A Sufficient Criterion for Divisibility of Quantum Channels

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.