Free-Space Quantum Channel Modeling

• Free-space quantum channel modeling is the rigorous study of quantum state transmission through optical links affected by atmospheric turbulence, scattering, and geometric losses.
• Modern modeling approaches use analytical, semi-analytical, empirical, and simulation-based frameworks to derive the probability distribution of transmittance and assess protocol performance.
• Key metrics such as average transmittance, variance, and outage probability inform system design choices like aperture sizing and adaptive optics to optimize secure quantum communication.

Free-space quantum channel modeling refers to the rigorous mathematical and physical treatment of the transmission of quantum states through atmospheric, urban, or satellite free-space optical links. The purpose is to accurately describe the impact of turbulence, scattering, absorption, geometric spreading, pointing instability, and loss on the quantum information capacity, fidelity, and security of such channels for quantum communication protocols, optical quantum networks, and experiments with nonclassical light. Free-space channels differ from fiber links by their strong spatial and temporal fluctuation characteristics, multi-modal propagation features, and nonstationary statistics. Modern modeling approaches encompass analytical, semi-analytical, empirical, and simulation-based frameworks, typically yielding the probability distribution of transmittance (PDT), fading laws, and support for specific quantum protocols such as QKD, squeezed-state transfer, and frequency-bin encoding.

1. Physical Principles and Channel Impairments

A free-space quantum channel is fundamentally characterized by the random transmission coefficient $\eta$ describing the fraction of quantum optical power delivered to a receiver through an aperture. Atmospheric turbulence introduces phase and amplitude fluctuations governed microscopically by the refractive-index structure constant $C_n^2$, Rytov variance $\sigma_R^2=1.23\,C_n^2\,k^{7/6}\,L^{11/6}$, and macroscopic coherence scales such as the Fried parameter $r_0$ (Jaouni et al., 2024). These fluctuations manifest as beam wander (random centroid displacement), beam broadening, shape distortion (ellipticity, speckle), and amplitude scintillation. Loss mechanisms also include geometric spreading, atmospheric absorption (e.g., $T_{\rm abs}(z)=\exp[-\alpha(\lambda)z]$), and pointing error (e.g., $\eta_{\rm pt}=\exp[-2(\sigma_p\,L)^2/D_r^2]$) (Karakosta-Amarantidou et al., 2024). Statistical models of these effects depend on both the optical parameters (wavelength, aperture, beam waist) and the turbulence profile of the propagation path.

2. Probability Distribution of Transmittance (PDT): Analytical and Empirical Models

The central modeling object for fluctuating free-space quantum channels is the PDT $\mathcal{P}(\eta)$, offering a complete statistical description required for protocol and security analysis (Vasylyev et al., 2018, Klen et al., 2023). Major classes of PDTs have been derived:

• Truncated Log-normal Model: Assumes $\ln\eta$ is Gaussian; suitable for strong turbulence, yields closed-form moments $\langle\eta^k\rangle = e^{k\mu+\frac12k^2\sigma^2}$ (Klen et al., 2023).
• Beam-wandering Model: Captures random centroid displacement—PDT given by a stretched log-negative Weibull law with Rayleigh statistics for the centroid (Vasylyev et al., 2018).
• Elliptic-beam Approximation: Treats the received beam as a multivariate Gaussian with correlated random spot axes and orientation, leading to a multidimensional integral representation of the PDT; widely used for moderate link lengths (Vasylyev et al., 2018).
• Circular-beam Approximation: Simplifies the elliptic model by imposing axis equality; parameters are determined by matching the first two moments of transmittance or spot size, accurate for weak-to-moderate turbulence with dramatic computational savings (Pechonkin et al., 17 Jul 2025).
• Empirical Beta-distribution Model: Accurately fits numerical/experimental PDTs for arbitrary aperture sizes and turbulence strengths by matching the first two moments, allowing robust implementation in link-budgets and quantum protocol simulators (Klen et al., 2023).

These models may formally be expressed as mixtures over physical parameters such as spot size $S$ (e.g., $\mathcal{P}(\eta)=\int P(S)\mathcal{P}(\eta|S)\,dS$ for the circular-beam approximation).

3. Numerical Simulation and Laboratory Emulation Techniques

Monte-Carlo phase-screen and split-step propagation simulations provide high-fidelity synthetic PDTs by integrating the stochastic paraxial wave equations over segmented atmospheric realizations (Klen et al., 2023). Laboratory-based emulation techniques replace the fluctuating channel by a convex combination of fixed-loss measurements weighted by the desired $\mathcal{P}(\eta)$, enabling precise benchmarking of nonclassical-state evolution, click-counting statistics, and post-selection protocols without the need for dynamically turbulent hardware (Bohmann et al., 2017).

PDT Model	Turbulence Regime	Computational Complexity
Log-normal	Strong	Low
Elliptic-beam	Moderate	High
Circular-beam	Weak–Moderate	Low
Beta Empirical	Any (with fitting)	Very low

These techniques facilitate exhaustive design-space exploration, including investigation into channel metrics such as $\langle\eta\rangle$, $\mathrm{Var}(\eta)$, QBER, and post-selected nonclassicality (Vasylyev et al., 2018, Bohmann et al., 2017).

