  We consider a free-fermion chain undergoing dephasing, described by two
different random-measurement protocols (unravelings): a quantum-state-diffusion
and a quantum-jump one. Both protocols keep the state in a Slater-determinant
form, allowing to address quite large system sizes. We find a bifurcation in
the distribution of the measurement operators along the quantum trajectories,
that's to say, there is a point where the shape of this distribution changes
from unimodal to bimodal. The value of the measurement strength where this
phenomenon occurs is similar for the two unravelings, but the distributions and
the transition have different properties reflecting the symmetries of the two
measurement protocols. We also consider the scaling with the system size of the
inverse participation ratio of the Slater-determinant components and find a
power-law scaling that marks a multifractal behaviour, in both unravelings and
for any nonvanishing measurement strength.
