A Quantum Mechanical model of measurement

[peteratcam]

Following a short discussion in another thread, I promised to start a new thread for this.
I asked myself the question "how would I describe the process of measurement in a fully quantum mechanical way?" and came up with this.

1) A model for the general measuring device
The aim is to have a QM system which describes a measuring device, with a finite dimensional space of states so that the whole thing can be written down with finite matrices.

I imagine a dial, with a pointer (see picture coming soon).
Any position of the pointer corresponds to a different quantum mechanical state for the pointer (much like a 1-d particle on a ring).
To make things easier, discretise the possible positions of the pointer, so that we have N possible pointer states, all equally spaced.
Assume that the Hamiltonian for the dial is H = 0 so there is no dynamics at all.
So the measuring apparatus is an N-state quantum system.

2) Operators for the measuring device.
There is only one operator we need for the measuring device: the operator that moves the pointer. By analogy with the momentum operator, we want the generator of pointer rotations, which I will call R.
Define it so that the unitary matrix exp(-iR) advances the pointer clockwise by pi radians.

So far so good, we have a device which we can start in the state pointing to 0, and we have an operator which rotates the pointer round by whatever angle we want.
We can calibrate the device by letting the maximum reading be 'm', corresponding to pi radians.

3) Performing a measurement of the number '4'.
Suppose we want to measure a number, called y, using our device, the way to do it is:
In units where hbar = 1, change the hamiltonian to H = R y/m for a duration of 1 unit of time, then change back to H = 0.
The time evolution operator will be exp(-i(y/m)R), which will swing the pointer round by (y/m) of a half circle.
Eg, we might choose m = 20, with N = 40, so the states on the dial run from -19,-18,...0,...19,20.
Acting on the initial state with exp(-i(4/20)R) will move the pointer to point at '4'. ie, we have measured the number 4 using our measuring apparatus.

[EDIT at 9am GMT: Despite numerous replies already, I will continue from here, but address concerns in a reply below]
4) Measuring the properties of another quantum system
So far we have shown how to measure a number on our dial. Now we want to measure the properties of another system.
Generalising what we did to measure a number, the following procedure will 'measure' the value of observable $$\hat O$$. (I've put the hat on it so it doesn't look like zero).
The full hamiltonian, which acts on the product space $${\mathcal H}_M\otimes{\mathcal H}_S$$ will
initially be $$H = \mathbf 1 \otimes H_S$$ since our measuring device has H=0.
Now turn on the measurement hamiltonian. Again, 'm' sets the maximum reading on the dial.
$$H_{meas} = \frac{1}{m\tau}R\otimes \hat O$$
After time tau, set the measurement hamiltonian back to H = 0. Ideally, our clever experimentalist is able to make 'tau' very short, so we have an instantaneous measurement.

4a) Explicit example, measuring the energy of an harmonic oscillator.
Suppose the maximum reading on my dial is 3, and I have N = 12. Label the states of the apparatus by $$|-2.5\rangle,|-2.0\rangle,|-1.5\rangle,\ldots|0.0\rangle,\ldots,|2.5\rangle,|3.0\rangle$$.
Also, suppose the system we are interested in is a Harmonic oscillator, currently in the state
$$\frac{1}{2}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle+\frac{1}{2}|2\rangle$$.

Therefore the inital state of the composite system is the product state:
$$|0.0\rangle \otimes [\frac{1}{2}|0\rangle+\frac{1}{\sqrt{2}}|1\rangle+\frac{1}{2}|2\rangle]$$
Now the experimentalist comes along, and attaches his measuring device to the harmonic oscillator in such a way as to measure the energy. The effect of this is that the measurement hamiltonian becomes
$$H_{meas} = \frac{1}{m\tau}R\otimes H_s$$.
After time tau (which is much shorter than the period of the oscillator), he disconnects the measuring device.
The final state of the composite system is
$$\frac{1}{2}|0.5\rangle \otimes |0\rangle+\frac{1}{\sqrt{2}}|1.5\rangle \otimes |1\rangle+\frac{1}{2}|2.5\rangle \otimes |2\rangle$$
The apparatus state is now entangled with the measured state.
*So the readings on the apparatus are the eigenvalues of the observable.
*Moreover, the state of the system is the eigenvector for the measured value.
Crucially, the apparatus states are orthogonal, so there is no interference possible between the two options for the system.
Applying the probability interpretation to the entangled system is consistent with applying it to just the system we measured originally, except the measuring process and apparatus are described in a QM way too now.

