  We explain how to apply a Gaussian-preserving operator to a fermionic
Gaussian state. We use this method to study the evolution of the entanglement
entropy of an Ising spin chain following a Lindblad dynamics with string
measurement operators, focusing on the quantum-jump unraveling of such
Lindbladian. We find that the asymptotic entanglement entropy obeys an area law
for finite-range string operators and a volume law for ranges of the string
which scale with the system size. The same behavior is observed for the
measurement-only dynamics, suggesting that measurements can play a leading role
in this context.
