  We study the dynamics of entanglement in the quantum Ising chain with
dephasing dissipation in a Lindblad master equation form. We consider two
unravelings which preserve the Gaussian form of the state, allowing to address
large system sizes. The first unraveling gives rise to a
quantum-state-diffusion dynamics, while the second one describes a specific
form of quantum-jump evolution, suitably constructed to preserve Gaussianity.
In the first case we find a crossover from area-law to logarithm-law
entanglement scaling and draw the related phase diagram. In the second case we
only find logarithm-law scaling, remarking the different entanglement behavior
for different unravelings of the same Lindblad equation. Finally, we compare
these outcomes with the predictions of a non-Hermitian Hamiltonian evolution,
finding conflicting results.
