  We study theoretically entanglement and operator growth in a spin system
coupled to an environment, which is modeled with classical dephasing noise.
Using exact numerical simulations we show that the entanglement growth and its
fluctuations are described by the Kardar-Parisi-Zhang equation. Moreover, we
find that the wavefront in the out-of-time ordered correlator (OTOC), which is
a measure for the operator growth, propagates linearly with the butterfly
velocity and broadens diffusively with a diffusion constant that is larger than
the one of spin transport. The obtained entanglement velocity is smaller than
the butterfly velocity for finite noise strength, yet both of them are strongly
suppressed by the noise. We calculate perturbatively how the effective time
scales depend on the noise strength, both for uncorrelated Markovian and for
correlated non-Markovian noise.
