A Unified Framework for UAV-Based Free-Space Quantum Links: Beam Shaping and Adaptive Field-of-View Control

Khalid Qaraqe; Mazen Hasna; Mohammad Taghi Dabiri; Saif Al-Kuwari

REVIEW 4 major objections 5 minor 36 references

A Unified Framework for UAV-Based Free-Space Quantum Links: Beam Shaping and Adaptive Field-of-View Control

T0 review · 4 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read UAV-to-ground quantum links need sub-10 cm beams and micro-radian tracking to exceed 2 Mbps key rates with QBER below 0.001; the paper's grid-based photon-capture model stays accurate where the classical wide-beam approximation fails.

desk verdict Useful, correct fix to the wide-beam pointing-error model for narrow-beam UAV quantum links, but the headline key rates are raw click rates, not QKD key rates. read the letter →

arxiv 2506.20336 v1 pith:Y7G3DZP5 submitted 2025-06-25 eess.SP quant-ph

Mohammad Taghi Dabiri , Mazen Hasna , Saif Al-Kuwari , Khalid Qaraqe This is my paper · ORCID

classification eess.SPquant-ph

keywords quantumkeydistributionUAV-to-groundlinksfree-spaceopticalcommunicationpointingerrorsbeamwaistoptimizationfield-of-viewcontrolbiterrorrateGamma-Gammaturbulence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tries to establish that the classical wide-beam approximation for pointing errors, which assumes the beam radius at the receiver is much larger than the aperture, becomes invalid in the quantum regime of UAV-to-ground links, where the beam waists that maximize photon collection are tightly focused (below 10 cm). The paper replaces that approximation with a grid-based discretization of the receiver aperture and derives analytical expressions for the average raw key generation rate and quantum bit error rate, folding in beam divergence, transmitter and receiver pointing errors, atmospheric attenuation, Gamma-Gamma turbulence, narrow field-of-view filtering, and background photon noise; a raw key bit is counted whenever exactly one photon is detected in a time slot, including slots where that photon is background light. On these modeling assumptions, optimized beam waists with 50 micro-radian transmitter tracking yield raw key rates above 2 Mbps and QBER below $10^{-3}$, while 2 milliradian tracking makes the link unusable with QBER above $10^{-1}$. The results matter because most existing UAV-QKD channel models inherit the wide-beam pointing-error formulation from classical free-space optics, and the paper shows that this inheritance fails exactly in the operating regime that makes airborne quantum key distribution feasible.

What carries the argument

The load-bearing object is the grid-based photon capture approximation (Proposition 1). For a Gaussian single-photon probability density and a circular receiver aperture of radius $r_a$, the exact capture probability is a two-dimensional integral over the aperture; the grid scheme splits the aperture into $N_g$ segments of width $\Delta x=2r_a/N_g$, evaluates the Gaussian at each segment center $x_i$, and folds the segment's vertical extent into the coefficient $c_i = (2\Delta x/\sqrt{2\pi} w_z)\,\mathrm{erf}(\sqrt{2/w_z^2}\sqrt{r_a^2-x_i^2})$. This turns the capture probability into a sum of one-dimensional Gaussian terms that remain accurate when the beam waist $w_z$ is comparable to or smaller than the aperture, which is precisely the regime where the classical wide-beam formula $\mu_p(r_d)\approx (2r_a^2/w_z^2)\exp(-2\|r_d\|^2/w_z^2)$ fails. The capture probability then enters the end-to-end mean detected photon count $\mu_q = \mu_t \eta_{atm} \mu_d \mu_p(r_d) \eta_{turb} \mu_{FoV}$, with $\mu_{FoV}$ a Bernoulli visibility factor determined by the receiver's angular acceptance cone. Averaging over the Rayleigh-distributed beam displacement $r_d$ and the Gamma-Gamma turbulence $\eta_{turb}$ reduces the key rate and QBER to single one-dimensional integrals, which is what makes the framework tractable enough for system-level optimization.

What would settle it

A direct check is to rerun the paper's own parameter set through a complete BB84-style protocol that discards roughly half the accepted detections for basis sifting, rejects background-only slots, and applies a decoy-state security analysis; if the resulting secret key rate no longer reaches the Mbps range or the QBER exceeds the protocol threshold, the central performance claim fails as stated. A complementary experimental test would measure the raw single-photon detection rate and QBER on a 1 km UAV-to-ground link with 50 micro-radian transmitter tracking error, beam waist 5–10 cm, and background radiance $10^{-6}$ W/m²/sr/nm, and compare the measured values with Propositions 4 and 5.

