A Sufficient Criterion for Divisibility of Quantum Channels
Frederik vom Ende
TL;DR
The paper addresses the divisibility problem for quantum channels, asking when a channel $\Phi$ can be written as a product $\Phi_1\Phi_2$ with non-trivial factors. It develops a dimension-independent approach by factoring out an elementary two-Kraus channel $\Psi_\lambda$ and examining when $\Phi\Psi_\lambda^{-1}$ remains completely positive, using the Kraus subspace $\mathcal{K}_\Phi$ and its orthogonal complement via the Choi–Jamiołkowski isomorphism. A central result provides a constructive sufficient criterion: if there exist orthogonal unit vectors $x,x^\perp$ such that $\langle x^\perp|K_j^\dagger G_k|x\rangle=0=\langle x|K_j^\dagger G_k|x\rangle$ for all Kraus indices $j$ and basis elements $G_k$ of $\mathcal{K}_\Phi^\perp$, then $\Phi$ is divisible and, when $\Phi(|x\rangle\langle x|)\neq\Phi(|x^\perp\rangle\langle x^\perp|)$, one obtains an explicit decomposition with a Kraus-rank-reducing factor. The paper provides a rigorous derivation via the kernel of the Choi matrix, several corollaries including qubit and block-diagonal cases, and concrete examples illustrating both the power and limits of the approach. Overall, the work offers a practical blueprint for decomposing channels and a step toward understanding divisibility in higher dimensions with potential impact on channel semigroups and quantum circuit synthesis.
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 $Φ=Φ_1Φ_2$. The idea is to first define an "elementary" channel $Φ_2$ and then to analyze when $ΦΦ_2^{-1}$ 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 $x,x^\perp$ such that $\langle x^\perp|\mathcal K_Φ\mathcal K_Φ^\perp|x\rangle=\langle x|\mathcal K_Φ\mathcal K_Φ^\perp|x\rangle=\{0\}$ where $\mathcal K_Φ$ is the Kraus subspace of $Φ$ and $\mathcal K_Φ^\perp$ 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 $Φ$ into in some sense "simple" channels. Finally, be aware that our techniques are not limited to the particular elementary channel we chose.