Application Of McDiarmid Inequality In Finite-Key-Length Decoy-State Quantum Key Distribution

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 I show how to directly compute a lower bound of $R$ via McDiarmid inequality in statistics. This method increases the provably secure key rate of realistic quantum channels by at least 30% when the raw key length is $\approx 10^5$ to $10^6$. More importantly, this 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.