Watch

Extended reading notes

Core claim

The central claim, stated in the paper's own terms, is that 'the wide-beam approximation in [24] becomes invalid' for UAV-to-ground quantum links, and that replacing that approximation is required to design links that actually work. The paper derives a grid-based approximation for the photon capture probability $\mu_p(r_d)$ that divides the receiver aperture into $N_g$ equal strips and represents each strip's contribution as a weighted Gaussian, so that $\mu_p(r_d) \approx \sum_{i=1}^{N_g} c_i \exp(-2(x_i-r_d)^2/w_z^2)$, with coefficients $c_i$ containing an error function; with $N_g=10$ it tracks exact numerical integration. This capture probability feeds a Poisson model for detected photons per quantum slot, and the paper's Propositions 2–5 produce one-dimensional integral expressions for the probability of detecting at least one photon, the raw key rate $R_{key}=R_q P(n_{\mathrm{eff}}=1)$, and the QBER. The headline quantitative results are that for 50 micro-radian transmitter tracking error the raw key rate surpasses 2 Mbps and QBER drops below $10^{-3}$, while for 2 milliradian tracking error the key rate falls below 100 kbps and QBER exceeds $10^{-1}$, and that the receiver field of view must be tuned as a trade-off between signal capture and background rejection.

Load-bearing premise

The load-bearing premise is that every time slot in which exactly one photon is detected counts as a valid raw key bit, even when that photon is stray background light, and no fraction of detections is set aside for the sender-receiver basis comparison that real quantum protocols perform; if those real-protocol steps are included, the reported key rates would fall.

Editorial extensions

If this is right

Existing UAV-QKD analyses that use the wide-beam pointing-error model of [24] should be re-examined, because the approximation is inaccurate for the sub-10 cm beam waists the paper identifies as optimal.
Link designers should treat receiver field of view as an adaptive parameter: wider FoV raises key rate under low background light, while narrower FoV is needed to keep QBER acceptable under daylight.
The analytical framework enables fast joint optimization of beam waist, tracking precision, photon emission rate, and FoV without full Monte Carlo simulation.
Milliradian-level tracking, which is acceptable for classical FSO links, makes UAV-to-ground QKD unusable in this model; micro-radian tracking is necessary.
The same modeling approach applies with minor modifications to ground-to-UAV and terrestrial short-range free-space QKD, as the paper states in its conclusion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A full protocol-level analysis with basis sifting and decoy states would likely reduce the quoted Mbps figures by a factor of two or more, and possibly much more, because the paper's raw-key definition counts background-only slots as key bits; this is an inference from the paper's Appendix D, not its stated conclusion.
The grid-based aperture model could be combined with an online estimator of background radiance and pointing-error variance to close the loop on FoV control; the paper proposes adaptive FoV tuning but does not specify a control algorithm.
For very strong turbulence or non-Gaussian beam distortions, the Gamma-Gamma and Gaussian-beam assumptions may need re-examination, so the quantitative predictions are safest in the moderate-turbulence regime the paper assumes.
A natural next step is to extend the framework to entangled-photon UAV links, where two correlated channels must be modeled jointly and background rejection matters even more; the paper mentions entanglement as future work.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The short version: the paper's core modeling point is right, and the grid-based capture approximation is a legitimate, validated fix for the narrow-beam regime that UAV QKD actually needs. But the headline "quantum key rate" in Fig. 4 and the abstract is a raw detector click rate, not a QKD key rate, and that distinction matters.

What's new: the observation that the Farid-Hranilovic wide-beam pointing-error model breaks down when wz is around 5–10 cm, and that this is exactly the regime you want for single-photon links, is correct and worth saying. The grid-based approximation in Proposition 1 is standard Riemann-sum quadrature, but it's implemented cleanly, validated against the exact 2D integral and Monte Carlo, and makes the subsequent analysis tractable. The design conclusions—sub-10 cm beam waists and micro-radian tracking—follow from the model and are plausible.

