  In this paper, we rigorously prove the intuition that in security proofs for
BB84 one may regard an incoming signal to Bob as a qubit state. From this
result, it follows that all security proofs for BB84 based on a virtual qubit
entanglement distillation protocol, which was originally proposed by Lo and
Chau [H.-K. Lo and H. F. Chau, Science 283, 2050 (1999)], and Shor and Preskill
[P. W. Shor and J. Preskill, Phys. Rev. Lett. 85, 441 (2000)], are all valid
even if Bob's actual apparatus cannot distill a qubit state explicitly. As a
consequence, especially, the well-known result that a higher bit error rate of
20% can be tolerated for BB84 protocol by using two-way classical
communications is still valid even when Bob uses threshold detectors. Using the
same technique, we also prove the security of the Bennett-Brassard-Mermin 1992
(BBM92) protocol where Alice and Bob both use threshold detectors.
