We analyze the operation of a switching-based detector that probes a qubit'sobservable that does not commute with the qubit's Hamiltonian, leading to anontrivial interplay between the measurement and free-qubit dynamics. In orderto obtain analytic results and develop intuitive understanding of the differentpossible regimes of operation, we use a theoretical model where the detector isa quantum two-level system that is constantly monitored by a macroscopicsystem. We analyze how to interpret the outcome of the measurement and how thestate of the qubit evolves while it is being measured. We find that the answersto the above questions depend on the relation between the different parametersin the problem. In addition to the traditional strong-measurement regime, weidentify a number of regimes associated with weak qubit-detector coupling. Anincoherent detector whose switching time is measurable with high accuracy canprovide high-fidelity information, but the measurement basis is determined onlyupon switching of the detector. An incoherent detector whose switching time canbe known only with low accuracy provides a measurement in the qubit's energyeigenbasis with reduced measurement fidelity. A coherent detector measures thequbit in its energy eigenbasis and, under certain conditions, can providehigh-fidelity information.
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