  We study quantum processes, as one parameter families of differentiable
completely positive and trace preserving (CPTP) maps. Using different
representations of the generator, and the Sylvester criterion for positive
semi-definite matrices, we obtain conditions for the divisibility of the
process into completely positive (CP-divisibility) and positive
(P-divisibility) infinitesimal maps. Both concepts are directly related to the
definition of quantum non-Markovianity. For the single qubit case we show that
CP- and P-divisibility only depend on the dissipation matrix in the master
equation form of the generator. We then discuss three classes of processes
where the criteria for the different types of divisibility result in simple
geometric inequalities, among these the class of non-unital anisotropic Pauli
channels.

Quantum Physics