I believe the points with (*) were the points in the other thread under point 2, which I wanted to address.

5) Generalisations.
This is completely general. My measuring apparatus will measure any observable you like and in performing the measurement, it is clear why only eigenvalues are allowed, and why eigenvectors is what you get afterwards.

[edit: more may need to be edited if it is unclear, but I thought it better to have all of it in one place if there is going to be a discussion]

 

[humanino]

There is no big breakthrough in this attempt, it is just decoherence

The discontinuous "wave function collapse" postulated in the Copenhagen interpretation to enable the theory to be related to the results of laboratory measurements now can be understood as an aspect of the normal dynamics of quantum mechanics via the decoherence process.

 

[SpectraCat]

There is no big breakthrough in this attempt, it is just decoherence

I don't necessarily think so ... this was spawned from a discussion in https://www.physicsforums.com/showthread.php?t=385341" where someone brought up the point (often made) that decoherence only solves part of the measurement problem, i.e. how to provide a description of the apparently irreversible resolution of a quantum state into the eigenstate of an operator for which a measurement was being performed, that is consistent with the inherently time-reversible laws that govern the evolution of quantum systems. Decoherence does indeed provide such an explanation, however, as far as I know it does not explain how that resultant eigenstate is registered by the measurement device.

This thread was started by peteratcam to give his perspective on that point, and I for one am eager to see it.

 

[conway]

I am pleased to agree with Spectracat (for a change). Could we not refrain from the scoffing and give the man a chance to make his point?

 

[humanino]

I do not agree that Bell's FAPP purpose argument holds. There is no established general mathematical theorem to prove either side of the argument. I will let Spectracat answer, as indeed it is not a short and straightforward argument, but it is discussed at length in the decoherence literature, for instance Omnes "The interpretation of quantum mechanics" (Princeton 1994) 7.7 where one can find the core of the discussion, preceded and followed by many explicit calculations and numerical estimates. Decoherence has also been measured in the lab, and agree with those estimates.

Bell's arguments holds at a quite general level, if one is willing to accept as a postulate that every observable can be measured in principle. Since it is quite an interesting discussion, but too long for me to go into right now, I will just copy below some parts of Omnes' conclusion in chapter 12 section 7 to the question "can one circumvent decoherence ?"

Not all aspects of decoherence are yet completely understood [...] Among these questions is the status of an axiom put forward long ago by von Neumann. It stated that every observable can be measured [...] This is of course not known to be a necessary axiom [...] Even if the reduced density operator has become diagonal, the full density operator still represents a quantum superposition if there was one initially (this being a consequence of the linearity of the Shrodinger equation). One can therefore always conceive of an observable that would show the existence of this superposition if it could be measured. The possibility of this measurement would then imply that decoherence is only a practical answer to the basic problem of quantum superpositions, not a fundamental one.

[...]

The argument was based on the idea that the measurement can only be useful if the probability for getting a significant result is larger than the probability that a large quantum fluctuation will produce the same final phenomenon [...] It cannot yet be considered as an indisputable proof, but its result is quite suggestive. [...] such a measurement is seen to be impossible and [...] its existence incompatible with general relativity (it would collapse immediately in a black hole).

[...]

• Not all observables can be measured, but only a few of them. This means that the majority of the properties one could think of, even in the case of a relatively small system, can never be confirmed or disproved by any experiment. They belong to the domain of conversational physics, not of real physics.
• Decoherence cannot be circumvented once it has occurred, as long as the environment is large enough. It therefore affords a sensible answer to the problem of macroscopic superpositions.
• Not all conceivable states can be prepared. In particular, one cannot prepare the time-reversal of the actual stage of a macroscopic assembly of particles when some evolution has already occurred. This remark opens an interesting connection between decoherence and the foundations of irreversible thermodynamics.
• The direction of time in logic, in decoherence and in thermodynamics should be the same

 

[Fredrik]

Suppose we want to measure a number, called y, using our device, the way to do it is:
In units where hbar = 1, change the hamiltonian to H = R y/m for a duration of 1 unit of time, then change back to H = 0.

