  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.

Quantum Physics