Soft spots, in order of importance. First, the key rate metric. Proposition 4 and Appendix D define a raw key bit as any slot with exactly one detected photon, including State 2 where the only photon is a background photon. That is not a QKD key bit. BB84 discards half the bits in basis sifting, and a background-only detection carries no information about the transmitted bit—it is an error, not a key bit. So the greater-than-2 Mbps claim in Fig. 4a and the abstract's "secure QKD" framing are unsupported. The QBER expression is also incomplete: it only counts State 2 as erroneous and ignores basis mismatch, dark counts, and multi-photon WCP events. The authors do label Proposition 4 as "raw key generation rate," but the abstract and conclusions do not keep that caveat. Second, the "adaptive FoV tuning strategies" promised in the abstract and contributions are not actually implemented; the paper sweeps FoV and shows the trade-off, which is fine, but the word "adaptive" oversells it. Third, minor: the Monte Carlo validation in Figs. 3–4 has no error bars, and the small-argument approximation in Appendix C (e^{-z cpt mu_p} ≈ 1 − z cpt mu_p) is not justified with numbers.

The math itself is sound as far as it goes—no fitted parameters, the grid model is checked against the exact integral rather than a fitted dataset, and the citation pattern is reasonable (self-citations are to the authors' own prior FSO work, which is relevant). This is not a case of a broken model; it is a case of a mismatched metric. The paper deserves a serious referee, but the performance claims need re-derivation with a real protocol model (sifting, decoy states, finite-size effects) before the secure-QKD conclusion can stand.

Recommendation: send it to review, but instruct the referee to focus on the key rate definition and require a protocol-level correction.

Referee Report

4 major / 5 minor

Summary. The paper presents an analytical framework for UAV-to-ground free-space quantum links, replacing the wide-beam Farid-Hranilovic pointing-error formula with a grid-based discretization of the exact Gaussian aperture-capture integral. It derives approximate closed-form expressions for single-photon detection probability, a quantity it calls the quantum key generation rate, and QBER, including transmitter-side FSM jitter, receiver angle-of-arrival misalignment, Gamma-Gamma turbulence, atmospheric attenuation, detector efficiency, FoV filtering, and background Poisson noise. The analytical results are validated against numerical integration of the exact integral and against Monte Carlo simulation over 10^6 time slots. The key design conclusions are that optimal beam waists lie below roughly 10 cm, that tracking precision must be at the microradian level, and that the receiver FoV must balance background rejection against misalignment tolerance; the paper reports Mbps-level key rates and QBER below 10^-3 for optimized parameters.

Significance. If accepted as a channel-capture model, the paper's grid-based approximation is a useful and credible contribution: the wide-beam approximation of Ref. [24] genuinely becomes inaccurate when the spot size is comparable to or smaller than the aperture, and the proposed discretization is shown to match the exact numerical integral and Monte Carlo with no fitted parameters. The paper also correctly identifies a real system-level trade-off between FoV, background noise, and receiver misalignment. However, the significance of the reported QKD performance numbers is currently compromised by the raw-click-rate definition of the key rate, by an algebraic inconsistency in the central formula, and by an unjustified linearization in the derivation of the detection probability. Until these are corrected, the abstract's 'secure QKD' and 'Mbps-level key rates' claims are not supported by the manuscript as written.

major comments (4)

[Appendix D, Eqs. (24)-(25), Fig. 4] The quantity called Rkey is not a QKD sifted or secret key rate. Appendix D explicitly defines a raw key bit as any slot with exactly one detected photon 'regardless of its source,' and State 2 (nq=0, nb=1) contributes PS2=μ_b e^{-μ_b}[1-∫P(nq≥1|rd)f_rd(rd)drd]. Eq. (24) also contains no factor of 1/2 for BB84 basis sifting. A background-only detection carries no information about the transmitted bit and would enter a QKD protocol only as an error, not as a valid key bit; the >2 Mbps value in Fig. 4(a) and the abstract's 'Mbps-level key rates' are therefore detector click rates, not secure key rates. The QBER expression in Eq. (26) likewise counts only State 2 as erroneous and omits basis-mismatch errors and multi-photon weak-coherent-pulse events. The performance claims need to be re-derived with a protocol-level model, including sifting, error correction and privacy amplification, and ideally decoy-state and finite-size analysis, before they can be reported as secure QKD rates.
[Eq. (25) versus Eqs. (43)-(45)] There is an algebraic inconsistency in the central key-rate formula. From Appendix D, PS1=e^{-μ_b}S and PS3=(1/2)μ_b e^{-μ_b}S, where S denotes the integral over rd, so the coefficient multiplying S should be e^{-μ_b}(1+μ_b/2). Equation (25), however, contains e^{-μ_b}-(1/2)μ_b e^{-μ_b}, i.e., e^{-μ_b}(1-μ_b/2). The sign should be corrected, and the QBER expression in Eq. (26) should be checked against the corrected denominator. Although the discrepancy is numerically small when μ_b≪1, the formula as printed is not the sum of the three stated disjoint events.
[Appendix C, Eqs. (22)-(23)] The linearization e^{-z cpt μ_p} ≈ 1 - z cpt μ_p is applied after a change of variable in which z ranges over (0,∞), not only over small arguments. For z > 1/(cpt μ_p) the approximation becomes negative and does not approximate the exponential. The agreement with Monte Carlo in Fig. 3 is encouraging for the tested parameter range, but the derivation does not establish the conditions under which Proposition 3 remains accurate. The authors should either state and justify the low-transmissivity regime in which this approximation is valid or use the exact moment expression.
[Eq. (26), Proposition 5] The QBER definition considers only background-only slots as erroneous and assumes that State 1 and State 3 are error-free. In a polarization-encoded BB84 system, basis mismatch alone produces errors in a substantial fraction of sifted slots unless reconciliation is accounted for, and detector dark counts and multi-photon pulses also contribute. As written, Proposition 5 is not a QBER in the protocol sense and cannot be used to infer QKD security thresholds. This quantity should be either relabeled as a background-error indicator or extended to the full detection statistics.