I thought you were going to end up with something very similar to von Neumann measurement theory, but the quote is a major difference that ensures that your model isn't compatible with QM. The Hamiltonian isn't changed at random into one of several possibilities by the interaction between the measuring device and the system.

 

[meopemuk]

I wrote some (possibly relevant) thoughts about the QM measurement problem in https://www.physicsforums.com/showpost.php?p=2618641&postcount=4

Eugene.

 

[peteratcam]

(I have now finished the original post)

There is no big breakthrough in this attempt, it is just decoherence

I'm not claiming a big breakthrough (perhaps a pedagogical one at most). But I've read some of the introductory reviews on decoherence and I didn't feel that it addressed what I've tryed to address by the toy model above. I feel that the decoherence program is worried more about irreversibility, but *all* I wanted to demonstrate to myself with the model above is that the mysterious postulates that "we always measure an eigenvalue of an operator, and in doing so, collapse the wavefunction to the associated eigenvector" can be understood through quantum dynamics of composite systems.

 

[peteratcam]

Fredrik's reply was written before I had finished the exposition in the original post, but I will reply to it.

I thought you were going to end up with something very similar to von Neumann measurement theory, but the quote is a major difference that ensures that your model isn't compatible with QM. The Hamiltonian isn't changed at random into one of several possibilities by the interaction between the measuring device and the system.

My model is compatible with QM. Unitary evolution of states according to a hamiltonian - what more do you want!
I don't claim the Hamiltonian is changed *at random*. Rather, a clever experimentalist arranges his levers, electrical wires, and filters, such that the hamiltonian which describes the interaction between the system and the apparatus is one which will measure the desired outcome. It is what happens all the time in labs.

 

[Fredrik]

I don't claim the Hamiltonian is changed *at random*. Rather, a clever experimentalist arranges his levers, electrical wires, and filters, such that the hamiltonian which describes the interaction between the system and the apparatus is one which will measure the desired outcome. It is what happens all the time in labs.

It sure sounded like you were describing a random change of the Hamiltonian. But now it sounds even worse, like an experimentalist can choose what outcome to get.

 

[peteratcam]

It sure sounded like you were describing a random change of the Hamiltonian. But now it sounds even worse, like an experimentalist can choose what outcome to get.

My word 'outcome' was a terrible choice. I mean to say that in an ideal world, an experimentalist can choose what to measure (be it energy, position, etc), but obviously he can't choose the result of the measurement. Choosing what property to measure equates to choosing a hamiltonian for the interaction between system and apparatus.

 

[Dmitry67]

So why outcome is different in identical setups, when people measure the same observables?

 

[peteratcam]

So why outcome is different in identical setups, when people measure the same observables?

Because that is what QM predicts, I don't really know how else to answer that. My model was to make an explicit example of the quantum dynamics of a measurement process, which I believe it does - unless there are any objections...

 

[Dmitry67]

No, QM itself does not predict it, because QM math and Decoherence are deterministic.

Now, when collapse int are ruled out, we have non-collapse interpetations (+macroscopic realism aka shut up and calculate).

Decoherence removes non-diagonal elements, but it does not tell us what outcome is "real". All non-collapse interpretations must provide a mechanism to resolve that issue. Using hidden variables (BM) or by claiming that all outcomes are real (MWI).

You can not, in principle, derive non-deterministic view of the world from deterministic QM without some sorts of additional assumtions (well, in MWI these assumptions are null, but anyway...)

 

[Dmitry67]

I tried to pinpoint the problem:

Applying the probability interpretation to the entangled system is consistent with applying it to just the system we measured originally, except the measuring process and apparatus are described in a QM way too now.

Saying "aha, I have calculated the result, now let’s apply the probability interpretation" is an attempt to resurrect the Mr. Wavefunction Collapse from it’s grave. You can't do it.
Born rule is either predicted (BM) or denied (but accepted FAPP) in MWI. Both theories are deterministic.

 

[Fredrik]

Peteratcam, your approach is just von Neumann measurement theory, as I initially suspected. The time evolution that you describe is sometimes called a "premeasurement". It's just an interaction that creates a correlation between states of the measuring device and states of the system. von Neumann didn't know this of course, but the actual measurement is a second interaction between the measuring device and its environment that makes the the system FAPP indistinguishable from a mixed state (i.e. a state that has "collapsed" into one of the alternatives).

