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Stringent Statistical Fluctuation Analysis for Quantum Key Distribution Considering After-pulse Contributions

Statistical fluctuation problems are faced by all quantum key distribution (QKD) protocols under finite-key condition. Most of the current statistical fluctuation analysis methods work based on independent random samples, however, the precondition cannot be always satisfied on account of different choice of samples and actual parameters. As a result, proper statistical fluctuation methods are required to figure out this problem. Taking the after-pulse contributions into consideration, we give the expression of secure key rate and the mathematical model for statistical fluctuations, focusing on a decoy-state QKD protocol (Sci Rep. 3, 2453, 2013) with biased basis choice. On this basis, a classified analysis of statistical fluctuation is represented according to the mutual relationship between random samples. First for independent identical relations, we make a deviation comparison between law of large numbers and standard error analysis. Secondly, we give a sufficient condition that Chernoff bound achieves a better result than Hoeffding's inequality based on independent relations only. Thirdly, by constructing the proper martingale, for the first time we represent a stringent way to deal with statistical fluctuation issues upon dependent ones through making use of Azuma's inequality. In numerical optimization, we show the impact on secure key rate, the ones and respective deviations under various kinds of statistical fluctuation analyses.