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.

18 pages, submitted to J. Math. Phys

References in corpus (4)

• Assessing non-Markovian dynamics
• On measures of non-Markovianity: divisibility vs. backflow of information
• Which Bath-Hamiltonians Matter for Thermal Operations?
• Quantum dynamics is not strictly bidivisible