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.
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