)·f ρ (ρ)dρ, (8)

[0105] The geometric variance of the pointing error at the receiver is approximately:

[0106] σ ρ 2 (z) = πtan 2 (δ / 2)z 2 , (9)

[0107] In the formula, δ represents the angular jitter error of the transmitter, and δ smaller than the preset threshold is approximately expressed as Assuming that the radial deviation and angular deviation of the OAM beam under the action of pointing error are ρ and α, the complex amplitude expression of the misaligned l-state OAM beam is expanded into the helical harmonic function:

[0108]

[0109] In the formula, represents the amplitude component of the m-state OAM beam, E l is the amplitude, represents the binomial coefficient, J m-n (·) represents the first kind of modified Bessel function; the energy of the received beam is expressed as:

[0110]

[0111] In the formula, represents the probability density of the OAM beam in the m-state, ε0 is the vacuum dielectric constant; the proportion of each spiral harmonic energy in different OAM states to the total beam energy is used as the weight factor of each harmonic component, and the probability of receiving the correct OAM state under the influence of pointing error is:

[0112]

[0113] In the formula, C q Represents the spiral harmonic energy of different OAM states under the influence of pointing error.

[0114] Specifically, in the case of lateral displacement deviation, according to the central limit theorem, we can assume that the random displacement deviation is caused by jitter, and use Rayleigh distribution to represent the jitter offset of the beam center, which is used to update the intensity distribution of OAM after transmission under the influence of pointing error. Combined with the finite receiving aperture effect, the signal transmission efficiency caused by the lateral displacement deviation is obtained.

[0115] Since the power of the OAM beam received by the receiving telescope in each circle is different in the case of misalignment, the polar coordinate method for calculating the received optical power is no longer used, and the point coordinate method is used to calculate the surface integral; since the jitter offset of the beam center obeys the Rayleigh distribution, the transmission efficiency under the influence of the pointing error is obtained; in order to be close to the actual scene, the lateral displacement deviation under different transmission distances is further analyzed to obtain the geometric variance of the pointing error at the receiving end. The pointing error is due to the lateral displacement deviation, which will cause energy divergence when receiving the OAM mode, resulting in received bit errors. The mode purity of the OAM state is reduced, resulting in the reception of erroneous information. Assuming the radial and angular deviations of the OAM beam under the action of the pointing error, the complex amplitude expression of the misaligned l-state OAM beam is expanded into a spiral harmonic function.

[0116] It can be seen from equation (10) that the pointing error causes the pure l-state OAM beam mode to expand, and the power spectrum of each mode is symmetrically distributed around m=l. 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, so that the proportional distribution of different modes can be represented according to the energy weight distribution.

[0117] The weight factor of each harmonic component represents the proportion of each spiral harmonic energy of different OAM states to the total energy of the beam. Therefore, the probability of receiving the correct OAM state under the influence of pointing error is calculated.

[0118] In addition, it can be seen from equations (7) and (10) that the transmission efficiency and detection probability caused by the pointing error are both related to the OAM index l, indicating the existence of the state-dependent pointing error effect.

[0119] Step S130, calculating the probability of receiving the correct OAM state under the influence of atmospheric turbulence according to the OAM beam characteristic parameters.

[0120] Specifically, atmospheric turbulence refers to the irregular movement in the atmosphere. The temperature fluctuations and airflow in the turbulent atmosphere will lead to the appearance of vortices, which will cause local refractive index fluctuations at any location in the atmosphere. Since atmospheric turbulence can cause phase distortion, this degradation effect is particularly obvious in OAM communication. Phase distortion can cause wavefront distortion of the OAM beam, causing the peak position of the OAM beam to shift, thereby interfering with the transmission of information, such as Figure 4 Shown is a diagram of phase distortion caused by atmospheric turbulence.

