Abstract
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^⊥|K_ΦK_Φ^⊥|x= x|K_ΦK_Φ^⊥|x={0} where K_Φ is the Kraus subspace of Φ and K_Φ^⊥ 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.