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We address the peculiarities of the quantum measurement process in the courseof a continuous weak linear measurement (CWLM). As a tool, we implement anefficient numerical simulation scheme that allows us to generate single quantumtrajectories of the measured system state as well as the recorded detectorsignal, and study statistics of these trajectories with and withoutpost-selection. In this scheme, a linear detector is modelled with a qubit thatis weakly coupled to the quantum system measured and is subject to projectivemeasurement and re-initialization after a time interval at each simulationstep. We explain the conditions under which the scheme provides an accuratedescription of CWLM.  We restrict ourselves to a qubit non-demolition measurement. The qubit isinitially in an equal-weight superposition of two quantum states. In the courseof time, the detector signal is accumulated and the superposition is destroyed.The times required to resolve the quantum states and to destroy thesuperposition are of the same order. We prove numerically a rather counterintuitive fact: the average detector output conditioned on the final state doesnot depend on time. It looks like from the very beginning, the qubit knows inwhich state it is. We study statistics of decision times where the decisiontime is defined as time required for the density matrix along a certaintrajectory to reach a threshold where it is close to one of the resultingstates. 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 tokeep the qubit in the equal-weight superposition. The detector readings areused to decide in which state the qubit is and which correction rotation toapply to bring it back to the superposition. We show how to optimize thefeedback parameters and move towards more efficient feedback schemes.

Conditioned outputs, distribution of decision times and  measurement-based feedback scheme for continuous weak linear measurement of a  simple quantum system