Does someone know of a clear (pedagogical) example where one can really see(with the math) where interaction and measurement are not synonymous in quantum mechanics?
I know that every measurement involves a certain interaction with the outside world (e.g. momentum gain from a photon), which results in the system collapsing into one of its eigenstates, meaning a pure state.
On the other hand, it is also known that not all interactions result in collapsing into eigenstates of the system, so they are in principle very different from what we call "measurement".
It would be definitely nice to also see a bit of the math behind, maybe for simplicity just restricting it to operator algebra and showing how measurement and interaction are defined, shedding a clear light on their difference. I must admit, from a purely physical point of view, I don't know their difference either.
Answer
This is not a settled question. Just as it is still debated whether or not there is wavefunction collapse, so is it debated what exactly we should understand by a measurement. In the following, we will go through the ideas behind the von Neumann measurement scheme, which is one way to try and talk about measurement in quantum mechanics.
An interaction happens every time when two quantum systems cannot be neatly separated anymore. Given two systems with Hilbert spaces of states $\mathcal{H}_1,\mathcal{H_2}$, they are interacting if entangled states between them are allowed, that is, if the physical Hilbert space of states of the whole system is not $\mathcal{H}_1 \times \mathcal{H}_2$, but $\mathcal{H}_1 \otimes \mathcal{H}_2$. A measurement is a special kind of interaction.
Let $\mathcal{H}_q$ be the Hilbert space of the object of which we want to measure some property, and $\mathcal{H}_m$ the Hilbert space of the apparatus which we will use to perform the measurement. Note that the von Neumann scheme treats the apparatus as a quantum object, just like everything else. The distinguishing quality of a suitable measurement apparatus is that the unitary time evolution of the combined system (perhaps + the environment, too) acts as
$$ \lvert \psi \rangle \otimes \lvert \phi \rangle \overset{\mathcal{U(\tau)}}{\mapsto} \sum_n c_n \lvert \psi_n \rangle \otimes \lvert \phi_n \rangle$$
where $\psi_n$ are some uniquely determined states of the measured system and $\phi_n$ are orthogonal states of the apparatus w.r.t. to the operator we want to measure that can be "read off macroscopically" (they are "pointer states"). Now, the apparatus is supposed to be "macroscopic", so we can determine the pointer state just by looking at it. So, each pointer state corresponds uniquely to a state $\psi_n$ of the system. This process is called measurement of the second kind, and of the first kind if the $\psi_n$ are eigenstates of an observable.
Note that the form the unitary evolution is supposed to take here is special. In general, it looks like $$\lvert \psi \rangle \otimes \lvert \phi \rangle \mapsto \sum_{i,j} c_{ij} \lvert \psi_i \rangle \otimes \lvert \phi_j \rangle$$
where we would not have such a correspondence between pointer states and object states. The $\lvert \psi_n \rangle \otimes \lvert \phi_n \rangle$ occurring in the time evolution of the measuring/decohering system are not a basis of the (tensor) space of states, while, in general, the time evolution should entail all basis vectors $\lvert \psi_i \rangle \otimes \lvert \phi_j \rangle$, of which the basis for the "special" time evolution we suppose for a measurement apparatus is just a subset (namely that where $i = j$). One can devise many arguments why it should take such a form when going to "classical sizes", this is the study of decoherence and einselection.
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