  Semi-quantum key distribution protocols are allowed to set up a secure secret
key between two users. Compared with their full quantum counterparts, one of
the two users is restricted to perform some "classical" or "semi-quantum"
operations, which makes them easily realizable by using less quantum resource.
However, the semi-quantum key distribution protocols mainly rely on a two-way
quantum channel. The eavesdropper has two opportunities to intercept the
quantum states transmitted in the quantum communication stage. It may allow the
eavesdropper to get more information and make the security analysis more
complicated. In the past ten years, many semi-quantum key distribution
protocols have been proposed and proved to be robust. But there are few works
concerned about their unconditional security. It is doubted that how secure the
semi-quantum ones are and how much noise can they tolerate to establish a
secure secret key. In this paper, we prove the unconditional security of a
single-state semi-quantum key distribution protocol proposed by $Zou$ et al. in
[Phys. Rev. A. 79]. We present a complete proof from information theory aspect
by deriving a lower bound of the protocol's key rate in the asymptotic
scenario. Using this bound, we figure out an error threshold value such that
all error rates are less than this threshold value, the secure secret key can
be established between the legitimate users definitely. Otherwise, the users
should abort the protocol. we make an illustration of the protocol under the
circumstance of the reverse quantum channel is a depolarizing one with
parameter $q$. Additionally, we compare the error threshold value with some
full quantum protocols and several existing semi-quantum ones whose
unconditional security proofs have been provided recently.
