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deneve

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as far as I can gather Von Neumann's measurements of the first kind can be modeled as below:

for system being measured S and apparatus A with states |S1>, |S2> and |A1>, |A2>,

|A0> (ready state), respectively, we have, assuming the measurement process is unitary

U|S1>|A0>----> |S1>|A1>

U|S2>|A0>----> |S2>|A2>

so assuming linearity, for a superposition:

U[|S1>|A0>+|S2>|A0>]----> |S1>|A1>+|S2>|A2> --------(1)

i.e. the apparatus has become entangled with the states of S. Now forgetting about collapse of the wave function and all of that, what I want to know is, what justification do we have that this ansatz can be applied to measurements of the second kind? I mean that equation (1) is accepted for measurements like spin in the z direction by a stern gerlach device because von neumanns approach correlates spin with position via an appropriate interaction hamiltonian and the spin state is not demolished by the measurement. If however the apparatus is a geiger counter or flourescent screen then the state of the electron spin after detection, if remeasured, would be non existent because it's now been absorbed into the measuring instrument. It's not like the idea that preparing an observable in an eigenstate guarantees that the corresponding eigenvalue for that eigenstate will recur if re-measured. In textbooks, equation (1) above is assumed to work for both standard measurements (of the first kind) and measurements which destroy the state of S (second kind - photomultipliers etc.).

For example suppose that S is a photo multiplier and I, as observer X watch the experiment, then the above argument extends to

U' [|S1>|A0>|X0+|S2>|A0>|X0]----> |S1>|A1>|X1>+|S2>|A2>|X2> --------(2)

How do we know that this approach works when the measurement devices like photomultipliers don't leave the state of the quantum system in its eigenstate. For example if the first part of (2) obtains then this says the electron has spin up,say and was detected and was observed, but the spin of that electron is now completely lost because the electron has been swallowed up in the apparatus (if it's a geiger counter for example).

I guess what I'm saying is, does anyone know what justification we have that we can treat measurement processes(and apparatus) of different kinds, including those that demolish the measured state by the method of unitary evolution described by equation (1) ?