  We consider a model of a detector of ballistic electrons at the edge of a
two-dimensional electron gas in the integer quantum Hall regime. The electron
is detected by capacitive coupling to a gate which is also coupled to a passive
RC circuit. Using a quantum description of this circuit, we determine the
signal over noise ratio of the detector in term of the detector
characteristics. The back-action of the detector on the incident wavepacket is
then computed using a Feynman-Vernon influence functional approach. Using
information theory, we define the appropriate notion of quantum limit for such
an "on the fly" detector. We show that our particular detector can approach the
quantum limit up to logarithms in the ratio of the measurement time over the RC
relaxation time. We argue that such a weak logarithmic effect is of no
practical significance. Finally we show that a two-electron interference
experiment can be used to probe the detector induced decoherence.
