Master equation approximation In the Lindbladian master equation, one of the approximations is assuming $\rho(t)=\rho_S(t)\otimes\rho_B(t)$.
However, if we try to solve the total state in a numerical way, using $\dot{\rho}(t)=-i[H_{SB},\rho(t)]\,\,(H_{SB}=S(t)\otimes B(t))$, we get,
$\rho(t_1)-\rho(0)=-i\delta t\big(S(0)\rho_S(0)\otimes B(t)\rho_B(0)-\rho_S(0)S(0)\otimes \rho_B(0)B(0)\big)$
$\rho(t_1)=\rho(0)-i\delta t\big(S(0)\rho_S(0)\otimes B(t)\rho_B(0)-\rho_S(0)S(0)\otimes \rho_B(0)B(0)\big)\neq \rho_S(t_1)\otimes \rho_B(t_1)$
The above total state for $t=t_1$ does not seem separable.
Therefore, I do not understand how to justify $\rho(t)=\rho_S(t)\otimes\rho_B(t)$ condition.
Where am I wrong?
 A: It is only a good approximation if your initial state at the beginning of the evolution is separable:
$$
\rho(t_0) = \rho_S(t_0) \otimes \rho_B(t_0) 
$$
The argument goes like this: if you have no coupling between the system and environment then you would certainly just have $\rho(t) = \rho_S(t) \otimes \rho_B(t)$ for all $t$ (as system and environment evolve separately in that case).
This means that any non-separable part is perturbatively small, and proportional to the coupling (say $\lambda$) in $H_{SB}$. You can convince yourself by using the ansatz
$$
\rho(t) = \rho_S(t) \otimes \rho_B(t) + \lambda \rho^{(1)}_{\mathrm{corr}} + \lambda^2 \rho^{(2)}_{\mathrm{corr}} + \ldots
$$
You'll find that the master equation at leading order in the coupling doesn't care about the higher order correlations (you could track them with some effort if you wish though).
A: The approximation $\rho(t) \approx \rho(t) \otimes \rho_B(t)$ is not done for arbitrary times. It is performed inside a time integral where there are autocorrelation functions of the baths. The assumption is that the autocorrelations decay on a time scale short enough that this approximation holds.
This assumption is not actually needed. I recommend the read of the derivation of the Lindblad master equation in the introduction section of this open-access paper, where the authors use the Zwanzing's and Nakajima's projection technique:
https://iopscience.iop.org/article/10.1088/1367-2630/12/11/113032/meta