[0121] Therefore, it is necessary to calculate the probability of receiving the correct OAM state under the influence of atmospheric turbulence based on the characteristic parameters of the OAM beam. Through steps S110 to S130, a channel model of pointing errors and turbulence effects related to the OAM quantum state is established. The atmospheric channel model is combined with OAM, and by analyzing the influence of state-related diffraction, misalignment and turbulence, it is conducive to the accurate estimation of the performance of high-dimensional quantum key distribution, avoiding the difference in receiving efficiency of OAM quantum states with developed pointing errors and turbulence effects in the atmospheric environment, resulting in detection efficiency mismatch and system security vulnerabilities.

[0122] Wherein, step S130 includes: using Kolmogorov turbulence theory to evaluate the random statistics of atmospheric turbulence, and expressing the probability of receiving m-state OAM light after l-state OAM light is transmitted as:

[0123]

[0124] In the formula, <·> represents the function of the statistical mean, represents the atmospheric coherence coefficient, The refractive index structure constant representing the atmospheric turbulence intensity, λ is the wavelength of light, △=|lm|; link transmission efficiency η atm It represents the light intensity loss at the receiving end caused by the atmospheric turbulence effect during the transmission of the normalized OAM initial light, which is used to describe the link attenuation and is:

[0125] η atm = <s0>·exp(-βz), (14)

[0126] In the formula, <s0>represents the probability of receiving l-state OAM light after the normalized OAM initial light is transmitted, β represents the link attenuation coefficient, and the probability of transmitting an erroneous signal is Spiral harmonics C under atmospheric turbulence l for:

[0127]

[0128] Then, under the influence of atmospheric turbulence, the probability of receiving the correct OAM state is:

[0129]

[0130] Specifically, the Kolmogorov turbulence theory is used to evaluate the random statistics of atmospheric turbulence, and the probability of receiving m-state OAM light after l-state OAM light is obtained. It is used to calculate the light intensity loss at the receiving end caused by the atmospheric turbulence effect during the transmission of the normalized OAM initial light, which is used to describe the link attenuation, further expressed as the link transmission efficiency, and the probability of transmitting an erroneous signal is calculated. The analysis of the received bit error is similar to the analysis of the spiral spectrum reception probability under the influence of the pointing error. The formula is The same applies to the atmospheric turbulence model, and finally the spiral harmonic expression under atmospheric turbulence is obtained. Therefore, the probability of receiving the correct OAM state under the influence of atmospheric turbulence is calculated. In addition, formula (13) and formula (15) show that the influence of atmospheric turbulence on the OAM state is related to the mode.

[0131] Step S140, based on the probability of receiving the correct OAM state under the influence of pointing error and atmospheric turbulence, the key of the decoyed state OAM encoding HD-QKD is estimated.

[0132] Specifically, in the OAM-coded HD-QKD system, the SUP basis and the OAM basis form a chain of mutually unbiased bases (MUBs) to ensure the unconditional security of QKD. Normally, the transmitter modulates the quantum state independently through a spatial light modulator (SLM), and then the weak dry pulse is attenuated to the single photon level. Another attenuation is used to implement the "weak + vacuum decoy state" method. After the OAM light is transmitted in the free space link, the photons carrying OAM are detected by the OAM classifier and the single-photon detector (SPD) at the receiving end. According to the probability of receiving the correct OAM state under the influence of pointing error and atmospheric turbulence, the key estimation of the decoy state OAM-coded HD-QKD is performed, and the final secure key distribution is completed after post-processing such as base screening, error estimation, key key negotiation, error correction verification and confidentiality amplification.

[0133] Wherein, step S140 includes: in an OAM-coded HD-QKD system, including two mutually unbiased bases SUP base and OAM base, wherein the OAM base includes |-l> and |l>, and the SUP base includes (n=0,1); According to the GLLP formula, the d-dimensional security key expression is:

[0134]

[0135] In the formula, q m represents the probability of selecting a basis, and They represent the gain and quantum bit error rate of the signal state, △1 represents the proportion of emitted single-photon signals, represents the error correction efficiency of the signal state, represents the bit error of the single photon state, H d (x) = -(1-x)log2(1-x)-xlog2[x / (d-1)] represents the Shannon entropy in d dimension. For the decoy state OAM coded HD-QKD system, the single photon signal transmission efficiency η includes the pointing error and the loss of atmospheric turbulence as well as the internal loss at the receiving end, which is expressed as:

