Performance analysis of quantum key distribution in underwater turbulence channels

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. In this paper, 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. Based on the derived bounds, we present the performance of BB84 protocol in different water types including clear, coastal and turbid water and under different atmospheric conditions such as clear, hazy and overcast. We further investigate the effect of system parameters such as aperture size and detector field-of-view on QBER and SKR performance metrics.

A Primer on Underwater Quantum Key Distribution

“…Tis section will now provide closed-form bounds for these parameters when applying the BB84 protocol in free-space underwater channels. We consider the impact of difraction, turbulence, path loss, and other important factors, as reported in studies [81,82]. We 10…”

“…By inserting ( 16) into (13), one can easily see that the transmission of the i-th input mode (i.e., �� � P T 􏽰 u 0 (r) � �� � P T 􏽰 f i (r)) originates the feld pattern u(ρ) � �������� A(L)μ i P T 􏽰 ϕ i (ρ) at P 1 , and the resulting fraction of captured photons (15) is c � A(L) • μ i . As shown in [77,81], to fnd a practical lower bound on the underwater quantum channel QBER, we can limit our eforts to evaluating a lower bound on the expected value of the maximum modal transmittivity μ 1 , i.e., a value μ turb such that E[μ 1 ] ≥ μ turb [81]. It can be shown that…”

“…where F � ((πd TX d RX )/4λL) 2 is the Fresnel number and J 1 (•) is the frst-order Bessel function of the frst kind [77,81]. Te wave structure function W(•, •) in ( 17) is associated with the spatial power spectrum of the refractive index [85].…”

“…Te wave structure function W(•, •) in ( 17) is associated with the spatial power spectrum of the refractive index [85]. For a spherical wave, a specifed transmission distance of L, and a set separation distance ρ between two points on the phase front perpendicular to the axis of propagation, it is expressed as [81,86] Quantum Engineering…”

“…Channel. Based on the lower and upper bounds for sift and error probabilities obtained in [77] over a turbulent channel, we can fnally obtain a lower bound on QBER, given by [81] QBER…”