What do commuting quantum channels look like?

Question

Consider two channels, $\Phi,\Psi\in\mathrm C(\mathcal X)$ acting on some space $\mathcal X$, and suppose they commute, that is, $$\Phi(\Psi(\rho))=\Psi(\Phi(\rho))$$ for all states $\rho$. Can anything be said about the structure, e.g. in terms of their Kraus operators, that this implies? For example, does there have to be a specific relation between their Kraus operators?

Suppose $\Phi(X)=\sum_a A_a X A_a^\dagger$ and $\Psi(X)=\sum_b B_b X B_b^\dagger$. Then the question relates to the channels with Kraus operators $\{A_a B_b\}_{ab}$ and $\{B_b A_a\}_{ab}$. There are a few cases in which the solution is obvious, e.g. if the Kraus operators commute with each others, $[A_a,B_b]=0$, but I'm not sure how to proceed in the more general case. Is there anything general that can be said about the problem?

Answer 1

Examples

The condition $[A_a, B_b]=0$ is sufficient, but not necessary for $\Phi$ and $\Psi$ to commute. Indeed, the standard Kraus representations of many commuting pairs of channels have non-commuting Kraus operators, e.g. bit- and phase-flip channel, depolarizing and unitary channel, amplitude damping and phase-flip channel etc. However, Kraus operators of commuting channels necessarily satisfy $(3)$ below.

Relation between Kraus operators of commuting channels

Let $K(\Phi)\in L(\mathcal{X}\otimes\mathcal{X})$ denote the unique linear operator such that

$$ K(\Phi)\mathrm{vec}(X) = \mathrm{vec}(\Phi(X)),\tag1 $$

for all $X\in L(\mathcal{X})$, c.f. equation $(2.61)$ on page $77$ in The Theory of Quantum Information by John Watrous. Then $\Phi$ and $\Psi$ commute if and only if $K(\Phi)$ and $K(\Psi)$ commute. By Proposition $2.20$ on page $80$, we have

$$ K(\Phi)=\sum_a A_a\otimes \overline{A_a}, \quad K(\Psi)=\sum_b B_b\otimes \overline{B_b}\tag2 $$

which means that $\Phi$ and $\Psi$ commute if and only if

$$ \sum_{a,b}A_aB_b \otimes \overline{A_aB_b} = \sum_{a,b}B_bA_a \otimes \overline{B_bA_a}.\tag3 $$

This allows us to state sufficient - but not necessary - conditions on Kraus operators that are somewhat more general than the requirement $[A_a,B_b]=0$. Specifically, if for every pair $a, b$ we have

$$ A_a B_b = e^{i\theta_{a,b}}B_b A_a\tag4 $$

for some $\theta_{a,b}\in[0,2\pi)$ then $\Phi$ and $\Psi$ commute.