The density matrices for qubits coupled to different spin chains may be written as:
$\rho_{ii'}^A(t)=\sum_{kk'}A_{ii'}^{kk'}\rho^{A}_{kk'}(0)$
$\rho_{jj'}^B(t)=\sum_{ll'}B_{jj'}^{ll'}\rho^{B}_{ll'}(0)$
Where all indices are either 0 or 1 and the matrices A and B are the dynamical maps for each density matrix.
For the density matrix of system $C=A+B$ one gets:
$\rho_{ii',jj'}^C(t)=\sum_{kk',ll'}A_{ii'}^{kk'}B_{jj'}^{ll'}\rho^{C}_{kk',ll'}(0)$
Can this be written as the tensor product of $A$ and $B$ acting on the density matrix written as a vector, something as below?
$\rho^C(t)=A\otimes B \rho^{C}(0)$
I understand how to calculate the separate cases however the combined case is giving me trouble especially when it comes to the ordering of the elements of the density matrices and maps.