  We present a simple, dimension-independent criterion which guarantees that
some quantum channel $\Phi$ is divisible, i.e. that there exists a non-trivial
factorization $\Phi=\Phi_1\Phi_2$. The idea is to first define an "elementary"
channel $\Phi_2$ and then to analyze when $\Phi\Phi_2^{-1}$ is completely
positive. The sufficient criterion obtained this way -- which even yields an
explicit factorization of $\Phi$ -- is that one has to find orthogonal unit
vectors $x,x^\perp$ such that $\langle x^\perp|\mathcal K_\Phi\mathcal
K_\Phi^\perp|x\rangle=\langle x|\mathcal K_\Phi\mathcal
K_\Phi^\perp|x\rangle=\{0\}$ where $\mathcal K_\Phi$ is the Kraus subspace of
$\Phi$ and $\mathcal K_\Phi^\perp$ is its orthogonal complement. Of course,
using linearity this criterion can be reduced to finitely many equalities.
Generically, this division even lowers the Kraus rank which is why repeated
application -- if possible -- results in a factorization of $\Phi$ into in some
sense "simple" channels. Finally, be aware that our techniques are not limited
to the particular elementary channel we chose.
