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

TL;DR

This paper compares differential, Renyi, and collision entropies in hybrid quantum channels, revealing their relationships and implications for quantum key distribution security.

Contribution

It provides a unified analysis of entropic measures in hybrid quantum noise, linking theoretical properties with practical QKD security assessments.

Findings

Differential entropy approaches collision entropy under certain conditions.

Quantum Renyi entropy aligns with differential entropy for order α=2.

Finite key analysis shows a 10% entropy approximation impacts security metrics.

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 $\alpha = 2$. 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.