[0136] η=η pe ·η atm ·η Bob ·η D , (18)

[0137] Where η Bob represents the internal optical loss at the receiving end, η D represents the detection efficiency of SPD, then for:

[0138]

[0139] Where, Y0 = 2p d represents the yield when the transmitter does not emit photons, p d represents the dark count of the detector, μ represents the average number of photons in the signal state; based on the influence of the receiving bit error, the pointing error and the mode crosstalk, It is expressed as:

[0140]

[0141] Where, e0 = (d-1) / d represents the dark count bit error rate of HD-QKD, and t = 1-P at ·P Pe represents the crosstalk probability, e d represents the misalignment error; using the weak and vacuum state deception methods, we estimate:

[0142]

[0143]

[0144] Where, Y1 OAM represents the yield of single-photon states, Y1 L,v,0 represents the lower bound of the yield of single-photon trapped states, μ represents the average photon intensity of the signal state, v represents the average photon intensity of the trapped state, and e v Indicates the bit error in the decoy state, e μ Indicates the bit error in the signal state, represents the gain of the decoy state under the OAM basis, Indicates the bit errors in the decoy state under the SUP basis.

[0145] Specifically, the performance degradation of free-space OAM-coded HD-QKD is mainly concentrated on the quantum secure key rate (SKR) and quantum bit error rate (QBER) under the mode-dependent effect. According to the GLLP (Gottesman-Lo-Lütkenhaus-Preskill) formula, the security key of d-dimensional BB84-QKD is obtained, and the yield of single-photon states, the proportion of emitted single-photon signals, and the bit error of single-photon states are estimated through the "weak + vacuum decoy state" method.

[0146] Step S150: According to the key estimation result, the finite-length security key estimation of the decoyed state OAM encoding HD-QKD is performed to obtain the finite-length security key rate of the HD-QKD system.

[0147] Specifically, due to the differences between HD-QKD and 2D-QKD in key analysis, the security of the HD-QKD system is analyzed in the asymptotic domain, especially in the limit of infinite key effect. In the asymptotic case, the yield and bit error rate of the single-photon state required by (17) can be accurately estimated by the OAM-coded QKD measurement values ​​and converge to the approximate true value, thereby securely obtaining the final key length of the OAM-coded QKD.

[0148] However, in actual HD-QKD systems, the number of quantum signal pulses emitted is limited, so there are statistical fluctuations between the observed values ​​and the true values ​​of the system parameters. When analyzing the security of HD-QKD systems, it is necessary to consider the impact of statistical fluctuations on the finite key effect response to the final security key.

[0149] Therefore, according to the key estimation result, the finite-length security key estimation of the decoyed state OAM encoded HD-QKD can be performed to obtain the finite-length security key rate of the HD-QKD system.

[0150] Wherein, step S150 includes: for the observed value ζ>0, using the Chernoff inequality to obtain the confidence interval of the expected value E[ζ] of the observed parameter:

[0151]

[0152] In the formula, E U [ζ] and E L [ζ] represents the upper and lower limits of the confidence interval, δ L and δ U for:

[0153]

[0154] In the formula, ε represents the failure probability used to estimate the expected value of the confidence interval. According to equations (24) and (25), the upper and lower bounds of the global gain and bit error rate of the signal state and the decoy state under statistical fluctuations are obtained. Combining equations (21)-(23), the upper and lower bounds of the total gain and bit error rate of the single-photon pulse are obtained. Then, the security key length of the OAM basis of the HD-QKD system under the finite length effect is:

[0155]

[0156] In the formula, K OAM,L is the security key length, H d represents the Shannon entropy in d dimension, is the lower limit of the number of single-photon pulses in the key after OAM sieving, is the total amount of OAM-based key after screening, is the upper limit of the phase error rate of the key after the single-photon signal state SUP basis sieve, is the upper limit of the quantum bit error rate of the key after the signal state OAM basis screening; let the number of transmitted quantum signal pulses be N, then the finite length security key rate of the HD-QKD system is:

[0157]

[0158] Specifically, the Chernoff inequality is used to obtain the confidence interval of the expected value of the observation parameter. Through the upper and lower bounds of the global gain and bit error rate of the signal state and the decoy state under statistical fluctuations, as well as the upper and lower limits of the total gain and bit error rate of the single-photon pulse, the security key length of the OAM basis of the HD-QKD system under the finite length effect is finally obtained. When the number of transmitted quantum signal pulses is obtained, the finite-length security key rate of the HD-QKD system is obtained, thereby generating a higher security key rate within a finite length.

