Statistical-fluctuation analysis for quantum key distribution with consideration of after-pulse contributions
Hongxin Li, Haodong Jiang, Ming Gao, Zhi Ma, Chuangui Ma, Wei Wang · Physical Review A · 2015
The statistical fluctuation problem is a critical factor in all quantum key distribution (QKD) protocols under finite-key conditions. The current statistical fluctuation analysis is mainly based on independent random samples, however, the precondition cannot always be satisfied because of different choices of samples and actual parameters. As a result, proper statistical fluctuation methods are required to solve this problem. Taking the after-pulse contributions into consideration, this paper gives the expression for the secure key rate and the mathematical model for statistical fluctuations, focusing on a decoy-state QKD protocol [Z.-C. Wei et al., Sci. Rep. 3, 2453 (2013)] with a 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, a deviation comparison is made between the law of large numbers and standard error analysis. Second, a sufficient condition is given that the Chernoff bound achieves a better result than Hoeffding's inequality based on only independent relations. Third, by constructing the proper martingale, a stringent way is proposed to deal issues based on dependent random samples through making use of Azuma's inequality. In numerical optimization, the impact on the secure key rate, the comparison of secure key rates, and the respective deviations under various kinds of statistical fluctuation analyses are depicted.