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 $\phi_2$.

without using the preview function. Please have a look at the guidelines as to what MO does and doesn't support, and how to get round them. $\endgroup$ – Yemon Choi Feb 11 '10 at 10:17