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Chinese Physics B

Free-space Continuous-Variable Quantum Key Distribution in moderate and strong atmospheric turbulence

Hailong Zhang, Xingran Chen, Jianhong Shi

Published May 11, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1674-1056/ae6b34

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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.
The increasing prevalence of satellite-to-ground laser communications necessitates unconditional security measures that so far can only be provided by quantum key distribution. However, in a free-space optical channel, the diffraction of laser beams along with turbulence effects can reduce the initially high secure key rates potentially provided by continuous variable quantum key distribution (CV-QKD). Using derived formulas from four-state discrete modulation CV-QKD with reverse reconciliation, we highlight parameters that yield key rates on the order of 50 Mbps, and then describe the possible factors that can cause its reduction in a free-space optical channel. Numerical beam propagation simulations are performed to supplement the analyses.
Abstract Free-space quantum key distribution (QKD) offers broader geographical coverage and more flexible system deployment compared to fiber-based systems. However, the free-space environment is highly complex, and various attenuation factors can significantly reduce key distribution efficiency or even lead to encoding failures. This paper discusses and analyzes the impact of turbulence and fog in mountainous environments on free-space discrete-variable quantum key distribution (DV-QKD). Through numerical simulation, the study examines the effects of altitude and visibility on transmittance and turbulence intensity, finding that turbulence intensity decreases with increasing altitude while transmittance increases; improvements in visibility also lead to increased transmittance. The beam wandering due to turbulence is also dominant, combining these factors, the effects on total transmittance and secret key rate are taken into consideration. Our main work would provide a certain reference for the deployment of practical QKD systems in actual mountainous environment.
Abstract Since the invention of the laser in the 60s, one of the most fundamental communication channels has been the free-space optical channel. For this type of channel, a number of effects generally need to be considered, including diffraction, refraction, atmospheric extinction, pointing errors and, most importantly, turbulence. Because of all these adverse features, the free-space channel is more difficult to study than a stable fiber-based link. For the same reasons, only recently it has been possible to establish the ultimate performances achievable in quantum communications via free-space channels. Differently from previous literature, mainly focused on the regime of weak turbulence, this work considers the free-space optical channel in the more challenging regime of moderate-to-strong turbulence. This regime may occur in long-distance free-space links on the ground, in uplink to high-altitude platform systems (HAPS) and, more interestingly, in downlink from near-horizon satellites. In such a regime we rigorously investigate ultimate limits for quantum communications and show that composable keys can be extracted using continuous variable quantum key distribution. In particular, we apply our results to downlink from satellites at large zenith angles, for which not only turbulence is strong but also refraction causes non-trivial effects in terms of trajectory elongation.
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 rrr (or temporal separation of signal and reference states ΔTΔTΔT) 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 ΔTΔTΔT), 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.
Continuous-variable quantum key distribution is a practical application of quantum information theory that is aimed at generation of secret cryptographic key between two remote trusted parties and that uses multi-photon quantum states as carriers of key bits. Remote parties share the secret key via a quantum channel, that presumably is under control of of an eavesdropper, and which properties must be taken into account in the security analysis. Well-studied fiber-optical quantum channels commonly possess stable transmittance and low noise levels, while free-space channels represent a simpler, less demanding and more flexible alternative, but suffer from atmospheric effects such as turbulence that in particular causes a non-uniform transmittance distribution referred to as fading. Nonetheless free-space channels, providing an unobstructed line-of-sight, are more apt for short, mid-range and potentially long-range (using satellites) communication and will play an important role in the future development and implementation of QKD networks. It was previously theoretically shown that coherent-state CV QKD should be in principle possible to implement over a free-space fading channel, but strong transmittance fluctuations result in the significant modulation-dependent channel excess noise. In this regime the post-selection of highly transmitting sub-channels may be needed, which can even restore the security of the protocol in the strongly turbulent channels. We now report the first proof-of-principle experimental test of coherent state CV QKD protocol using different levels Gaussian modulation over a mid-range (1.6-kilometer long) free-space atmospheric quantum channel. The transmittance of the link was characterized using intensity measurements for the reference but channel estimation using the modulated coherent states was also studied. We consider security against Gaussian collective attacks, that were shown to be optimal against CV QKD protocols . We assumed a general entangling cloner collective attack (modeled using data obtained from the state measurement results on both trusted sides of the protocol), that allows to purify the noise added in the quantum channel . Our security analysis of coherent-state protocol also took into account the effect of imperfect channel estimation, limited post-processing efficiency and finite data ensemble size on the performance of the protocol. In this regime we observe the positive key rate even without the need of applying post-selection. We show the positive improvement of the key rate with increase of the modulation variance, still remaining low enough to tolerate the transmittance fluctuations. The obtained results show that coherent-state CV QKD protocol that uses real free-space atmospheric channel can withstand negative influence of transmittance fluctuations, limited post-processing efficiency, imperfect channel estimation and other finite-size effects, and be successfully implemented. Our result paves the way to the full-scale implementation of the CV QKD in real free-space channels at mid-range distances.
Abstract Free-space continuous-variable quantum key distribution (CV-QKD) links provide greater information capability and broader geographical coverage. A major challenge for the practical use of free-space quantum systems is that the signal-to-noise ratios of the channels are relatively low and are sensitive to atmospheric disturbance. The atmospheric channel has several negative characteristics that diminish the secret key rate, such as beam extinction and various turbulence effects. The effect of beam wandering on the quantum properties is mainly considered, and the probability distribution of the transmission coefficient is derived to establish the channel model. Low-density parity-check (LDPC) codes operated in the reconciliation procedure are utilized to deal with information interactivity. However, the LDPC codes need to be chosen from a range of codes, where the code rate is considered in low signal-to-noise ratio free-space links. The accumulate–repeat–accumulate code is adopted to improve the decoding threshold of a code. It is found that the multi-edge feature of extended codes is beneficial to the performance of short- and medium-length codes. And numerical results show the proposed scheme provides enhancement in the reconciliation efficiency. LDPC codes containing an additional structure that facilitates decoding could be suitable for free-space CV-QKD protocols when weak atmospheric turbulence is considered.