[Paper Review] Performance Analysis of Quantum Key Distribution in Underwater Channels

Amir Hossein Fahim Raouf|arXiv (Cornell University)|Aug 20, 2022

Energy Harvesting in Wireless Networks4 citations

TL;DR

This paper analyzes the performance of BB84 and decoy BB84 quantum key distribution (QKD) protocols in underwater channels, accounting for absorption, scattering, and turbulence using a modified Beer-Lambert path loss model and wave structure function for turbulent propagation. Key results show that achievable distances range from 7 m (180° FOV in daylight) to 71 m (20 cm aperture at night), with relay-assisted QKD extending reach under low-noise, clear-water conditions.

ABSTRACT

The current literature on quantum key distribution (QKD) is mainly limited to the transmissions over fiber optic, atmospheric or satellite links and are not directly applicable to underwater environments with different channel characteristics. Absorption, scattering, and turbulence experienced in underwater channels severely limit the range of quantum communication links. In the first part of this thesis, we analyze the quantum bit error rate (QBER) and secret key rate (SKR) performance of the well-known BB84 protocol in underwater channels. As path loss model, we consider a modified version of Beer-Lambert formula which takes into account the effect of scattering. We derive a closed-form expression for the wave structure function to determine the average power transfer over turbulent underwater path and use this to obtain an upper bound on QBER as well as a lower bound on SKR. In the second part of this thesis, as a potential solution to overcome range limitations, we investigate a multi-hop underwater QKD where intermediate nodes between the source and destination nodes help the key distribution. We consider the deployment of passive relays which simply redirect the qubits to the next relay node or the receiver without any measurement. Based on the near-field analysis, we present the performance of relay-assisted QKD scheme in terms of QBER and SKR in different water types and turbulence conditions. In the last part of this thesis, we investigate the fundamental performance limits of decoy BB84 protocol over turbulent underwater channels and provide a comprehensive performance characterization. Based on near field analysis, we utilize the wave structure function to determine the average power transfer over turbulent underwater path and use this to obtain a lower bound on key generation rate. Based on this bound, we present the performance of decoy BB84 protocol in different water type.

Motivation & Objective

• To evaluate the performance of BB84 and decoy BB84 QKD protocols in underwater environments, which differ significantly from fiber, atmospheric, or satellite channels.
• To identify the dominant impairments in underwater QKD, including absorption, scattering, and turbulence, and quantify their impact on QBER and secret key rate (SKR).
• To investigate the effects of system parameters—aperture size and detector field-of-view (FOV)—on QKD performance under varying environmental conditions.
• To assess the feasibility of multi-hop relay-assisted QKD as a solution to extend transmission range in turbulent underwater channels.
• To establish fundamental performance limits of decoy BB84 in turbulent underwater links using a lower bound on key generation rate derived from wave structure function analysis.

Proposed method

• Uses a modified Beer-Lambert law that incorporates scattering effects to model path loss in underwater channels.
• Employs the wave structure function to model average power transfer over turbulent underwater paths, enabling derivation of statistical bounds on QBER and SKR.
• Derives a closed-form expression for the wave structure function to compute upper bounds on QBER and lower bounds on SKR for BB84 and decoy BB84 protocols.
• Applies near-field analysis to model relay-assisted QKD with passive relays that retransmit qubits without measurement, minimizing error accumulation.
• Evaluates system performance across different water types (clear, coastal, turbid) and atmospheric conditions (clear, hazy, overcast, daylight, night with moon).
• Quantifies the trade-off between increased collected signal photons and increased background noise due to larger aperture and FOV, using numerical simulations and bounds.

Experimental results

Research questions

• RQ1 How does turbulence, absorption, and scattering affect the QBER and secret key rate in underwater BB84 QKD?
• RQ2 What is the impact of aperture size and detector field-of-view (FOV) on the achievable distance and key generation rate in underwater QKD systems?
• RQ3 Can relay-assisted QKD significantly extend the end-to-end transmission range in underwater environments, and under what conditions is it beneficial?
• RQ4 How does the performance of the decoy BB84 protocol compare to standard BB84 in turbulent underwater channels, and what are its fundamental performance limits?
• RQ5 What system parameter configurations (aperture size, FOV) maximize performance under different environmental conditions (day vs. night, clear vs. turbid water)?

Key findings

• In clear ocean with weak turbulence and night conditions (full moon), achievable distance increases from 62 m (5 cm aperture) to 71 m (20 cm aperture), due to higher photon collection without significant background noise increase.
• With FOV increasing from 10° to 180°, achievable distance drops from 56 m to 7 m in daylight due to increased background noise, highlighting the critical role of FOV in high-irradiance environments.
• At night, FOV has negligible impact on performance, as background noise remains low; thus, performance is primarily limited by path loss and turbulence.
• In daylight, increasing aperture size reduces achievable distance due to disproportionate increase in background noise, while at night, larger apertures improve performance by collecting more signal photons.
• Relay-assisted QKD improves end-to-end distance when noise is minimized (small FOV, small aperture, low environment irradiance), with optimal relay count increasing as FOV or aperture size decreases.
• The decoy BB84 protocol outperforms standard BB84 in terms of key generation rate under turbulent conditions, with a lower bound on key rate derived via wave structure function analysis, confirming its robustness in dynamic underwater channels.