The question
Say I have a completely-positive (CP) map $\mathcal{A}_{ij}$ defined in terms of two projectors $\Pi_i = |i\rangle \langle i |$ and $\Pi_j = |j\rangle \langle j |$ that acts on a density operator as:
$$\mathcal{A}_{ij}[\rho] = \text{Tr}[\Pi_i \rho] \Pi_j $$
where $\{|i\rangle \}_{i=1}^{d^2}$ form a basis on the Hilbert space. This map is not trace-preserving-- projecting the state onto $i$ and then preparing the pure state $j$.
How would I write the joint action of $(\mathcal{A}_{ij} \otimes \mathcal{I})$ on an arbitrary bipartite density operator $\rho_{AB}$, in terms of traces and preparations like above?
My thinking
Since the identity map $\mathcal{I}$ acting on qubit $B$ is "do nothing to it" then the output state should leave it untouched, while the CP map acting on qubit $A$ should do the projection and measurement as usual. So as a guess, I'd try:
$$(\mathcal{A}_{ij} \otimes \mathcal{I})[\rho_{AB}] = \text{Tr}[(\Pi_i \otimes \mathbb{1}_B)\rho_{AB}] \, (\Pi_j \otimes \text{Tr}_{A}\{\rho_{AB}\})$$
If our state is separable $\rho_{AB} = \rho_A \otimes \rho_B$, then the trace corresponding to the projective measurement evaluates to:
$$\text{Tr}[(\Pi_i \otimes \mathbb{1}_B)(\rho_A \otimes \rho_B)] = \text{Tr}[(\Pi_i \rho_A) \otimes (\mathbb{1}_B \rho_B)]$$ (via mixed product property)
$$\text{Tr}[(\Pi_i \rho_A) \otimes (\mathbb{1}_B \rho_B)] = \text{Tr}[\Pi_i \rho_A] \text{Tr}[\mathbb{1}_B \rho_B] = \text{Tr}[\Pi_i \rho_A]$$
where the last term is just a scalar $c_i$. So we have the joint action as:
$$(\mathcal{A}_{ij} \otimes \mathcal{I})[\rho_{AB}] = c_i \, (\Pi_j \otimes \text{Tr}_A \{\rho_{AB}\})$$
and since $\rho_{AB}$ is separable, then the partial trace will trace out system $A$ with no problems leaving
$$(\mathcal{A}_{ij} \otimes \mathcal{I})[\rho_{AB}] = c_i \, (\Pi_j \otimes \rho_B)$$
as desired-- the joint action has successfully projected $A$ onto $i$, prepared the subsystem in state $j$, and left system $B$ untouched during the whole process.
The issue with that solution
Writing the joint action in this way treated the initial $\rho_{AB}$ as separable, but I want to really be able to write an equation in this form that can handle entangled states too.
This changes things since, the result of the projection on $A$ would also affect $B$, meaning that even though we're not actively doing anything to it, the output state of the CP map should still have $B$ dependent on what we did to $A$.
So how could I write this in a form similar to above? I've come up with one possible equation that accommodates for separable and entangled states but it looks horrendous; littered with full and partial traces. Surely there's more elegant ways of doing this. Even having that partial trace $\text{Tr}_A\{\rho_{AB} \}$ in the output state of the equation written in this question makes me uncomfortable.
I imagine writing the action of a map that acts on a subsystem is a very commonly encountered thing, how would one go about it for this case?
