mp.mathematical physics - The Quantum Operations On The Bipartite Systems

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
$mathcal H=mathcal H_1otimesmathcal H_2$. Density operators
$Winmathcal D(mathcal H)$, and effects
$Finmathcal E(mathcal H)$. Similarly, there are corresponding
symbols $W_iinmathcal D(mathcal H_i), F_iinmathcal E(mathcal H_i)$ 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:

$$begin{diagram} node{mathcal D(mathcal H_1)} arrow[4]{e,t}{phi_1}node[4]{mathcal D(mathcal H_1)}\ node{}\ node{mathcal D(mathcal H_1otimesmathcal H_2)} arrow[2]{n,l}{Tr_2} arrow[4]{e,t}{Phi} arrow[2]{s,l}{Tr_1} node[4]{mathcal D(mathcal H_1otimesmathcal H_2)} arrow[2]{s,r}{Tr_1} arrow[2]{n,r}{Tr_2} \ node{}\ node{mathcal D(mathcal H_2)} arrow[4]{e,b}{phi_2} node[4]{mathcal D(mathcal H_2)} end{diagram}$$
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
$phi_2$.

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