  We use Majorana operators to study entanglement dynamics under random free
fermion unitary evolution and projective measurements in one dimension. For
certain choices of unitary evolution, namely those which swap neighboring
Majorana operators, and measurements of neighboring Majorana bilinears, one can
map the evolution to the statistical model of completely packed loops with
crossings (CPLC) and study the corresponding phase diagram. We generalize this
model using the language of fermionic Gaussian states to a general free fermion
unitary evolution acting on neighboring Majorana operators, and numerically
compute its phase diagram. We find that both the Goldstone and area law phases
persist in this new phase diagram, but with a shifted phase boundary. One
important qualitative aspect of the new phase boundary is that even for the
case of commuting measurements, the Goldstone phase persists up to a finite
non-zero measurement rate. This is in contrast with the CPLC, in which
non-commuting measurements are necessary for realizing the Goldstone phase. We
also numerically compute the correlation length critical exponent at the
transition, which we find to be near to that of the CPLC, and give a tentative
symmetry based explanation for some differences in the phase transition line
between the CPLC and generalized models.
