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.
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