[0159] In this embodiment, Laguerre-Gaussian modes are used as orthogonal modal basis sets to obtain OAM beam characteristic parameters, and based on the OAM beam characteristic parameters, the probability of receiving the correct OAM state under the influence of pointing error and the probability of receiving the correct OAM state under the influence of atmospheric turbulence are calculated respectively; according to the probability of receiving the correct OAM state under the influence of pointing error and atmospheric turbulence, the key estimation of the decoyed state OAM-encoded HD-QKD is performed, and finally the finite-length security key estimation of the decoyed state OAM-encoded HD-QKD is performed to obtain the finite-length security key rate of the HD-QKD system, and by calculating the influence of the OAM quantum state degradation caused by pointing error and atmospheric turbulence, the high-dimensional quantum key distribution performance is accurately estimated, and the effective long security key rate of the HD-QKD system is obtained, thereby avoiding detection efficiency mismatch and system security vulnerabilities, thereby smoothly generating the security key of OAM-QKD.

[0160] In one embodiment, MATLAB is used for experimental simulation, targeting a 10 km free space link scenario, and the high-dimensional quantum key distribution performance estimation method proposed in the present invention is simulated and analyzed. The main parameters of the simulation are shown in Table 1:

[0161] Table 1. Parameters of OAM-QKD numerical simulation

[0162]

[0163] In order to evaluate the transmission characteristics of OAM under the influence of pointing error, the relationship between transmission distance and efficiency is simulated. Considering the selection of transmitter jitter error, the variation of divergence angle is numerically analyzed. According to the relationship between divergence angle and beam waist radius The divergence angle of the beam is θ≈98.8μrad. The jitter error δ is determined by the divergence angle, and the requirements for the tracking and aiming system are evaluated from it. -6 , 10 -5 , 10 -4 The three values ​​are simulated to analyze the signal transmission efficiency caused by pointing errors under different jitters.

[0164] As shown in Figure 5(a), smaller angular jitter error has greater advantages in short-distance (z < < 3 km) transmission and significantly improves signal transmission efficiency; in long-distance transmission, the impact of angular jitter error can be ignored. Considering the system cost and the performance requirements for the tracking and aiming system, δ = 10 is selected. -5 rad is used as the angular jitter error for the next analysis. Figure 5(b) verifies that the pointing error has a mode-dependent effect on the OAM transmission efficiency.

[0165] In order to measure the impact of pointing error on the bit error at the receiving end, the received spiral spectrum distribution is analyzed at 3 km. The OAM index of the transmitted light is set to m = 3, and the angular index p = 0. The horizontal axis of Figure 6 (a) represents the OAM index l of each spiral harmonic after the optical beam passes through the pointing error, and the vertical axis represents the weight of the energy occupied by the spiral harmonic component. The OAM index of the incident light is m = 3, and the OAM components of the pointing error are mainly distributed at l = 2, 3, 4, and 5. The OAM spiral spectrum caused by the pointing error is unevenly distributed and concentrated around l = 3. Obviously, low-order OAM light is easier to identify and receive than high-order OAM light. As can be seen from Figure 6 (b), different OAM modes will also affect the probability of the received spiral spectrum distribution caused by pointing error. When the received OAM index is different from the incident light, it means that a bit error is received, which in turn affects the performance of the system.

[0166] In order to demonstrate the effect of atmospheric turbulence on transmission efficiency and provide data for security key rate analysis, the present invention conducts middle weak The transmission efficiency under turbulent conditions is simulated and analyzed. Figure 7 It can be seen that under weak turbulence conditions, signals can be transmitted efficiently.

