A Sufficient Criterion for Divisibility of Quantum Channels

Abstract

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 $⟨x^{\perp }|{\mathcal{K}}_{\Phi }{\mathcal{K}}_{\Phi }^{\perp }|x⟩=⟨x|{\mathcal{K}}_{\Phi }{\mathcal{K}}_{\Phi }^{\perp }|x⟩=\{0\}$ 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.

arXiv comment18 pages, submitted to J. Math. Phys

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