What's the outcome state if I make a measurement with a projector orthogonal to that of the initial state? Let my initial state be $\rho = |i \rangle\langle i| = \Pi_i$ for some orthonormal basis $|i\rangle$.
If I make a projective measurement with $\Pi_j$, such that $\Pi_i \Pi_j = \delta_{ij} \Pi_i = \delta_{ij} \Pi_j$, what's the update rule for this quantum dynamics? If I set
$$\rho \mapsto \rho' = \frac{\Pi_j \Pi_i \Pi_j}{\operatorname{tr}\Pi_j \Pi_i}$$
I clearly get into trouble when $i$ is strictly different from $j$, $i \neq j$. What am I missing? The operator $0$ is not a state, but I couldn't justify (and there should be notes pointing this out, if this is the case) either $\Pi_i$ or $\Pi_j$ being the final state.
An argument that would solve this problem to me is to consider a measurement of an observable, in such away that these projectors never come in isolation, since the observable's spectral decomposition involves a complete set of them, and not all of them can be orthogonal to each other (and therefore to my initial $\rho$). So somehow the above expression should perhaps be
$$\rho \mapsto \rho' = \frac{1}{\operatorname{tr}(\sum_j\Pi_j \Pi_i)} \sum_j \Pi_j \Pi_i \Pi_j$$
in such a way that $\rho' = \sum_j \delta_{ij} \Pi_j = \sum_j \delta_{ij} \Pi_i = \Pi_i$. How sensible is that? What is the true physical outcome?
 A: Mathematically, a projective measurement is a complete set of projectors $\{\Pi_j\}$, such that $\sum_j \Pi_j=1$. If your initial density matrix is $\rho = \Pi_i$, then your equations simply tell you that the probability of outcome $j$ is
$$p_j = {\rm Tr} [\rho \Pi_j] = \delta_{ij},$$
i.e. outcome $i$ occurs with probability one. The state after measurement given that outcome $i$ was obtained is then
$$\rho' =  \frac{\Pi_i \rho \Pi_i}{p_i} = \Pi_i.$$
More generally (i.e. if $\rho\neq \Pi_i$), you could obtain another final state $\Pi_{j\neq i}$ with probability $p_j$. For this particular initial state, however, only one outcome is possible.
The lesson is that "measuring projector $\Pi_j$ in isolation" is not an operation that makes sense. A measurement always includes a complete set of projectors (or positive operators, more generally), encoding the fact that at one of the possible measurement outcomes is always obtained. You don't get to choose this outcome, Nature decides! Otherwise measurements wouldn't give you any information.
