Fringe visibility and distinguishability in two-path interferometer with an asymmetric beam splitter

We study the fringe visibility and the distinguishability of a general Mach-Zehnder interferometer with an asymmetric beam splitter. Both the fringe visibility V and the distinguishability D are affected by the input state of the particle characterized by the Bloch vector S =( S _x, S _y, S _z) and the second asymmetric beam splitter characterized by the paramter β. For the total system is initially in a pure state, it is found that the fringe visibility reaches the upper bound and the distinguishability reaches the lower bound when cosβ=-S_x. The fringe visibility obtain the maximum only if S_x=0 and β=π/2 when the input particle is initially in a mixed state. The complementary relationship V²+D² ≤ 1 is proved in a general Mach-Zehnder interferometer with an asymmetric beam splitter, and the conditions for the equality are also presented.