  We develop a notion of dephasing under the action of a quantum Markov
semigroup in terms of convergence of operators to a block-diagonal form
determined by irreducible invariant subspaces. If the latter are all
one-dimensional, we say the dephasing is maximal. With this definition, we show
that a key necessary requirement on the Lindblad generator is bistochasticity,
and focus on characterizing whether a maximally dephasing evolution may be
described in terms of a unitary dilation with only classical noise, as opposed
to a genuine non-commutative Hudson-Parthasarathy dilation. To this end, we
make use of a seminal result of K\"{u}mmerer and Maassen on the class of
commutative dilations of quantum Markov semigroups. In particular, we introduce
an intrinsic quantity constructed from the generator, which vanishes if and
only if the latter admits a self-adjoint representation and which quantifies
the degree of obstruction to having a classical diffusive noise model.