[0167] As shown in Figure 8(a), the spiral spectrum distribution of the OAM beam after the incident light with OAM index m = 3 is transmitted through 3km of atmospheric turbulence. The OAM components of the received beam are mainly distributed in l = 0, 1, 2, 3, 4, 5, 6. Unlike the pointing error, the incident OAM light is diffused by atmospheric turbulence, and the beam component is equal to the original OAM, which means that the probability of receiving other modes is a symmetrical distribution centered on the correct mode. Figure 8(b) simulates the relationship between the probability of receiving the correct OAM mode and the transmission distance under different OAM modes, confirming the existence of the mode-dependent turbulence effect, and providing data support for subsequent HD-QKD key analysis.

[0168] Through the above simulation, the theoretical model and simulation data of OAM light transmission in the atmospheric channel were obtained. In order to further reflect the application value of theoretical analysis, the performance of OAM-coded HD-QKD was evaluated. In the practical process of HD-QKD, the analysis of finite length effect cannot be ignored. Therefore, a finite transmission pulse is selected to analyze the relationship between the security key rate and the transmission distance in different dimensions. It is worth noting that the quantum bit error rate analysis of HD-QKD takes into account the influence of pointing error and atmospheric turbulence, and the probability of sending and receiving each OAM mode is equal.

[0169] from Fig. 9 It can be seen that in the context of short-distance transmission, high-dimensional QKD has a great advantage in the generation of secure key rates. In the case of long-distance transmission, 2D-QKD should be selected for key transmission, and 2D-QKD has a longer transmission distance than high-dimensional QKD. The above simulation results show that there is no direct relationship between the choice of QKD dimension and system performance. In actual scenarios, the appropriate dimension should be selected to implement QKD according to the task requirements.

[0170] According to the above simulation results, it can be seen that the appropriate dimension can be selected to implement QKD according to the actual task requirements. HD-QKD based on OAM coding provides ideas for effective information transmission between local hot spots between ground and air links, especially on airborne platforms, which effectively promotes the practical application of quantum communication in free space transmission. Therefore, a more comprehensive model can be developed to improve system performance, optimize parameter selection, and further promote the practical application of the system using machine learning technology.

[0171] In the prior art, OAM, as a high-dimensional encoding method of light field freedom, has been widely used in the field of quantum communication. Although the latest breakthrough experiments have proved the feasibility of HD-QKD links based on OAM encoding, the potential statistical characteristics of atmospheric channels in high-dimensional quantum systems have not been well explained and experimentally verified. The present invention proposes a method for estimating the performance of high-dimensional quantum key distribution with orbital angular momentum encoding under mode-dependent effects, which improves the completeness of HD-QKD theory. By combining the atmospheric channel model with OAM, the influence of state-dependent diffraction, misalignment and turbulence is analyzed. At the same time, high-dimensional coding is incorporated into the security key rate analysis, and the influence of reception errors on system performance under finite key effects is considered; it has important reference value for future research on free-space quantum communication channel modeling based on OAM encoding.

[0172] The above contents are further detailed descriptions of the present invention in combination with specific implementation methods, and it cannot be determined that the specific implementation of the present invention is limited to these descriptions. For ordinary technicians in the technical field to which the present invention belongs, several simple deductions or substitutions can be made without departing from the concept of the present invention, which should be regarded as falling within the scope of protection of the present invention.

Claims

1. A method for estimating the performance of high-dimensional quantum key distribution encoded with orbital angular momentum, characterized in that: The following steps are involved: Using Laguerre-Gaussian modes as the orthogonal mode basis set, the characteristic parameters of the OAM beam are obtained; According to the OAM beam characteristic parameters, calculating the probability of receiving a correct OAM state under the influence of pointing error; Calculating the probability of receiving a correct OAM state under the influence of atmospheric turbulence according to the OAM beam characteristic parameters; Based on the probability of receiving the correct OAM state under the influence of pointing error and atmospheric turbulence, the key estimation of decoy OAM coded HD-QKD is performed. According to the key estimation results, the finite-length security key estimation of the decoyed state OAM encoded HD-QKD is performed to obtain the finite-length security key rate of the HD-QKD system.

