  The phenomenological dissipation of the Bloch equations is reexamined in the
context of completely positive maps. Such maps occur if the dissipation arises
from a reduction of a unitary evolution of a system coupled to a reservoir. In
such a case the reduced dynamics for the system alone will always yield
completely positive maps of the density operator. We show that, for Markovian
Bloch maps, the requirement of complete positivity imposes some Bloch
inequalities on the phenomenological damping constants. For non-Markovian Bloch
maps some kind of Bloch inequalities involving eigenvalues of the damping basis
can be established as well. As an illustration of these general properties we
use the depolarizing channel with white and colored stochastic noise.
