A sufficient criterion for divisibility of quantum channels

We present a simple, dimension-independent criterion which guarantees that some quantum channel Φ is divisible, i.e., that there exists a non-trivial factorization Φ = Φ1Φ2. The idea is to first define an “elementary” channel Φ2 and then to analyze when ΦΦ2−1 is completely positive. The sufficient criterion obtained this way—which even yields an explicit factorization of Φ—is that one has to find orthogonal unit vectors x, x⊥ such that 〈x⊥|KΦKΦ⊥|x〉=〈x|KΦKΦ⊥|x〉={0} where KΦ is the Kraus subspace of Φ and KΦ⊥ 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 Φ into in some sense “simple” channels. Finally, be aware that our techniques are not limited to the particular elementary channel we chose.