  Quantum measurement is essential to both the foundations and practical
applications of quantum information science. Among many possible models of
quantum measurement, feedback measurements that dynamically update their
physical structure are highly interesting due to their flexibility which
enables a wide range of measurements that might otherwise be hard to implement.
Here we investigate by detector tomography a measurement consisting of a
displacement operation combined with photon detection followed by a real time
feedback operation. We design the measurement in order to discriminate the
superposition of vacuum and single photon states -- the single-rail qubit --
and find that it can discriminate the superposition states with a certainty of
96\%. Such a feedback-controlled photon counter will facilitate the realization
of quantum information protocols with single-rail qubits as well as the
non-locality test of certain entangled states.