I think you would find http://www.hep.princeton.edu/~mcdonald/examples/QM/zurek_prd_24_1516_81.pdf interesting, at least the first few pages. I'm too tired right now to try to explain the consequences of what he's saying there, so you'll have to read it for yourself (if you're interested).

 

[meopemuk]

So why outcome is different in identical setups, when people measure the same observables?

This is the biggest mystery of nature. Nobody knows the reason for this randomness.

QM itself does not predict it, because QM math and Decoherence are deterministic.

QM formalism does not predict/explain the random behavior of nature. This behavior is simply accepted as a given physical fact, and the entire formalism is built upon this assumption. QM math is fully deterministic, but this math is directed at calculation of probabilities. QM cannot say what exactly will be measured. It can only predict probabilities of different possible outcomes. The random behavior of nature and statistical QM formalism are in perfect agreement with each other.

Eugene.

 

[Dmitry67]

meopemuk, I agree with you: different non-collapse Int explain randomness differently. But returning to the beginning of the discussion, unfortunately, "point 2" (we started from it) is not explained by OP.

 

[Fredrik]

I assume you mean the "point 2" that's mentioned here:

2) How does the above interaction place a detection device into a "state" (i.e. definite outcome) where its value reflects the eigenvalue of the eigenstate into which the quantum state in part 1 has been resolved?

I've been saying this a lot, but I don't think I've been nagging too much about it yet: The above is only a problem and a mystery if we assume that QM is a description of reality (even at times between state preparation and measurement) rather than just a set of rules that tells us how to calculate probabilities of possible results of experiments. There is no measurement problem in the ensemble interpretation.

(I think there's a version of the MWI that has a good chance of being free from measurement problems as well, but I don't understand it well enough to say for sure that it is).

 

[meopemuk]

I've been saying this a lot, but I don't think I've been nagging too much about it yet: The above is only a problem and a mystery if we assume that QM is a description of reality (even at times between state preparation and measurement) rather than just a set of rules that tells us how to calculate probabilities of possible results of experiments. There is no measurement problem in the ensemble interpretation.

This is exactly right. QM talks only about results (probabilities) of measurements. This is not a weakness of QM, but its strength. Because according to the scientific method we are only allowed to talk about things that can be directly verified in experiments. All discussions about "reality", "hidden mechanisms", etc. belong to metaphysics/philosophy/religion.

Eugene.

 

[Dmitry67]

meopemuk, I respect your view - positivism leading to macroscopic realism (BTW macroscopic realism does not have measurement problem) but it is not the only view. yes, discussions like "are virtual particles real?" does not make any sense, but there is another side: Max Tegmark MUH, mathematical = physical = real. I don't think MUH is classified as purely 'metaphysics/philosophy/religion'

 

[SpectraCat]

I assume you mean the "point 2" that's mentioned here:
I've been saying this a lot, but I don't think I've been nagging too much about it yet: The above is only a problem and a mystery if we assume that QM is a description of reality (even at times between state preparation and measurement) rather than just a set of rules that tells us how to calculate probabilities of possible results of experiments. There is no measurement problem in the ensemble interpretation.

(I think there's a version of the MWI that has a good chance of being free from measurement problems as well, but I don't understand it well enough to say for sure that it is).

This is exactly right. QM talks only about results (probabilities) of measurements. This is not a weakness of QM, but its strength. Because according to the scientific method we are only allowed to talk about things that can be directly verified in experiments. All discussions about "reality", "hidden mechanisms", etc. belong to metaphysics/philosophy/religion.

Eugene.

I guess I see the issue differently ... if the postulates of QM are correct (and they certainly seem to be), then there must be a way to understand this phenomenon in the context of laboratory experiments. As an experimentalist, I am particularly interested in such explanations, because they help to evolve my understanding of the entire system.