2. The method for estimating the performance of high-dimensional quantum key distribution encoded by orbital angular momentum according to claim 1, characterized in that: The method uses the Laguerre-Gaussian mode as an orthogonal mode basis set to obtain the OAM beam characteristic parameters, including: Using the Laguerre-Gaussian mode as the orthogonal mode basis set, the complex amplitude of the OAM beam in the Laguerre-Gaussian mode is: In the formula, l and p represent the angular distribution and radial distribution respectively, is the beam radius, which indicates the spot size of the LG beam transmitted to the propagation distance z, w0 is the beam waist radius, represents the Rayleigh distance, k = 2π / λ, λ is the wavelength of light, is the generalized Laguerre polynomial, (r,θ,z) is the polar coordinates; When p = 0, the LG mode intensity distribution of the lowest order radial mode is: In the formula, A l,p (r, θ, z) represents the amplitude of the OAM beam. Normalizing equation (2) yields ∫∫ D I l (r,θ,z)rdrdθ=1, where D represents the transmission distance.

3. The method for estimating the performance of high-dimensional quantum key distribution encoded by orbital angular momentum according to claim 2, characterized in that: The calculating, according to the OAM beam characteristic parameters, the probability of receiving a correct OAM state under the influence of the pointing error comprises: Under lateral deviation, according to the central limit theorem, it is assumed that the random displacement deviations x and y caused by jitter have a mean of 0 and a variance of σ ρ 2 The Gaussian random variable of ρ represents the jitter offset of the beam center. The jitter offset ρ is represented by Rayleigh distribution: The LG mode intensity distribution of OAM after transmission under the influence of pointing error is re-expressed as: Combined with the finite receiving aperture effect, the signal transmission efficiency caused by the lateral displacement is: Where P R Represents the received optical power, P T Indicates the transmitted optical power; Use the point coordinate method to find the surface integral: L pe (ρ)=L pe (△x,△y)=∫ S I l (x+△x,y+△y,z)dxdy, (6) Where, (△x) 2 +(△y) 2 =ρ 2 ; The telescope area S is approximated as a square aperture π(D r / 2) 2 , then the update formula (6) is: In the formula, Based on the Rayleigh distribution, the transmission efficiency under pointing error is expressed as: The geometric variance of the pointing error at the receiver is approximately: s ρ 2 (z)=πtan 2 (δ / 2)z 2 , (9) In the formula, δ represents the angular jitter error of the transmitter, and δ smaller than the preset threshold is approximately expressed as Assuming that the radial deviation and angular deviation of the OAM beam under the action of pointing error are ρ and α, the complex amplitude expression of the misaligned l-state OAM beam is expanded into the helical harmonic function: In the formula, represents the amplitude component of the m-state OAM beam, E l is the amplitude, represents the binomial coefficient, J m-n (·) represents the modified Bessel function of the first kind; The energy of the received beam is expressed as: In the formula, represents the probability density of the OAM beam in the m-state, ε0 is the vacuum dielectric constant; The ratio of each spiral harmonic energy of different OAM states to the total beam energy is used as the weight factor of each harmonic component. The probability of receiving the correct OAM state under the influence of pointing error is: In the formula, C q Represents the spiral harmonic energy of different OAM states under the influence of pointing error.

4. The method for estimating the performance of high-dimensional quantum key distribution encoded by orbital angular momentum according to claim 3, characterized in that: The calculating, according to the OAM beam characteristic parameters, the probability of receiving a correct OAM state under the influence of atmospheric turbulence comprises: The Kolmogorov turbulence theory is used to evaluate the random statistics of atmospheric turbulence, and the probability of receiving m-state OAM light after transmitting l-state OAM light is expressed as: In the formula, <·> represents the function of the statistical mean, represents the atmospheric coherence coefficient, The refractive index structure constant that represents the intensity of atmospheric turbulence, λ is the wavelength of light, △ = |lm|; Link transmission efficiency η atm It represents the light intensity loss at the receiving end caused by the atmospheric turbulence effect during the transmission of the normalized OAM initial light, which is used to describe the link attenuation and is: or atm = <s0> ·exp(-βz), (14) In the formula, <s0>represents the probability of receiving l-state OAM light after the normalized OAM initial light is transmitted, β represents the link attenuation coefficient, and the probability of transmitting an erroneous signal is Spiral harmonics C under atmospheric turbulence l for: Then, under the influence of atmospheric turbulence, the probability of receiving the correct OAM state is:

