  Quantum decoherence arises due to uncontrollable entanglement between a
system with its environment. However the effects of decoherence are often
thought of and modeled through a simpler picture in which the role of the
environment is to introduce classical noise in the system's degrees of freedom.
Here we establish necessary conditions that the classical noise models need to
satisfy to quantitatively model the decoherence. Specifically, for
pure-dephasing processes we identify well-defined statistical properties for
the noise that are determined by the quantum many-point time correlation
function of the environmental operators that enter into the system-bath
interaction. In particular, for the exemplifying spin-boson problem with a
Lorentz-Drude spectral density we show that the high-temperature quantum
decoherence is quantitatively mimicked by colored Gaussian noise. In turn, for
dissipative environments we show that classical noise models cannot describe
decoherence effects due to spontaneous emission induced by a dissipative
environment. These developments provide a rigorous platform to assess the
validity of classical noise models of decoherence.
