How is measurement on system in a Hilbert space seen? I am a bit confused about different kinds of measurements on a system in state $W$  where $W$ is the density operator in Hilbert space $H$. A general measurement can be given by POVM's, let $E_1,E_2....E_n$ be the POVM's then I have the following cases


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*Each $E_i$ is a projection operator and all projection operators are pairwise orthogonal. I am familiar with this case, we can assign an outcome with each projection operator as after measurement (in case $W$ is a pure state) the state vector is eigen state of exactly one projection operator (as they are pairwise orthogonal) and thus we have a unique outcome.

*Each $E_i$ is a projection operator and all projection operators are not pairwise orthogonal. In this case say if the initial state $W$ is pure and is simultaneous eigen vector of $E_i$ and $E_j$ say after measurement state remains same, the outcome corresponding to which projection operator has occurred ( corresponding to both )?

*In case $E_i$ are not projection operators. I don't know how measurement is perceived in this case.

 A: Looking at simple cases might help. You can have a vector $\vert\Psi\rangle$ in a Hilbert Space and represent it as a pure state such as $\vert\Psi\rangle\langle\Psi\vert$ and you could do the same for the vector $\vert\Phi\rangle$ in a Hilbert Space and represent it as a pure state such as $\vert\Phi\rangle\langle\Phi\vert.$
You could also imagine that instead of preparing a pure state like $\vert\Psi\rangle\langle\Psi\vert$ or a pure state like $\vert\Phi\rangle\langle\Phi\vert$ you play a game of pretend where you mathematically pretend that you had a probability of preparing $\vert\Psi\rangle\langle\Psi\vert$ or $\vert\Phi\rangle\langle\Phi\vert$ and consider the mixed state $\cos^2\theta\vert\Psi\rangle\langle\Psi\vert+\sin^2\theta\vert\Phi\rangle\langle\Phi\vert.$ Or alternatively get a mixed state by summing over things that are actually there by pretending that they are not there.
OK, so you really could prepare pure states, and you can pretend to have mixed states. So pretending is an obvious generalization since you can clearly pretend to do things you can really do and also pretend to do other things.
OK, so you can do measurements, in reality they are normal interactions with actual devices, amplifiers, and environments. Governed by actual Hamiltonians. Each of which ends up being an entangled vector in the Hilbert space that is a sum of products of eigenstates of the subsystem, the device state and the environment state. You can make a mixed state out of that result by ignoring the environment and the device too. So ignoring all those things that really go on you can pretend a measurement puts the state into a mixed state that is a sum of pure states associated with eigenstates of the subsystem.
One way to represent that is with Projection Valued Measures. But remember how we ignored parts of the actual system so that our game of pretend applied to the subsystem looked like the whole story? Well, what if you allowed a Projection Valued Measure on the actual system (or just something larger than the subsystem in question) then this can allow operations on the subsystem beyond what you can make out of PVMs on the subsystem itself. Just like you can pretend to have a mixed state for the subsystem by ignoring parts of the actual system that are outside the subsystem.
In a sense, Positive Operator Valued Measures (POVMs) are to PVMs as mixed states are to pure states. Both are essential to reproduce the possible statistics of a subsystem if you want to falsely pretend it is the whole system. You have to make changes if you want to play pretend with the universe and not get caught making incorrect statistical predictions. Sure, you could always study the actual system and use a PVM (just like you could study the actual pure state instead of ignoring parts of the actual Hilbert Space vector and then making a mixed state with the same statistics), and then you say you understand. But if you want to ignore the actual system you need a generalization, the POVM.
There is no simple story about how a measurement works when you ignore real things that actually go on for instance if you had the vector in your Hilbert Space, $\vert\Psi\rangle\otimes\vert A\rangle + \vert\Phi\rangle\otimes\vert B\rangle$ and you ignore that pesky A and B that are really there then instead of the pure state corresponding to what you actually had in reality you have the mixed  state $a\vert\Psi\rangle\langle\Psi\vert+b\vert\Phi\rangle\langle\Phi\vert$ where the real constants $a$ and $b$ are chosen so that you get the right statistics when you ignored A and B.
But this affects measurements and even something as simple as a projection operator too. For instance if you project $\Psi$ and $\Phi$ with a self-adjoint projection operator then you can't add the results of projecting those two vectors since there wasn't really two vectors, there was one $\vert\Psi\rangle\otimes\vert A\rangle + \vert\Phi\rangle\otimes\vert B\rangle.$ And so instead you have to project the mixed state $a\vert\Psi\rangle\langle\Psi\vert+b\vert\Phi\rangle\langle\Phi\vert$ which just gives another mixed state.
You can imagine a projection on the actual larger vector in the larger Hilbert space or instead imagine composing the density matrix operator with a projection on the smaller Hilbert space. Similarly you could have a PVM on the larger Hilbert Space or have a POVM on the smaller Hilbert Space. The PVM has a dimension that is the dimension of the space, a POVM can have more factors because it is like a PVM in the larger Hilbert space with a larger dimension. The PVM requires that they be self adjoint projections, the POVM only requires they be Non-negative Bounded Operators. But what does the action of a projection look like to a subspace?
Just imagine that you measured a thing in a larger Hilbert Space and then only looked at the mixed state version of it where pure states of different vectors in the larger space are represented by mixed states, and you should be fine and be able to keep on pretending that things like devices and environments don't exist and get the same statistics as if you didn't play pretend.
Well, the same statistics as assuming those measurements didn't take time to perform like they actually do. You'll still be wrong about that, there is no getting around that if you use projections instead of modelling the actual interactions your subsystem had with actual devices. But the rest can be mitigated.
