# Liouville superoperators of multi-qubit quantum channels : Need for Numerics

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.

• You only need a permutation of the tensor factors $X\otimes I \otimes X \otimes I \simeq X\otimes X \otimes I \otimes I = \mathcal{U} \otimes \mathcal{I}$ Commented Mar 13 at 12:44