  We investigate quantitatively the wave-particle duality in a general
Mach-Zehnder interferometer setup with an asymmetric beam splitter. The
asymmetric beam splitter introduces additional a priori which-path knowledge,
which is different for a particle detected at one output port of the
interferometer and a particle detected at the other. Accordingly, the fringe
visibilities of the interference patterns emerging at the two output ports are
also different. Hence, in sharp contrast with the symmetric case, here we
should concentrate on one output port and distinguish two possible paths taken
by the particles detected at that port among four paths. It turns out that two
nonorthogonal unsharp observables are measured jointly in this setup. We apply
the condition for joint measurability of these unsharp observables to obtain a
trade-off relation between the fringe visibility of the interference pattern
and the which-path distinguishability.
