The role of averages in CV-QKD over fast fading channels

TL;DR

This paper analyzes CV-QKD over fast-fading channels by contrasting two security models: Holevo bound average (HBA) and covariance matrix average (CMA). It derives analytic expressions for SKR under both models: HBA computes $R=I_{AB}^{T_{min}}-\langle I_{BE}\rangle$, while CMA uses averaged covariance matrices leading to $R=\langle I_{AB}\rangle-\chi_{BE}(\widetilde{\lambda}_1,\widetilde{\lambda}_2,\widetilde{\lambda}_3)$ with effective parameters $T_{eff}$ and $\chi_{eff}$. The results show that treating channel fluctuations differently yields markedly different SKR behaviors, with HBA more robust in high-variance regimes and CMA requiring optimization of the modulation variance, highlighting practical implications for free-space and satellite CV-QKD. The study emphasizes the importance of selecting a model that accurately captures channel fluctuations and suggests CMA aligns more closely with experimental data, though both approaches have tradeoffs.

Abstract

This work presents a study of continuous-variable quantum key distribution (CV-QKD) protocols over fast-fading channels, typically found in free-space communication links. Two eavesdropping models are considered to evaluate their security under collective attacks: \textit{Holevo bound average} (HBA) and \textit{covariance matrix average} (CMA). In the HBA approach, the Holevo bound is averaged over the channel transmittance. In contrast, the CMA method calculates the Holevo bound from the average covariance matrix. Analytical expressions are developed for both strategies. The two methods also differ in how they calculate the mutual information between the legitimate parties. The results demonstrate that the SKR is significantly influenced by how you treat channel fluctuations, highlighting the importance of choosing the model that better describes the actual implementation of the protocol.

Figures (5)

• Figure 1: This is the scheme of the system, where Alice prepares a two-mode coherent state and sends half through a quantum channel with transmittance $T$ and noise variance $\chi$. Eve can interact with this mode by injecting extra noise and storing the state information in a quantum memory (Q-M). Bob receives the output mode and measures one of the quadratures in a homodyne detector.
• Figure 2: This graph presents four panels that evaluate the performance of two distinct approaches in the low-variance regime, $V = 10$. The HBA approach is shown in blue, while the CMA approach is shown in green. The figure compares the SKR for both approaches under three different excess noise levels: $0\%$ (solid lines), $0.5\%$ (dashed lines), and $3\%$ (dotted lines). Results are displayed for two transmittance ranges: $\Delta T = 0.2$ (left panels) and $\Delta T = 0.6$ (right panels). The top row displays the SKR as a function of the minimum transmittance value and the bottom row as a function of the mean transmittance.
• Figure 3: Secret key rate as a function of the modulation variance for different values of $T_{min}$. (right) The blue lines represent the HBA approach, while the horizontal black lines correspond to the SKR calculated in the high variance regime $V\gg 1$. This confirms that the optimal SKR is obtained for $V\rightarrow\infty$. (left) The green lines show the CMA approach. In this case, the optimal variance varies depending on the transmittance value and must be reduced as $T_{min}$ approaches $0$. The SKR is computed considering $\Delta T=0.2$.
• Figure 4: (Left) Holevo bound for the HBA (Eq. \ref{['eq:23']}), plotted in blue with variance fixed at $V=10^3$. CMA results are shown in green: the lower curves correspond to the optimal variance $V=V_{\mathrm{opt}}$ (Eq. \ref{['eq:34']}), while the upper curves use $V=10^3$. (Right) Mutual information for the HBA (Eq. \ref{['eq:17']}) and CMA (Eq. \ref{['eq:25']}), using the same color scheme (blue for HBA; green for CMA).
• Figure 5: This Figure presents the key distribution as a function of the average transmittance $\langle T \rangle$, at the optimal variance for the CMA approach (green lines) and at a fixed variance of $V=10^3$ for the HBA (blue lines). The widths of the probability distribution are $\Delta T=0.2$ (left) and $\Delta T=0.6$ (right).