4. Quantum Protocol Modeling over Free-Space Channels

Quantum Key Distribution protocols (BB84, decoy-state, CV-QKD) are fundamentally impacted by fluctuating free-space losses. Key rate expressions, e.g., $R=Q_\mu[1-h_2(e_\mu)]-Q_\mu f\,h_2(E_\mu)$, require statistical averaging over the PDT (Scriminich et al., 2021, Hassan et al., 2023). Real-time post-selection, such as the prefixed-threshold method (P-RTS), discards low-transmittance events to improve key rates and SNR, optimizing the trade-off between throughput and security (Hassan et al., 2023). Adaptive optics, temporal/spectral filtering, and direct turbulence forecasting enable active channel mitigation; e.g., a GRU-based RNN (TAROCCO) forecasts $C_n^2$ for optimal scheduling and routing (Jaouni et al., 2024).

Continuous-variable QKD over free-space necessitates hybrid noise modeling: combined quantum Poissonian shot noise and classical AWGN define the achievable SKR as a function of the fluctuating transmission and noise parameters (Chakraborty et al., 2024, Liao et al., 13 Mar 2025). Time-dependent auxiliary quantum modes permit real-time estimation of $\eta(t)$, phase drift, and frequency offset, directly raising attainable SKRs in dynamic environments (Liao et al., 13 Mar 2025).

5. Channel Models for Specialized Scenarios

Satellite-based and balloon-based quantum channels have specific geometric, atmospheric, and pointing loss models. For aerial platforms, channel transmissivity is modeled as $$\eta(L)=\eta_{\rm geo}(L)\times\eta_{\rm atm}(L)\times\eta_{\rm pt}$$, with well-defined dependencies on altitude, slant path, and aperture size (Karakosta-Amarantidou et al., 2024). Stochastic models for metropolitan quantum networks employ Gaussian-beam geometric terms, atmospheric extinction, and turbulence-induced log-normal or gamma-gamma distributions for beam wander and scintillation (Nada et al., 20 Aug 2025).

Non-line-of-sight channels, utilizing scattered photons, are accurately modeled as dephasing-plus-loss quantum channels with angle-dependent transmission and coherence (visibility), supporting time-bin encoded quantum communication across diffuse or indirect paths (Sajeed et al., 2021).

Multi-channel models, as in atomic array platforms, treat each transverse mode as an independent 1D quantum channel and characterize inter-channel quantum correlations and blockade-enhanced nonlinearities (Solomons et al., 2021). Free-space interference quantum channels are described by multi-input bosonic Gaussian channels, subject to passivity and thermal noise constraints, with achievable information-theoretic rates depending on encoding and detection strategy (homodyne, heterodyne, joint detection) (Guha et al., 2011).

6. Metrics, Security, and Design Implications

The algorithmic calculation, description, and design of free-space quantum links depend on channel metrics:

• Average Transmittance and Variance: $\langle\eta\rangle$, $\mathrm{Var}(\eta)$ determine gain, QBER, SKR.
• Outage Probability: $\Pr(\eta<\eta_{\rm th})$ critical for reliability and network routing (Pechonkin et al., 17 Jul 2025).
• Interference Visibility: $$V_{\rm turb}=\frac{\langle\sqrt{\eta}\rangle}{\sqrt{\langle\eta\rangle}}$$ sets phase-encoded QBER.
• Nonclassicality Transfer: Output photon counting (Mandel $Q$) and squeezing metrics are analytic functions of the PDT.
• Post-selection and Adaptive Strategies: Channel-aware conditional use raises security and nonclassicality (e.g., thresholds $\eta_{PS}\sim0.5$, achieving negative eigenvalues in detection matrices) (Bohmann et al., 2017).

These metrics feed directly into system-level decisions—such as aperture sizing, beam focusing, adaptive optics order, filtering bandwidth, and temporal gating—which collectively optimize quantum communication performance under realistic atmospheric conditions.

7. Outlook and Future Directions

Contemporary free-space quantum channel modeling is transitioning toward fully integrated, modular frameworks interfacing turbulence physics, machine learning-based forecasting, empirical/simulation-based PDT determination, and quantum protocol optimization. Opportunities for advancement include hybrid noise models with non-Gaussian statistics (Chakraborty et al., 2024), real-time turbulence-aware control (Jaouni et al., 2024), and efficient multi-parameter PDT representations (e.g., circular or Beta approximations (Pechonkin et al., 17 Jul 2025, Klen et al., 2023)). Additionally, scalable emulation techniques, multimode and non-line-of-sight models (Sajeed et al., 2021), and system-level integration for diverse platforms (balloon, satellite, urban, atomic array) (Karakosta-Amarantidou et al., 2024, Nada et al., 20 Aug 2025, Solomons et al., 2021) are shaping the landscape for high-fidelity, secure quantum communications and distributed quantum information processing across atmospheric environments.