Given two distinct and noninteracting quantum mechanical systems $\mathfrak{S}_1$ and $\mathfrak{S}_2$ with state spaces $\mathcal H_1$ and $\mathcal H_2$, respectively, the state space of the combined system $\mathcal S_1+\mathcal S_2$ is the tensor product Hilbert space $\mathcal H=\mathcal H_1\otimes\mathcal H_2$. Density operators $W\in\mathcal D(\mathcal H)$, and effects $F\in\mathcal E(\mathcal H)$. Similarly, there are corresponding symbols $W_i\in\mathcal D(\mathcal H_i), F_i\in\mathcal E(\mathcal H_i)$ for subsystems $\mathfrak{S}_i(i=1,2)$, respectively.
Given any quantum operation, $\Phi: \mathcal D(\mathcal H)\rightarrow \mathcal D(\mathcal H)$, of the composite system $\mathcal S_1+\mathcal S_2$.
Problem: (1) Do there exist whether or not two quantum operation $\phi_1$ and $\phi_2$, of the subsystems $\mathfrak{S}_1$ and $\mathfrak{S}_2$, respectively, such that the following diagram is commutative:
$$ \begin{diagram} \node{\mathcal D(\mathcal H_1)} \arrow[4]{e,t}{\phi_1}\node[4]{\mathcal D(\mathcal H_1)}\ \node{}\ \node{\mathcal D(\mathcal H_1\otimes\mathcal H_2)} \arrow[2]{n,l}{Tr_2} \arrow[4]{e,t}{\Phi} \arrow[2]{s,l}{Tr_1} \node[4]{\mathcal D(\mathcal H_1\otimes\mathcal H_2)} \arrow[2]{s,r}{Tr_1} \arrow[2]{n,r}{Tr_2} \ \node{}\ \node{\mathcal D(\mathcal H_2)} \arrow[4]{e,b}{\phi_2} \node[4]{\mathcal D(\mathcal H_2)} \end{diagram} $$ i.e. $$\begin{eqnarray} Tr_2(\Phi(W))&=&\frac{tr(\Phi(W))}{tr(\phi_1(Tr_2(W)))}\phi_1(Tr_2(W)),\ Tr_1(\Phi(W))&=&\frac{tr(\Phi(W))}{tr(\phi_2(Tr_1(W)))}\phi_2(Tr_1(W)), \end{eqnarray} $$ where $\phi_i: \mathcal D(\mathcal H_i)\rightarrow \mathcal D(\mathcal H_i)(i=1,2)$ and $Tr_i: \mathcal D(\mathcal H)\rightarrow \mathcal D(\mathcal H_i)$ is a partial trace with respect to the subsystem $\mathfrak{S}_i(i=1,2)$.
(2) If quantum operation $\phi_1$ and $\phi_2$ exist, give the relationship among the quantum operations $\Phi, \phi_{1}$ and $\phi_2$.
