We address the peculiarities of the quantum measurement process in the course of a continuous weak linear measurement (CWLM). As a tool, we implement an efficient numerical simulation scheme that allows us to generate single quantum trajectories of the measured system state as well as the recorded detector signal, and study statistics of these trajectories with and without post-selection. In this scheme, a linear detector is modelled with a qubit that is weakly coupled to the quantum system measured and is subject to projective measurement and re-initialization after a time interval at each simulation step. We explain the conditions under which the scheme provides an accurate description of CWLM. We restrict ourselves to a qubit non-demolition measurement. The qubit is initially in an equal-weight superposition of two quantum states. In the course of time, the detector signal is accumulated and the superposition is destroyed. The times required to resolve the quantum states and to destroy the superposition are of the same order. We prove numerically a rather counter intuitive fact: the average detector output conditioned on the final state does not depend on time. It looks like from the very beginning, the qubit knows in which state it is. We study statistics of decision times where the decision time is defined as time required for the density matrix along a certain trajectory to reach a threshold where it is close to one of the resulting states. This statistics is useful to estimate how fast a decisive CWLM can be. Basing on this, we devise and study a simple feedback scheme that attempts to keep the qubit in the equal-weight superposition. The detector readings are used to decide in which state the qubit is and which correction rotation to apply to bring it back to the superposition. We show how to optimize the feedback parameters and move towards more efficient feedback schemes.
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