Mean Field Theory for Lindblad Equations

I am working through the Appendix A.1 and A.2 of the paper Topology by Dissipation and try to learn something about the final result.

The initial situation is a Lindblad-equation. Then one does a mean-field-approximation by doing a product-ansatz

$$
rho stackrel{text{Ansatz}}{approx} bigotimes_{ k in K / text{~} } rho_{k}
$$

where $K = text{“allowed reciprocal wavevectors within the FBZ”}$ and $k_1 text{~} k_2 : Leftrightarrow k_1 = -k_2$. Therefore the density $rho_k$ describes the mode-pair ${ -k, +k }$.

Now one calculates an effective Lindblad-equation for the mode-pair ${ -k, +k }$ as partial traces become evaluated with respect to some reference state and one obtains

$$operatorname{Tr}_{neq k} mathcal{L} [rho] = ldots approx sum_{sigma = pm} kappa_k left( bar{L}_{sigma k} rho_k bar{L}_{sigma k}^{dagger} – frac{1}{2} { bar{L}_{sigma k}^{dagger} bar{L}_{sigma k} , rho_k } right) =: mathcal{L}_k [ rho_k ] = partial_t rho_k $$

with new quantum jump operators $bar{L}_{sigma k}$ that are linear in the creation and annihilation operators.

Question: Is it now possible to rewrite the effective Lindblad equations for the different modes as a Lindblad equation for the density $rho$?

So is it possible to write

$$
partial_t rho = kappa sum_{ k in Q / text{~} } left( tilde{l}_k rho tilde{l}_k^{dagger} – frac{1}{2} { tilde{l}_k^{dagger} tilde{l}_k , rho } right)
$$

or something like that?

I was thinking about differentiating the l.h.s. of the equation and applying the effective Lindblad operators for each derivative appearing in the products, i.e.

$$
partial_t rho = sum_{ i = 1}^{M} rho_{q_1} otimes cdots otimes rho_{q_{i – 1}} otimes partial_t rho_{q_{i}} otimes rho_{q_{i + 1}} otimes cdots otimes rho_{q_M} = sum_{ i = 1}^{M} rho_{q_1} otimes cdots otimes rho_{q_{i – 1}} otimes mathcal{L}_{q_i} [rho_{q_{i}}] otimes rho_{q_{i + 1}} otimes cdots otimes rho_{q_M}
$$

for a counting ${ q_1, ldots, q_M}$ of $Q / text{~} $ but I struggle a little bit in proceeding. Is there a possibility to "pull" the quantum jump operators out of the products?

I am pretty sure that

$$
rho_{q_1} otimes cdots otimes rho_{q_{i – 1}} otimes mathcal{L}_{q_i} [rho_{q_{i}}] otimes rho_{q_{i + 1}} otimes cdots otimes rho_{q_M}
neq sum_{sigma = pm} kappa_k left( bar{L}_{sigma k} rho bar{L}_{sigma k}^{dagger} – frac{1}{2} { bar{L}_{sigma k}^{dagger} bar{L}_{sigma k} , rho } right)
$$

but maybe one can get a usual Lindblad equation at the cost of some extra terms or some standard-trick?