opaque and quite large, hence, choosing infinitesimal generators of a particular form is necessary in practice.

Following the second [[constraint]{}]{} above, let us denote $Y_t(\omega)$ the corresponding SDE on ${\mathbb{C}}^r$ whose infinitesimal generator is ${\mathcal{A}}_{\theta,t}$. Consider the following family of stochastic dynamics with ${{\boldsymbol{G}}}$, $A$ and $B$’s to be chosen $$\label{eqn::sdeY}
\begin{split}
  \dot{{{\boldsymbol{U}}}}(t) &=  i {{\boldsymbol{G}}}(t) {{\boldsymbol{U}}}(t),\qquad 
  \text{with } {{\boldsymbol{G}}}^{\dagger}(t) = {{\boldsymbol{G}}}(t),\  {{\boldsymbol{G}}}(t) = {\mathcal{Q}}_{{{\boldsymbol{U}}}(t)} {{\boldsymbol{G}}}(t) {\mathcal{P}}_{{{\boldsymbol{U}}}(t)} + {\text{h.c.}}\\
  {\,\mathrm{d}}Y &= A(Y,t) {\,\mathrm{d}}t + \sum_{j=1}^{M} B_j(Y,t) {\,\mathrm{d}}W_j. \\
  \end{split}$$ Define $X_{{LR}, t}(\omega) := {{\boldsymbol{U}}}(t) Y_t(\omega)$ and denote $\mu_{{LR}, t}$ the probability measure induced by random variable $X_{{LR},t}(\omega)$. It is straightforward to check that $X_{{LR},t}(\omega)$ satisfies the SDE $${\,\mathrm{d}}X_{{LR}} = a_{{LR}}\left(X_{{LR}}, t\right){\,\mathrm{d}}t + \sum_{j=1}^{M}b_{{LR},j} \left(X_{{LR}}, t\right){\,\mathrm{d}}W_j,
  \label{eqn::sde_lowrank}$$ where, for $1\le j\le M$, $$\begin{cases}
    \begin{aligned}
      a_{{LR}}(X_{{LR},t}, t) & =  
      i {{\boldsymbol{G}}}(t) X_{{LR},t} + {{\boldsymbol{U}}}(t) A({{\boldsymbol{U}}}^{\dagger}(t) X_{{LR},t}, t),
    \end{aligned} \\
    b_{{LR}, j}(X_{{LR},t}, t)= {{\boldsymbol{U}}}(t) B_j({{\boldsymbol{U}}}^{\dagger}(t) X_{{LR},t}, t).
  \end{cases}$$ Note that $M$ does not have to be the same as $N$ and ${\mathrm{d}}W_j$ in [Eq. ]{} are not necessarily the same Brownian motions in [Eq. ]{}; we use ${\mathrm{d}}W_j$ for both to save notation. Since we are interested in the error in the weak sense, how the randomness is achieved does not matter; what 