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It is known that the coupling lemma provides a useful tool in the study ofprobability theory and its related areas. It describes the relation between thevariational distance of two probability distributions and the probability thatoutcomes from the two random experiments associated with each distribution arenot identical. In this paper, the failure probability interpretation problemthat has been presented by Yuen and Hirota is discussed from the viewpoint ofthe application of the coupling lemma. First, we introduce the coupling lemma,and investigate properties of it. Next, it is shown that the claims for thisproblem in the literatures are justified by using the coupling lemma.Consequently, we see that the failure probability interpretation is notadequate in the security analysis of quantum key distribution.

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