Coupling Lemma and Its Application to The Security Analysis of Quantum Key Distribution

It is known that the coupling lemma provides a useful tool in the study of probability theory and its related areas. It describes the relation between the variational distance of two probability distributions and the probability that outcomes from the two random experiments associated with each distribution are not identical. In this paper, the failure probability interpretation problem that has been presented by Yuen and Hirota is discussed from the viewpoint of the application of the coupling lemma. First, we introduce the coupling lemma, and investigate properties of it. Next, it is shown that the claims for this problem in the literatures are justified by using the coupling lemma. Consequently, we see that the failure probability interpretation is not adequate in the security analysis of quantum key distribution.