Abstract

The Lambert–Tsallis ${W_q}$ function has found applications in several areas of physics, such as quantum optics, astronomy, and semiconductor physics. In this work, we discuss some applications of ${W_q}$ in quantum key distribution (QKD). Three problems are considered: (i) the quantum access network, (ii) analysis of an on-chip ${{\rm SiO}_2}$ amplitude modulator used in continuous-variable QKD (CV-QKD), and (iii) parameter estimation of a stochastic quantum channel. In quantum access networks, QKD and classical data travel on the same optical fiber. In this case, an increase of the quantum bit error rate is caused mainly by the spontaneous Raman scattering (SRS) produced by classical data. The amount of SRS produced depends on the fiber length. In the present work, we use the Lambert–Tsallis ${W_q}$ function to calculate analytically the fiber length needed to produce a given level of SRS. Thus, our formula allows the calculation of the QKD channel’s length when the probability of a click on the receiver side without having any incident photons from the quantum transmitter is defined a priori. On the other hand, a crucial step in the security of CV-QKD is the correct channel transmissivity and excess noise estimations. These parameters can be overestimated or underestimated when real devices do not behave as predicted by their models. In this direction, using the ${W_q}$ function, we provide an equation for calculation of the fluctuation of the concentration of free carriers in an integrated amplitude modulator and another equation for calculation of the parameter that models a stochastic quantum channel. These equations are useful in building strategies to avoid quantum hacking.

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