Security proof of the two-way quantum secure direct communication with channel loss and noise

Quantum secure direct communication is one of the important mode of quantum communication, which sends secret information through a quantum channel directly without setting up a prior key. Over the past decade, numerous protocols have been proposed, and part of them have been experimentally demonstrated. However, the strict security proof is still in suspense. Two-way protocol is seen as one of the most practical protocol, in this paper, the security of the two-way quantum secure direct communication protocol will be proved against general attacks when the noisy and lossy channel is taken into account. There are two steps included in the proof, firstly, we need to prove that the mutual information between Alice and Bob is larger than the mutual information shared between Alice and the eavesdropper when the error rate in control mode is less than the predetermined threshold. Secondly, once the error rate is lower than the threshold, one can always find out a forward error correction code to guarantee the security of information transmission.