# 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 $\mathcal H=\mathcal H\_1\otimes\mathcal H\_2$. 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 $\phi_2$.

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I have edited the slab of LaTeX which you seemed to have pasted in 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. –  Yemon Choi Feb 11 '10 at 10:17
@Yemon: Your (heroic!) edit doesn't seem too successful to me. I am getting this text: Unknown environment "diagram". –  Harald Hanche-Olsen Feb 11 '10 at 13:58
@Harald: if you saw what happened before my edit, I think the present version is a considerable improvement:) I had insufficient zeal/charity for trying to work out what the diagram was supposed to depict and then recoding it –  Yemon Choi Feb 11 '10 at 19:56
@Yemon: Yes, I had looked at the edit history. That's why I said “heroic”. Not complaining, really. –  Harald Hanche-Olsen Feb 12 '10 at 12:34

If I understand the question right, essentially what is being asked is: for a general quantum operation on a system with two subsystems, can you learn what happens to subsystem 1 just by looking at the initial state of subsystem 1, and what happens to subsystem 2 just by looking at the initial state of subsystem 2. If this is what is being asked, the answer is No: consider the case where the two subsystems have the same dimension and the quantum operation $\Phi$ which swaps the subsystems.