  We introduce a classification scheme for the generators of open fermionic
Gaussian dynamics. We simultaneously partition the dynamics along the following
four lines: (1) unitary versus non-unitary, (2) active versus passive, (3)
state-dependent versus state-independent, and (4) single-mode versus
multi-mode. We find that only nine of these 16 types of dynamics are possible.
Using this partition we discuss the consequences of imposing complete
positivity on fermionic Gaussian dynamics. In particular, we show that
completely positive dynamics must be either unitary (and so can be implemented
without a quantized environment) or active (and so must involve particle
exchange with an environment).
