equivalent to minimize ${\left\lVert{{\boldsymbol{C}}}\right\rVert}_{{HS}}^2 = {\left\langle{{\boldsymbol{C}}}, {{\boldsymbol{C}}}\right\rangle}_{{HS}}$. The first order stationary conditions of ${\left\langle{{\boldsymbol{C}}}, {{\boldsymbol{C}}}\right\rangle}_{{HS}}$ with respect to $A$, $B_j$ ($1\le j \le M$) and ${{\boldsymbol{G}}}(t)$ gives $$\label{eqn::statn_codn}
\left\{
\begin{split}
0 &= {\left\langle{{\boldsymbol{U}}}^{\dagger}(t) {{\boldsymbol{C}}}{{\boldsymbol{U}}}(t), \ {\mathbb{E}}_{\theta_t}\bigl[(\delta A) y^{\dagger} + y (\delta A)^{\dagger}\bigr]\right\rangle}_{{HS}} \\
0 &= {\left\langle{{\boldsymbol{U}}}^{\dagger}(t) {{\boldsymbol{C}}}{{\boldsymbol{U}}}(t), \ {\mathbb{E}}_{\theta_t}\bigl[(\delta B_j) B_j^{\dagger} + B_j (\delta B_j)^{\dagger}\bigr]\right\rangle}_{{HS}} \\
0 &= \operatorname{tr}\left( {\left[{{\boldsymbol{C}}},  {\mathbb{E}}_{\mu_{{LR},t}}[ x x^{\dagger}]\right]_{-}} \delta {{\boldsymbol{G}}}\right),
\end{split}\right.$$ where $\delta A$ and $\delta B_j$ are perturbations of functions $A$ and $B_j$ respectively and $\delta {{\boldsymbol{G}}}$ is a Hermitian matrix as perturbation of ${{\boldsymbol{G}}}$ and ${\left[\cdot, \cdot\right]_{-}}$ is the commutator.

Consider the third condition in [Eq. ]{}. Let us complete the basis of ${{\mathbb{C}}^{{n}}}$ by extending $\text{Ran}({{\boldsymbol{U}}}(t))$, denoted by $\{U_1, U_2, \cdots, U_r, U_{r+1}, \cdots \cdots\}$. Since $\delta {{\boldsymbol{G}}}$ is arbitrary among all possible perturbations, consider the special choice $\delta {{\boldsymbol{G}}}= \lambda U_j {\left\langleU_k, \cdot\right\rangle} + {\text{h.c.}}$ where $j \le r$ and $k > r$. Note that when both $j, k \le r$ or $j, k > r$, by [[our restriction in [Eq. ]{}]{}]{}, ${\left\langleU_j, {{\boldsymbol{G}}}(t) U_k\right\rangle} = 0$. Hence as the perturbation of ${{\boldsymbol{G}}}$, $\delta {{\boldsymbol{G}}}$ should preserve this property. That means, we cannot choose $\delta {{\boldsymbol{G}}}$ with nonzero entries for $j, 