Comparative Analysis of Differential and Collision Entropy for Finite-Regime QKD in Hybrid Quantum Noisy Channels

TL;DR

The paper develops a unified entropy framework for hybrid quantum channels with mixed discrete-continuous noise modeled by Gaussian mixtures, comparing differential entropy, Rényi entropy (α=2), and collision entropy. It shows that for well-separated HQN components, the entropies satisfy H(Z) ≈ H_2(Z) + log_2 R_eff, with an additive term (d/2) log_2 e that becomes negligible at large effective rank R_eff, providing a practical link between continuous- and discrete-variable entropy measures. In finite-key CV-QKD, maintaining entropy estimation errors within 10% is shown to be crucial for preserving composable secrecy and accurate key-rate predictions, highlighting the operational relevance of the entropy framework. The work combines analytical approximations with numerical phase-space analyses (p,q) to validate the asymptotic relations and demonstrates implications for entropy-based security assessment and resource optimization in hybrid quantum communications.

Abstract

In this work, a comparative study between three fundamental entropic measures, differential entropy, quantum Renyi entropy, and quantum collision entropy for a hybrid quantum channel (HQC) was investigated, where hybrid quantum noise (HQN) is characterized by both discrete and continuous variables (CV) noise components. Using a Gaussian mixture model (GMM) to statistically model the HQN, we construct as well as visualize the corresponding pointwise entropic functions in a given 3D probabilistic landscape. When integrated over the relevant state space, these entropic surfaces yield values of the respective global entropy. Through analytical and numerical evaluation, it is demonstrated that the differential entropy approaches the quantum collision entropy under certain mixing conditions, which aligns with the Renyi entropy for order $α= 2$. Within the HQC framework, the results establish a theoretical and computational equivalence between these measures. This provides a unified perspective on quantifying uncertainty in hybrid quantum communication systems. Extending the analysis to the operational domain of finite key QKD, we demonstrated that the same $10\%$ approximation threshold corresponds to an order-of-magnitude change in Eves success probability and a measurable reduction in the secure key rate.

Figures (3)

• Figure 1: (a) 3D representation of Hybrid Quantum Noise modeled using GMM, (b) 3D representation of the differential entropic function associated with HQN, (c) 3D representation of the quantum Rényi entropic function of HQN.
• Figure 2: (a) 3D representation of the quantum collision entropic function of HQN, (b) Relative error between $H(Z) - H_2(Z) = \log_2 R_{\text{eff}} + \tfrac{d}{2}\log_2 e$ and its approximation $\log_2 R_{\text{eff}}$ versus effective rank $R_{\text{eff}}$ for $d=2$. The red dashed line marks the 10% error threshold where the approximation is accurate, (c) Comparison of $H(Z) - H_2(Z)$ with its approximation $\log_2 R_{\text{eff}}$ for varying $R_{\text{eff}}$ at $d=2$. The approximation approaches the exact value as $R_{\text{eff}}$ increases, confirming its validity in high-rank regimes.
• Figure 3: Finite-regime entropy approximation for secure key rate $R_{N}$ under different block size $N$ for true $H(Z)$ and 10% underestimated $H_{2}(Z)$.