Obtaining the reduced density matrices for both subsystems of a bipartite system

Question

If we have a single copy of a bipartite quantum system with density matrix $\rho$, is it possible to extract the reduced density matrices of the constituent subsystems separately, i.e. to achieve the following transformation:

$$\rho \longrightarrow \mathrm{Tr}_A \rho \otimes \mathrm{Tr}_B \rho $$

where $\mathrm{Tr}_A \rho$ and $\mathrm{Tr}_A \rho$ are the reduced density matrices of the constituent subsystems $A$ and $B$.

If we just want $\mathrm{Tr}_B \rho$ we can just "ignore" the other subsystem: say we perform a measurement subsystem $A$ and discard it without noting the outcome, to obtain $\mathrm{Tr}_B \rho$. But is it possible to get both $\mathrm{Tr}_A \rho$ and $\mathrm{Tr}_B \rho$ in one shot?

Answer 1

No, the desired transformation is not physical. Consider its action on a basis consisting of pure product states. It sends every such state to itself, so it would seem that it's the identity. Clearly, it's not the identity. The contradiction arises because the transformation isn't linear$^1$. Hence, it isn't physical.

That said, it is possible to realize $$ \rho\otimes\rho \longrightarrow \mathrm{Tr}_A \rho \otimes \mathrm{Tr}_B \rho $$ by performing suitable partial trace.

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^{$^1$ And thus it is not uniquely determined by its action on a basis.}