5. The method for estimating the performance of high-dimensional quantum key distribution encoded by orbital angular momentum according to claim 4, characterized in that: The key estimation of the decoyed state OAM coded HD-QKD based on the probability of receiving the correct OAM state under the influence of the pointing error and the atmospheric turbulence includes: In the OAM-coded HD-QKD system, two mutually unbiased bases SUP base and OAM base are included, where the OAM base includes |-l> and |l>, and the SUP base includes (n=0,1); According to the GLLP formula, the security key expression of d dimension is: In the formula, q m represents the probability of selecting a basis, and They represent the gain and quantum bit error rate of the signal state, △1 represents the proportion of emitted single-photon signals, represents the error correction efficiency of the signal state, represents the bit error of the single photon state, H d (x) = -(1-x)log2(1-x)-xlog2[x / (d-1)] represents the Shannon entropy in d dimension; For the decoyed state OAM coded HD-QKD system, the single-photon signal transmission efficiency η includes the pointing error and the loss of atmospheric turbulence as well as the internal loss at the receiving end, which is expressed as: the=the pe ·or atm ·or Bob ·or D , (18) Where η Bob represents the internal optical loss at the receiving end, η D represents the detection efficiency of SPD, then for: Where, Y0 = 2p d represents the yield when the transmitter does not emit photons, p d represents the dark count of the detector, μ represents the average number of photons in the signal state; Based on the influence of receiving bit error, pointing error and mode crosstalk, It is expressed as: Where, e0 = (d-1) / d represents the dark count bit error rate of HD-QKD, and t = 1-P at ·P Pe represents the crosstalk probability, e d represents misalignment error; Using the weak and vacuum state deception methods, we estimate: Where, Y1 OAM represents the yield of single-photon states, represents the lower bound of the yield of single-photon trapped states, μ represents the average photon intensity of the signal state, v represents the average photon intensity of the trapped state, and e v Indicates the bit error in the decoy state, e μ Indicates the bit error in the signal state, represents the gain of the decoy state under the OAM basis, Indicates the bit errors in the decoy state under the SUP basis.

6. The method for estimating the performance of high-dimensional quantum key distribution encoded by orbital angular momentum according to claim 5, characterized in that: The method of performing a finite-length security key estimation of a decoyed state OAM-encoded HD-QKD system according to the key estimation result to obtain a finite-length security key rate of the HD-QKD system includes: For observed values ​​ζ>0, the Chernoff inequality is used to obtain the confidence interval of the expected value of the observed parameter E[ζ]: In the formula, E U [ζ] and E L [ζ] represents the upper and lower limits of the confidence interval, δ L and δ U for: Where ε represents the failure probability used to estimate the expected value of the confidence interval; According to equations (24) and (25), the upper and lower bounds of the global gain and bit error rate of the signal state and the decoy state under statistical fluctuations are obtained. Combining equations (21)-(23), the upper and lower bounds of the total gain and bit error rate of the single-photon pulse are obtained. The security key length of the OAM basis of the HD-QKD system under the finite length effect is: In the formula, K OAM,L is the security key length, H d represents the Shannon entropy in d dimension, is the lower limit of the number of single-photon pulses in the key after OAM sieving, is the total amount of OAM-based key after screening, is the upper limit of the phase error rate of the key after the single-photon signal state SUP basis sieve, is the upper limit of the quantum bit error rate of the key after the signal state OAM basis sieving; Let the number of transmitted quantum signal pulses be N, then the finite length security key rate of the HD-QKD system is:

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