Neural network-based finite-sample transmission coefficient estimation and uncertainty quantification in satellite-to-ground CV-QKD

Satellite-to-ground continuous variable quantum key distribution (CV-QKD) is constrained by rapidly varying free-space channels and short transmission windows, which limit the amount of data available for parameter estimation within each channel interval. Under finite-sample conditions, parameter point estimates are subject to statistical uncertainty and may even yield an overly favorable description of the true channel state, while errors in the estimated transmission coefficient directly affect the covariance matrix used in the security analysis. In this work, we investigate the finite-sample effects of transmission coefficient estimation in dynamic free-space CV-QKD. A neural network is introduced to estimate the transmission coefficient, and the Delta method is used to quantify the uncertainty in its predictions. The proposed method is compared with the standard maximum likelihood estimation method. Confidence intervals for the transmission coefficient are constructed for different sample sizes, and the effects of the statistical uncertainties on the secret key rate are analysed. Under the considered Gaussian-channel simulation conditions, the neural network can provide a more favorable nominal lower bound on the transmission coefficient than MLE in certain sample-limited regimes, leading to a higher parameter-estimation-limited key-rate metric. This work establishes a statistical framework that connects neural network-based transmission coefficient estimation and predictive uncertainty quantification with covariance matrix construction and secret key rate analysis. It provides a basis for machine learning-assisted parameter estimation in free-space CV-QKD systems operating with finite samples.