  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.
