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arXiv:2610.38808

Key Evaluation of a Relativistic Quantum Key Distribution Over a Turbulent Free-space Optical Channel

G. Bebrov

quant-phSubmitted Oct 28, 2026• Announced Oct 30, 2026• 0 citations
arXivHTMLPDF

Abstract

We investigate the performance of relativistic quantum key distribution (RQKD) over turbulent free-space optical channels for both single-photon and coherent-state implementations. A theoretical model is developed to describe the effect of atmospheric turbulence on interferometric visibility and secret-key generation by incorporating turbulence-induced phase fluctuations into the key-rate analysis. Numerical simulations are performed under realistic channel conditions to evaluate the achievable communication distance and secret-key rate. The results show that atmospheric turbulence degrades the protocol performance by reducing interference visibility, with the severity of the degradation determined by the ratio between the interferometer-arm separation $r$ (or temporal separation of signal and reference states $ΔT$) and Fried's parameter. The single-photon protocol remains operational over all investigated turbulence regimes, whereas the coherent-state implementation exhibits a critical dependence on the interferometer geometry (especially on $ΔT$), leading to a threshold beyond which secure key generation is no longer possible. These findings quantify the impact of atmospheric turbulence on relativistic quantum key distribution and provide practical guidelines for the design and optimization of free-space RQKD systems.

AI Summary

The paper studies how atmospheric turbulence affects secret-key generation in single-photon and weak-coherent-state relativistic QKD over free-space optical links, focusing on interferometric phase noise and arm geometry. The analysis combines atmospheric attenuation, turbulence-induced beam spreading, detector imperfections, and Gaussian phase fluctuations. Averaging the relative phase produces a visibility factor $e^{-\sigma_\phi^2}$, with phase variance related to the arm separation $r$ and Fried parameter $r_0$. For single photons, the proposed rate is $R=Y_1p_{\mathrm{det}}[1-\chi-fH(e)]$, using $\chi\simeq0.601$ from the Holevo information of the two transmitted phase-encoded states. The coherent-state calculation uses source replacement, a detection POVM incorporating imperfections, and the Devetak–Winter formula. Numerical evaluations use an $850\,\mathrm{nm}$ wavelength, $5\,\mathrm{cm}$ transmitter beam radius and receiver aperture radius, detector efficiency $0.5$, and dark-count probability $10^{-10}$. Single-photon rates remain nonzero over finite distance ranges for all three tested geometries, $r/r_0=1/10,1/15,1/20$, with stronger phase noise reducing achievable range. For coherent states, the reported rate vanishes at every nonzero distance when $r/r_0=1/10$; smaller ratios, including $1/30$, restore positive rates and extend the range. The treatment is theoretical and numerical, assumes asymptotically many rounds, and uses collective-attack security analysis for coherent states. Phase fluctuations are modeled as independent Gaussian contributions, and the reported thresholds concern the selected channel, detector, and geometry parameters. No experimental validation or finite-key analysis is presented.

Little sign of AI-generated textIncrementalMissing comparisonStrong assumptionsPossible error

Central claim may not hold: For r/r0=1/10 and 1/15, the paper's printed single-photon equations give negative secret fractions even with ideal error correction, contradicting its claim of positive rates in every investigated scenario. The single-photon calculation uses an unconditional erroneous-click probability instead of the sifted QBER, and its proposed optical projection cannot select the classical phase-choice labels. The noisy coherent-state Holevo calculation reuses a two-pure-state formula although erroneous clicks introduce additional ensemble components and generally mixed conditional states, without establishing a valid replacement security bound. The coherent-state turbulence calculation substitutes mean trigonometric factors inside nonlinear detection exponentials without justifying this as an approximation to phase averaging. The numerical scans vary physical arm separation with distance by fixing r/r0, while the asserted temporal-separation dependence lacks a quantitative conversion or temporal turbulence model.

Written by gpt-6.1-sol from the paper's full text. Automated: it can be wrong.

