Complementarity via Minimum Error Measurement in a Two-Path Interferometer

We study the fringe visibility and the which-path information (WPI) of a general Mach–Zehnder interferometer with an asymmetric beam splitter (BS). A minimum error measurement in the detector is used to extract the WPI. Both the fringe visibility V and the WPI I_\rm path are affected by the initial state of the photon and the second asymmetric BS. The condition in which the WPI takes the maximum is obtained. The complementarity relationship V^2+I_\rm path^2\leq 1 is found, and the conditions for equality are also presented.