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arXiv:2606.25807

A Candidate Framework for Free-Space Quantum Key Distribution based on Geometrical-Configuration Modulation

Yu-Ming Bai, Yu-Xuan Liu, Ming-Han Ding, Jun-Lin Li

quant-phphysics.app-phphysics.opticsSubmitted Jun 24, 2026• Announced Jun 25, 2026• 0 citations
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Abstract

This paper proposes a candidate framework for free-space quantum key distribution (QKD) based on geometrical-configuration modulation (GM). In the minimal implementation considered here, Alice coherently splits a single photon emitted from one source into two spatial output modes with a tunable separation, and uses the source separation $R$ as the GM variable that defines the prepared single-photon spatial superposition state. Bob records the single-photon detection coordinate in the far field or Fourier plane, providing the correlated data used for soft-input information reconciliation. Based on this physical mechanism, we first establish an $R-x$ protocol model in which the source separation $R$ and the single-photon detection coordinate $x$ are random variables, and further propose an $R-Δx$ extension based on the difference variable $Δx$ between adjacent accepted detection events to mitigate slowly varying center drift in free-space links. The framework specifies state preparation, far-field conditional probabilities, soft-input information generation, parameter estimation, reconciliation, and asymptotic candidate key-rate formulas. A complete composable security analysis further requires derive an explicit computable upper bound on Eve's information from experimentally observed parameters, together with finite-key analysis and experimental validation under free-space conditions. The proposed candidate framework (GM-QKD) provides a modulation approach based on spatial degrees of freedom in which the source geometry serves as the modulation variable.

Comments
22 pages, 3 figures

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This paper proposes a candidate framework for free-space quantum key distribution (QKD) based on geometrical-configuration modulation (GM). In the minimal implementation considered here, Alice coherently splits a single photon emitted from one source into two spatial output modes with a tunable separation, and uses the source separation RRR as the GM variable that defines the prepared single-photon spatial superposition state. Bob records the single-photon detection coordinate in the far field or Fourier plane, providing the correlated data used for soft-input information reconciliation. Based on this physical mechanism, we first establish an R−xR-xR−x protocol model in which the source separation RRR and the single-photon detection coordinate xxx are random variables, and further propose an R−ΔxR-ΔxR−Δx extension based on the difference variable ΔxΔxΔx between adjacent accepted detection events to mitigate slowly varying center drift in free-space links. The framework specifies state preparation, far-field conditional probabilities, soft-input information generation, parameter estimation, reconciliation, and asymptotic candidate key-rate formulas. A complete composable security analysis further requires derive an explicit computable upper bound on Eve's information from experimentally observed parameters, together with finite-key analysis and experimental validation under free-space conditions. The proposed candidate framework (GM-QKD) provides a modulation approach based on spatial degrees of freedom in which the source geometry serves as the modulation variable.
Continuous-variable quantum key distribution is a compelling framework for scalable quantum networks due to its seamless integration with existing optical communication infrastructure. However, a fundamental gap persists between theoretical protocols requiring ideal Gaussian modulation and the constrained, discrete-modulated signals dictated by practical high-speed hardware. Current security proofs for discrete modulation rely on semidefinite programming, which suffers from prohibitive computational overhead for high-order constellations and lacks direct physical insight into non-Gaussian modulation.In this Letter, we overcome this limitation by developing a security framework that obviates semidefinite programming in favor of an approach grounded fundamentally in the Heisenberg uncertainty principle. By introducing a multi-mode entanglement-source model to characterize non-Gaussian state preparation, we establish an explicit mapping between constellation geometry and the secret key rate. This framework effectively quantifies the security implications of hardware-limited, finite state preparation, enabling both numerical and analytical security analysis under high-order constellations. We experimentally validate our method on both discrete-component and integrated photonic platforms, demonstrating that a quadrature amplitude modulation format with 256 constellation points can asymptotically approach the Gaussian capacity limit. Beyond quantum key distribution, the principle of tightening uncertainty-constrained bounds via source-mode expansion offers a paradigm for exploring the information-theoretic properties of complex non-Gaussian systems.
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We consider a continuous-variable (CV) quantum key distribution (QKD) protocol over free-space channels, which is simpler and more robust than typical CV QKD protocols. It uses a bright laser, squeezed and modulated in the amplitude quadrature, and self-homodyne detection. We consider a scenario, where the line of sight is classically monitored to detect active eavesdroppers, so that we can assume a passive eavesdropper. Under this assumption, we analyze security of the QKD protocol in the composable finite-size regime. Proper modulation of the squeezed laser to the shot-noise level can completely eliminate information leakage to the eavesdropper and also eliminate the turbulence-induced noise of the channel in the amplitude quadrature. Under these conditions, estimation of the eavesdropper's information is no longer required. The protocol is extremely robust to modulation noise and highly loss-tolerant, which makes it a promising candidate for satellite-based continuous-variable quantum communication.
Quantum Key Distribution (QKD) provides a secure way of generating shared cryptographic keys between a sender (Alice) and a receiver (Bob). The original BB84 protocol uses a two-dimensional polarization basis, limiting the information content of a single photon to 1 bit. Using the transverse position of single photons as one basis and the Fourier space as a second basis, one can construct a pair of mutually unbiased higher-dimensional bases. This improves not only the security of the protocol, but also the key generation rate. We present experimental results with a Spatial-Light-Modulator-based encoding scheme employing two nearly orthogonal alphabets with on the order of 103symbols each and an information content of about 10 bit.
Using spatial modes for quantum key distribution (QKD) has become highly topical due to their infinite dimensionality, promising high information capacity per photon. However, spatial distortions reduce the feasible secret key rates and compromise the security of a quantum channel. In an extreme form such a distortion might be a physical obstacle, impeding line-of-sight for free-space channels. Here, by controlling the radial degree of freedom of a photon's spatial mode, we are able to demonstrate hybrid high-dimensional QKD through obstacles with self-reconstructing single photons. We construct high-dimensional mutually unbiased bases using spin-orbit hybrid states that are radially modulated with a non-diffracting Bessel-Gaussian (BG) profile, and show secure transmission through partially obstructed quantum links. Using a prepare-measure protocol we report higher quantum state self-reconstruction and information retention for the non-diffracting BG modes as compared to Laguerre-Gaussian modes, obtaining a quantum bit error rate (QBER) that is up to 3× lower. This work highlights the importance of controlling the radial mode of single photons in quantum information processing and communication as well as the advantages of QKD with hybrid states.