  Characterizing those quantum channels that correspond to Markovian time
evolutions is an open problem in quantum information theory, even different
notions of quantum Markovianity exist. One such notion is that of infinitesimal
Markovian divisibility for quantum channels introduced in
arXiv:math-ph/0611057. Whereas there is a complete characterization for
infinitesimal Markovian divisible qubit channels, no necessary or sufficient
criteria are known for higher dimensions, except for necessity of
non-negativity of the determinant.
  We describe how to extend the notion of infinitesimal Markovian divsibility
to general linear maps and closed and convex sets of generators. We give a
general approach towards proving necessary criteria for (infinitesimal)
Markovian divisibility that involve singular values of the linear map. With
this approach, we prove two necessary criteria for infinitesimal divisibility
of quantum channels that work in any finite dimension $d$: an upper bound on
the determinant in terms of a $\Theta(d)$-power of the smallest singular value,
and in terms of a product of $\Theta(d)$ smallest singular values. Our criteria
allow us to analytically construct, in any given dimension, a set of channels
that contains provably non infinitesimal Markovian divisible ones.
  We also discuss the classical counterpart of this scenario, i.e., stochastic
matrices with the generators given by transition rate matrices. Here, we show
that no necessary criteria for infinitesimal Markovian divisibility of the form
proved for quantum channels can hold in general. However, we describe subsets
of all transition rate matrices for which our reasoning can be applied to
obtain necessary conditions for Markovian divisibility.
