Measurements of the second kind and unitary treatment

[deneve]

I'm struggling to figure out whether measurements of the second kind can be treated using the unitary ansatz.
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) ?

 

[Demystifier]

I'm not sure I understand what exactly is bothering you, but let me try to answer it.
A demolishion of a state (e.g., a destruction of photon) is also a unitary process. See e.g. Sec. 2.1 of
http://xxx.lanl.gov/pdf/1112.2034.pdf

 

[deneve]

Hi Demystifier


Many thanks for your reply. The paper you quote I think will help - in more ways than one. Thank you so much for bringing it to my attention.

I hope I am understanding this correctly though regarding the status of the k prime in the paper you quote.

In the paper it says (their eq 1)

U|K>|phi_0> = |K'>|phi_k>
comparing this with my query gives

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

Ok so if the apparatus (A) is a photomultiplier, should I read this as spin was initially up (say) and measuring apparatus now records this, but electron is now in some other (unknown) spin state |S1'> thus the whole process is unitary?

I hope I've got this right and would be really grateful if you could confirm that I have the right end of the stick here.

Kind regards.

 

[Demystifier]

Deneve, I confirm that your post above is right, except for one little correction. If the operator U is known, then |S1'> is known as well. The unitary processes in QM are deterministic and therefore predictable.