As greatly described here, a density matrix can be vectorized in column-major order such that a (unitary) channel can be written
$$ \rho \rightarrow |\rho\rangle = vec(\rho) \\ \rho' = U \rho U^{\dagger} \rightarrow \mathcal{U} = U^*\otimes U \\ |\rho'\rangle = \mathcal{U} |\rho\rangle \leftrightarrow \rho' = unvec(|\rho'\rangle) $$
Is there a simple way to represent the multi-qubit Liouville superoperators using its constituents? Say,we have single qubit $U = X \otimes I$, and the $\mathcal{U} = X^*\otimes X$,
$$ \mathcal{\tilde{U}}_2 = U^*\otimes U = X \otimes I \otimes X \otimes I $$
Is there a way of writing $\mathcal{\tilde{U}}_2$ in terms of the single qubit superoperator $\mathcal{U}$ and identities? Numerically, if it is possible, I assume there needs to be a ton of reshaping and index shuffling is involved.