minor comments (5)

[Section III, Propositions 2-3] The symbol cpt is used in Propositions 2 and 3 before it is defined; it is first introduced in Appendix B. Define cpt = μ_t η_atm μ_d before Eq. (20).
[Table II and Section IV] Table II sweeps wz directly, but Eq. (3) relates the received beam radius to the transmitter waist w0 and distance Lz. Clarify whether w0 is the optimized quantity and wz is computed from it, or whether the simulations independently set the spot radius at the receiver.
[Title and Section IV] The title and abstract promise 'adaptive FoV tuning strategies,' but Section IV only studies the FoV trade-off; there is no concrete adaptive algorithm or control law. Either add such a strategy or soften the claim.
[Appendix D] The notation n_t for the number of transmitted photons and n_tot for the total number of detected photons is easy to confuse; use distinct symbols such as n_sig and n_tot.
[References] Reference [18] is cited as a TechRxiv preprint; if a peer-reviewed version exists, it should be cited instead of or in addition to the preprint.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the grid-based capture model is validated against the exact integral and Monte Carlo, and the performance metrics follow from stated model assumptions.

full rationale

The derivation chain is self-contained. The paper's central modeling contribution, the grid-based approximation in Proposition 1, is derived from the exact two-dimensional aperture integral (15) and validated against that integral and against Monte Carlo simulation (Figs. 2 and 3), rather than fitted to the target key-rate or QBER results. Propositions 2 and 3 follow from the stated conditional Poisson model, Gamma-Gamma turbulence statistics, and the Gaussian/Rayleigh pointing-error models, and Propositions 4 and 5 are explicit algebraic evaluations of the slot-level photon-counting scenarios defined in Appendices D and E. The claimed invalidity of the wide-beam approximation (16) is demonstrated by comparing it with the exact integral, not assumed. Self-citations (e.g., [25], [26], [34], [35]) are used as standard FSO context or baseline models and are not load-bearing for the core derivation. The raw-key-rate and QBER definitions in Eqs. (24)-(26) may be debatable as QKD protocol metrics (background-only slots are counted as key bits and no basis-sifting factor is applied), but that is a modeling/security-correctness concern, not a circular reduction: the reported numbers are outputs of the paper's stated definitions, not inputs that were fitted or assumed in order to produce the claimed results.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The model's conclusions are driven by the assumed numerical values in Table II (cpt=0.12, sigma_theta_e=50 microrad, sigma_AoA=50 microrad) and by several modeling choices: binary FoV acceptance, Poisson thinning, no basis sifting, and a small-argument linearization. None of these numbers is fitted, so the circularity burden is low, but the quantitative headline depends on them.

free parameters (4)

Effective per-pulse path efficiency cpt = mu_t * eta_atm * mu_d = 0.12 (Table II: mu_t=0.5, eta_atm=0.4, mu_d=0.6)
Chosen input, not fitted; all absolute key-rate numbers scale linearly with it.
Transmitter FSM tracking error sigma_theta_e = 50 microrad (baseline; swept 50 microrad to 2 mrad)
The headline conclusion that micro-radian tracking is required is read off curves generated at this assumed jitter level.
Receiver angular misalignment sigma_AoA = 50 microrad (baseline; swept 50 to 200 microrad)
Controls the FoV trade-off and QBER; chosen rather than measured.
Gamma-Gamma turbulence shape parameters alpha, beta = 2.1, 1.8
Chosen in Table II to represent a turbulence condition; not tied to specific weather or altitude data.