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30 pages, 7 figues

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The paper studies how atmospheric turbulence affects secret-key generation in single-photon and weak-coherent-state relativistic QKD over free-space optical links, focusing on interferometric phase noise and arm geometry.
The analysis combines atmospheric attenuation, turbulence-induced beam spreading, detector imperfections, and Gaussian phase fluctuations. Averaging the relative phase produces a visibility factor e−σϕ2e^{-\sigma_\phi^2}e−σϕ2​, with phase variance related to the arm separation rrr and Fried parameter r0r_0r0​. For single photons, the proposed rate is R=Y1pdet[1−χ−fH(e)]R=Y_1p_{\mathrm{det}}[1-\chi-fH(e)]R=Y1​pdet​[1−χ−fH(e)], using χ≃0.601\chi\simeq0.601χ≃0.601 from the Holevo information of the two transmitted phase-encoded states. The coherent-state calculation uses source replacement, a detection POVM incorporating imperfections, and the Devetak–Winter formula.
Numerical evaluations use an 850 nm850\,\mathrm{nm}850nm wavelength, 5 cm5\,\mathrm{cm}5cm transmitter beam radius and receiver aperture radius, detector efficiency 0.50.50.5, and dark-count probability 10−1010^{-10}10−10. Single-photon rates remain nonzero over finite distance ranges for all three tested geometries, r/r0=1/10,1/15,1/20r/r_0=1/10,1/15,1/20r/r0​=1/10,1/15,1/20, with stronger phase noise reducing achievable range. For coherent states, the reported rate vanishes at every nonzero distance when r/r0=1/10r/r_0=1/10r/r0​=1/10; smaller ratios, including 1/301/301/30, restore positive rates and extend the range.
The treatment is theoretical and numerical, assumes asymptotically many rounds, and uses collective-attack security analysis for coherent states. Phase fluctuations are modeled as independent Gaussian contributions, and the reported thresholds concern the selected channel, detector, and geometry parameters. No experimental validation or finite-key analysis is presented.
We devoted to analyzing information transfer in quantum channels in terms of the performance of quantum key distribution (QKD) free-space optical (FSO) communication systems through Bennett and Brassard 84 (BB84) protocol by taking into account the combined effect of attenuation, turbulence, pointing error, and angle of arrival (AOA) fluctuations. Path loss is modeled by Beer-Lambert law, and the turbulent channel is assumed to be Fisher-Snedecor F distributed. The pointing error and AOA fluctuations are modeled by Hoyt and Rayleigh distributions, respectively. The error and sift probabilities, quantum bit error rate, and secret key rate of a QKD FSO system are obtained in analytical form, and results are presented depending on various parameters. It is shown that increased beam waist, beam deviation, AOA fluctuations, and propagation distance cause performance degradation and less secure information transmission. However, an increase in the ratio of horizontal and vertical beam misalignments, field of view, average photon number per pulse, visibility of the atmospheric environment, and optical beam wavelength impact the performance of QKD FSO systems positively. The effect of the turbulence structure constant is observed to be modest on the performance of the QKD FSO system.
Quantum key distribution is a quantum communication protocol which seeks to address potential vulnerabilities in data transmission and storage. One of the main challenges in the field is achieving high rates of secret key in lossy and turbulent free-space channels. In this scenario, most experimental demonstrations have used the polarization of photons as their qubit carriers, due to the relative robustness of polarization in free-space propagation. Time-bin or phase-based protocols are considered less practical due to the wave-front distortion caused by atmospheric turbulence. However, demonstrations of novel free-space interferometer designs are enabling interferometers to measure multimodal signals with high visibility. That means it is now viable to consider the prospects of implementing time-bin or phase-based protocols, which have demonstrated high key rates and long transmission distances in optical fiber. In this work, we present the possibilities of implementing time-bin protocols in turbulent free-space channels, using the coherent one-way protocol as the example. We present an analysis of the secret key rate and quantum bit error rate of the system, providing the errors due to noise counts, and the extinction ratio of the pulses. Finally, we developed a model to quantify the expected losses for a turbulence free-space channel, specifically for a free-space satellite-to-ground station channel.
