  The complete positivity (CP) of a quantum dynamical map (QDM) is, in general,
difficult to show when its master equation (ME) does not conform to the
Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form. The GKSL ME describes the
Markovian dynamics, comprising a unitary component with time-independent
Hermitian operators and a non-unitary component with time-independent Lindblad
operators and positive time-independent damping rates. Recently, the
non-Markovian dynamics has received growing attention, and the various types of
GKSL-like MEs with time-dependent operators are widely discussed; however,
rigorous discussions on their CP conditions remain limited. This paper presents
conditions for QDMs to be CP, whose MEs take the GKSL-like form with arbitrary
time dependence. One case considered is where its ME takes the time-local
integro-differential GKSL-like form, which includes CP-divisible cases. Another
case considered is where the ME is time-non-local but can be approximated to be
time-local in the weak-coupling regime. As a special case of the time-non-local
case, the same discussion holds for the time-convoluted GKSL-like form, which
should be compared to previous studies.