assumptions (5)

standard math Single-photon transverse position follows a Gaussian beam probability distribution with waist wz (Eq. 2).
Standard Gaussian optics model for TEM00 sources; used throughout the capture probability derivation.
domain assumption Atmospheric turbulence transmissivity follows a Gamma-Gamma distribution with alpha=2.1 and beta=1.8.
Common model for moderate-to-strong turbulence; parameters are chosen in Table II, not measured for this link.
ad hoc to paper Receiver acceptance is binary: any arrival angle within theta_FOV is accepted with probability one, any outside with probability zero (Eq. 12).
Ignores the smooth dependence of single-mode-fiber coupling efficiency on angle and any correlation between transmitter tilt and arrival angle.
ad hoc to paper Detected photons in each time slot follow a Poisson distribution via independent thinning, and the raw key bit rate counts any slot with exactly one detected photon, including background-only slots, with no basis sifting.
This defines the paper's key-rate metric; standard QKD raw keys apply basis sifting and protocol-specific event selection, so the reported rate is a click rate.
ad hoc to paper The derivation approximates 1 - exp(-z * cpt * mu_p) by z * cpt * mu_p for small arguments (Appendix C).
No validity range is stated; the approximation error grows as cpt * mu_p approaches order one, which can occur for tightly focused beams.

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Cite this review

Pith. "Pith review of A Unified Framework for UAV-Based Free-Space Quantum Links: Beam Shaping and Adaptive Field-of-View Control." pith.science (2026). https://pith.science/paper/Y7G3DZP5

@misc{pith2026250620336,
  author       = {Pith},
  title        = {Pith review of: A Unified Framework for UAV-Based Free-Space Quantum Links: Beam Shaping and Adaptive Field-of-View Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y7G3DZP5}},
  note         = {Machine review of arXiv:2506.20336}
}

abstract

This paper develops a comprehensive analytical framework for modeling and performance evaluation of unmanned aerial vehicles (UAVs)-to-ground quantum communication links, incorporating key physical impairments such as beam divergence, pointing errors at both transmitter and receiver, atmospheric attenuation, turbulence-induced fading, narrow field-of-view (FoV) filtering, and background photon noise. To overcome the limitations of conventional wide-beam assumptions, we introduce a grid-based approximation for photon capture probability that remains accurate under tightly focused beams. Analytical expressions are derived for the quantum key generation rate and quantum bit error rate (QBER), enabling fast and reliable system-level evaluation. Our results reveal that secure quantum key distribution (QKD) over UAV-based free-space optical (FSO) links requires beam waists below 10 cm and sub-milliradian tracking precision to achieve Mbps-level key rates and QBER below $10^{-3}$. Additionally, we highlight the critical role of receiver FoV in balancing background noise rejection and misalignment tolerance, and propose adaptive FoV tuning strategies under varying illumination and alignment conditions. The proposed framework provides a tractable and accurate tool for the design, optimization, and deployment of next-generation airborne quantum communication systems.

Figures

Figures reproduced from arXiv: 2506.20336 by the authors.

**Figure 2.** illustrates the photon capture probability µp(rd) for two different spot sizes, wz = 5 cm and wz = 10 cm. These values are representative of the practical settings used in our simulation section, where it is shown that such beam sizes are more suitable for UAV-based quantum communication scenarios aiming to achieve higher key generation rates. As 2 4 6 8 10 12 14 16 18 20 0.2 0.4 0.6 0.8 1 [PITH_FULL_IMAGE:figures/… view at source ↗

**Figure 3.** End-to-end single-photon detection probability ve [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Impact of beam waist wz on system performance for various UAV tracking errors σθe = 50 µrad, 200 µrad, and 2 mrad. Receiver misalignment is fixed at σAoA = 50 µrad, and background radiance is Bλ = 10−6 W/m2 /sr/nm. (a) Quantum key generation rate. (b) QBER. rate drops …

**Figure 6.** Figure 6: Effect of background radiance Bλ on system performance for σθe = 100 µrad and σAoA = 50 µrad. (a) Quantum key generation rate and (b) QBER versus θFoV for Bλ = 10−6 , 10−5 , and 10−4 W/m2 /sr/nm. comes at the cost of a very high QBER, rendering the key unusable. To ach…

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