  The faraday mirror (FM) plays a very important role in maintaining the
stability of two way plug-and-play quantum key distribution (QKD) system.
However, the practical FM is imperfect, which will not only introduce
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 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 with the degree of the
imperfection of FM slightly.
