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

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.

💡 Research Summary

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The paper presents a comprehensive comparative study of three fundamental entropy measures—differential entropy, Rényi entropy of order α = 2 (collision entropy), and general Rényi entropy—in the context of hybrid quantum channels (HQCs) that experience both discrete (Poissonian) and continuous (Gaussian) noise components. The authors model the hybrid quantum noise (HQN) as a Gaussian mixture model (GMM), where the mixture weights follow a Poisson distribution and each Gaussian component captures the continuous‑variable (CV) noise statistics (mean vector μ_i and covariance matrix Σ_i). This formulation enables a unified treatment of the discrete‑continuous noise interplay that is typical in realistic quantum communication scenarios such as satellite‑based QKD or hybrid quantum‑classical repeaters.

Methodology
The HQN probability density is expressed as
(f_Z(z)=\sum_{i=0}^{K} w_i \mathcal N(z;\mu_i,\Sigma_i))
with (w_i=e^{-\lambda}\lambda^i/i!). The three entropy measures are then derived:

1. Differential Entropy
   (H(Z) = -\int f_Z(z)\log f_Z(z)dz). By exploiting the mixture structure, the authors approximate it as a weighted sum of the individual Gaussian differential entropies plus the Shannon entropy of the weight vector:
   (H(Z) \approx \sum_i w_i \bigl