Application of an improved version of McDiarmid inequality in finite-key-length decoy-state quantum key distribution

Abstract

In practical decoy-state quantum key distribution, the raw key length is finite. Thus, deviation of the estimated single photon yield and single photon error rate from their respective true values due to finite sample size can seriously lower the provably secure key rate R. Current method to obtain a lower bound of R follows an indirect path by first bounding the yields and error rates both conditioned on the type of decoy used. These bounds are then used to deduce the single photon yield and error rate, which in turn are used to calculate a lower bound of the key rate R. Here we report an improved version of McDiarmid inequality in statistics and show how use it to directly compute a lower bound of R via the so-called centering sequence. A novelty in this work is the optimization of the bound through the freedom of choosing possible centering sequences. The provably secure key rate of realistic 100 km long quantum channel obtained by our method is at least twice that of the state-of-the-art procedure when the raw key length ℓ raw is ≈105–106. In fact, our method can improve the key rate significantly over a wide range of raw key length from about 105 to 1011. More importantly, it is achieved by pure theoretical analysis without altering the experimental setup or the post-processing method. In a boarder context, this work introduces powerful concentration inequality techniques in statistics to tackle physics problem beyond straightforward statistical data analysis especially when the data are correlated so that tools like the central limit theorem are not applicable.