On the Poisson Approximation to Photon Distribution for Faint Lasers

TL;DR

This paper rigorously demonstrates that the photon number distribution of faint lasers can be accurately approximated by a Poisson distribution, providing a solid mathematical foundation for its use in quantum cryptography.

Contribution

It offers a quantitative proof and error estimation for the Poisson approximation of photon distributions in faint lasers, supporting a widely used assumption in quantum cryptography.

Findings

Poisson approximation closely matches binomially attenuated photon distributions

Error bounds for the approximation are quantitatively established

Numerical tests confirm theoretical predictions

Abstract

It is proved, that for a certain kind of input distribution, the strongly binomially attenuated photon number distribution can well be approximated by a Poisson distribution. This explains why we can adopt poissonian distribution as the photon number statistics for faint lasers. The error of such an approximation is quantitatively estimated. Numerical tests are carried out, which coincide with our theoretical estimations. This work lays a sound mathematical foundation for the well-known intuitive idea which has been widely used in quantum cryptography.