Secret Key Rate Analysis of Distribution Matching Algorithms for Discrete-Modulated CV-QKD

Continuous variable quantum key distribution protocols (CV-QKD) with discrete modulation have been intensively investigated to bridge the gap between ideal Gaussian modulation and modern coherent optical communication systems. To mitigate the penalty of discrete modulation, probabilistic constellation shaping (PCS) is applied to the modulation format and is typically performed by distribution matching (DM) algorithms. In this paper, we address the application of DM algorithms to perform PCS in CV-QKD protocols. We investigate the impact of approximating optimized Maxwell-Boltzman distributions with DM algorithms based on Huffman (HDM) and constant composition (CCDM) codes on the protocol's secret key rate (SKR) and tolerance to excess noise. Our results show that specifically symbol-by-symbol HDM degrades the SKR by at least 30\%, whereas CCDM matches the optimal SKR with code length of $10^3$ or more symbols. Furthermore, we also provide a statistical analysis of symbol dependence for both approaches, showing that CCDM must operate with blocks of at least $10^5$ symbols for the correlations become negligible. Finally, we propose an algorithm to generate independent symbols following near-optimal distributions.