Security of a single-state semi-quantum key distribution protocol

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.