Achieving the positivity of the secret key in a BB84 like quantum key distribution protocol

Woodhead [Phys. Rev. A \textbf{88}, 012331 (2013)] derived the lower bound of the secret key rate for a Bennett-Brassard (BB84) like quantum key distribution protocol under collective attacks. However, this lower bound does not always assure the generation of the secret key and thus the protocol may have to be aborted sometimes. Thus, we modify the Woodhead's lower bound of the secret key rate in such a way that the secret key is always generated in a BB84 like quantum key distribution protocol. Exploiting the obtained modified lower bound of the secret key rate, we analyze two state dependent quantum cloning machines such as (i) Wootters-Zurek QCM and (ii) Modified Buzek-Hillery QCM constructed by fixing the cloning machine parameters of Buzek Hillery quantum cloning machine (QCM), which may be used by the eavesdropper to extract information from the intercepted state. We, thereafter, show that it is possible for the communicating parties to distill a secret key, even in the presence of an eavesdropper. Moreover, we also discuss the effect of the efficiency of the QCM on the generation of the secret key for a successful key distribution protocol.