  We report on a study of complementarity in a two-terminal "closed-loop"
Aharonov-Bohm interferometer. In this interferometer, the simple picture of
two-path interference cannot be applied. We introduce a nearby quantum point
contact to detect the electron in a quantum dot inserted in the interferometer.
We found that charge detection reduces but does not completely suppress the
interference even in the limit of perfect detection. We attribute this
phenomenon to the unique nature of the closed-loop interferometer. That is, the
closed-loop interferometer cannot be simply regarded as a two-path
interferometer because of multiple reflections of electrons. As a result, there
exist indistinguishable paths of the electron in the interferometer and the
interference survives even in the limit of perfect charge detection. This
implies that charge detection is not equivalent to path detection in a
closed-loop interferometer. We also discuss the phase rigidity of the
transmission probability for a two-terminal conductor in the presence of a
detector.
