Passive Faraday-mirror attack in a practical two-way quantum-key-distribution system

The Faraday mirror (FM) plays a very important role in maintaining the stability of two-way plug-and-play quantum key distribution (QKD) systems. However, the practical FM is imperfect, which will not only introduce an additional quantum bit error rate (QBER) but also leave a loophole for Eve to spy the secret key. In this paper we propose a passive Faraday mirror attack in two-way QKD system based on the imperfection of FM. Our analysis shows that if the FM is imperfect, the dimension of Hilbert space spanned by the four states sent by Alice is three instead of two. Thus Eve can distinguish these states with a set of Positive Operator Valued Measure (POVM) operators belonging to three-dimension space, which will reduce the QBER induced by her attack. Furthermore, a relationship between the degree of the imperfection of FM and the transmittance of the practical QKD system is obtained. The results show that the probability that Eve loads her attack successfully depends on the degree of the imperfection of FM rapidly, but the QBER induced by Eve's attack changes slightly with the degree of the FM imperfection.