  We use a noisy signal with finite correlation time to drive a spin
(dissipative impurity) in the quantum XY spin chain and calculate the dynamics
of entanglement entropy of a bipartition of spins, for a stochastic quantum
trajectory. We compute the noise averaged entanglement entropy of a bipartition
of spins and observe that its speed of spreading decreases at strong
dissipation, as a result of the Zeno effect. We recover the Zeno crossover and
show that noise averaged entanglement entropy can be used as a proxy for the
heating and Zeno regimes. Upon increasing the correlation time of the noise,
the location of the Zeno crossover shifts at stronger dissipation, extending
the heating regime.