Secure data transmission is essential in today's world, and quantum key distribution (QKD) is widely recognized as the technology of choice to enable information-theoretical secure symmetric key generation. While optical fibers are typically used as a transmission channel, wireless transmission offers the flexibility to reach every part of the Earth; moreover, for Earth-Satellite QKD, wireless communication naturally becomes a necessity. In this study, we deployed a fiber-wireless optical system that transmitted a 64-QAM 400 Gbps classical signal for high-rate data exchange and a 1 MHz quantum signal for QKD. The impact of turbulence in the quantum system is evaluated by transmitting single-polarization quantum states through a turbulent free-space channel and by calculating the respective intrinsic quantum bit error rate (QBER). From this, we estimate the system's secret key rate (SKR) and its dependence on turbulence. The effects of atmospheric turbulence on the optical wireless signal were emulated and investigated by using a custom-made atmospheric chamber to induce different turbulent scenarios. The atmospheric chamber allowed us to vary the turbulence in a Rytov variance range from 0.06 to 2.52. The classical signal showed an overall average reliability of 82% for the mentioned turbulence regimes. The results of the quantum signal showed that for Rytov variances up to 1.3, and under the conditions tested, secret key generation is possible, reaching a maximum SKR of 8.7×10−4\mathbf {\times 10^{-4}}×10−4bits per pulse during the produced turbulence regimes.
Abstract In this paper, we analyze the performance of free-space optical quantum key distribution (FSO-QKD) using the BB84 protocol with BPSK modulation under realistic channel conditions. The Málaga distribution is employed to model the composite atmospheric turbulence, considering pointing errors. We derive closed-form expressions for the average quantum bit error rate (QBER) and the secure key rate (SKR), incorporating both turbulence and misalignment effects. Numerical results demonstrate the impact of turbulence parameters and beam misalignment on system performance. The findings provide insights for the robust design of FSO-QKD systems under practical deployment conditions.
Continuous variable quantum key distribution (CVQKD) is the sharing of a secret key between parties using the continuous quadratures of light, the phase and amplitude quadratures, which can infer the presence of an eavesdropper using fundamental quantum mechanics. A significant performance inhibitor of free-space CVQKD is turbulence which gives rise to optical wavefront distortions consisting of random phase and amplitude fluctuations (scintillation). In this work, we experimentally study the negative effects of turbulence on coherent states in a free-space channel for CVQKD. We demonstrate during coherent state transmissions from a continuous-wave laser in a turbulent channel, the interferometric visibility between a local oscillator and signal fluctuates and decreases with higher degrees of scintillation. This leads to the breaking of the phase-lock between the local oscillator and quantum signal and a decrease in the secret key rate (SKR). By incorporating an adaptive optics system, the degraded optical wavefront from turbulence can be corrected using a closed-feedback loop. This leads to the stabilisation and reduction in the decrease of the interferometric visibility in a turbulent channel. The improvement provided by adaptive optics leads to the increased performance of CVQKD, resulting in positive SKRs, which would have otherwise been infeasible in a turbulent channel.
Abstract Quantum Key Distribution (QKD) has been widely applied due to its theoretical one-time pad encryption. Typically employing laser beam as carrier, quantum states are transmitted through optical fibers or free-space channels, with the latter being more commonly adopted. However, compared to optical fibers, free-space links are more susceptible to external environmental factors, particularly atmospheric conditions. Beam propagation through the atmosphere experiences diffraction, refraction, and intensity attenuation, which can degrade key distribution efficiency and even prevent key generation at the receiver. The atmospheric turbulence poses the most significant challenge in all influencing factors, especially for the commonly encountered moderate-to-strong turbulence regimes. Therefore, in this work, we develop a phase screen model and turbulence-dominated transmittance model to comprehensively investigate the impact of various key parameters on optical beam propagation through turbulent atmospheres. Furthermore, we derive the key rate for the CV-QKD protocol using a TMSV (two-mode squeezed vacuum) state and perform numerical simulations.