That said, I agree that the measurement "problem" is mis-named ... I think it should be called "that thing about actual physical measurements where we don't really understand exactly how it happens but seems to be correct anyway" ... I guess that is too much of a mouthful though

 

[peteratcam]

Peteratcam, your approach is just von Neumann measurement theory, as I initially suspected. The time evolution that you describe is sometimes called a "premeasurement". It's just an interaction that creates a correlation between states of the measuring device and states of the system. von Neumann didn't know this of course, but the actual measurement is a second interaction between the measuring device and its environment that makes the the system FAPP indistinguishable from a mixed state (i.e. a state that has "collapsed" into one of the alternatives).

I think you would find http://www.hep.princeton.edu/~mcdonald/examples/QM/zurek_prd_24_1516_81.pdf interesting, at least the first few pages. I'm too tired right now to try to explain the consequences of what he's saying there, so you'll have to read it for yourself (if you're interested).

Ok, fine I can believe that. Let's take the first equation of that paper you linked to, which basically says under the unitary evolution of 'premeasurement':
$$|A_0\rangle\otimes|S_O\rangle \rightarrow \sum_s |A_s\rangle\otimes|s\rangle$$

I've seen stuff like that before, and you're right, it is the same idea I have in the original post. But I look at that and wonder how exactly I should describe the $$\rightarrow$$. And where does the fact that the result of the measurement are the eigenvalues come into it? So my toy model of an apparatus is precisely to understand how the "$$\rightarrow$$" works (so entirely in the realm of 'premeasurement' if you like).

To plug it into the bigger picture, I suppose the pointer states in my model should be thought of as the preferred apparatus states referred to by Zurek. I think the dial is a better system to play with than the bit-by-bit measurements in that paper, which are in the same spirit.

So now to the infamous point (2), which Dmitry67 says I haven't addressed. I think I've covered the bit in bold.
(2)How does the above interaction place a detection device into a "state" (i.e. definite outcome) where its value reflects the eigenvalue of the eigenstate into which the quantum state in part 1 has been resolved.

I don't own the book by von Neumann, so I can't compare what I've said to what he says there.

 

[meopemuk]

Max Tegmark MUH, mathematical = physical = real. I don't think MUH is classified as purely 'metaphysics/philosophy/religion'

I respectfully disagree.

 

[Fredrik]

But I look at that and wonder how exactly I should describe the $$\rightarrow$$.

The → is just the time evolution during the measurement, so exp(-iHt) acting on the state on the left will produce the state on the right, but you probably understood that. If you're concerned about what exactly the H looks like, I'm sure there are lots of toy models in the literature, but I haven't studied any of them (except for a quick look at section II in that paper...I didn't try to understand all the details).

And where does the fact that the result of the measurement are the eigenvalues come into it?

I'm not sure if you're asking where Zurek uses that fact, or why it is a fact at all. I don't think there's an explanation for the latter, except possibly some sort of anthropic argument in a many-worlds interpretation, and this article doesn't cover that. I'll return to the other interpretation of your question below (when I mention the reduced density matrix).

The article's goal is only to explain which basis appears in the final mixed state (not to explain why the final state is a mix of eigenstates), and I think it succeeds with that. Note that (1.1) and (1.3) are the same state expressed in different bases, and that it's not at all clear at this point which one of them corresponds to possible measurement results. Zurek's conclusion is (if I understand it correctly) that interactions between the measuring device and its environment will always put the combined system (measured system + measuring device) into a mixture of states of the form $|A_s\rangle\otimes|s\rangle$ such that the $|A_s\rangle$ are eigenstates of an operator that commutes with the Hamiltonian that describes the interaction between the system and the measuring device.

(I need to read that article more carefully. This stuff is pretty interesting).

So now to the infamous point (2), which Dmitry67 says I haven't addressed. I think I've covered the bit in bold.
(2)How does the above interaction place a detection device into a "state" (i.e. definite outcome) where its value reflects the eigenvalue of the eigenstate into which the quantum state in part 1 has been resolved.

Have you really? The question appears to be this: "How does the measurement process, which is a physical interaction between the system, the measuring device and its environment, put the system into an eigenstate of an observable determined by the measuring device?" I don't see what the answer is. I think that arguments like the one Zurek is making in that article only leads to the conclusion that there's a specific relationship between the Hamiltonian and the reduced density matrix for the system. In my opinion, this does not really answer the question, because the only way to justify the interpretation of the reduced density matrix as "either that state or that state, with these probabilities" is to assume the things we'd like to prove.

I don't own the book by von Neumann, so I can't compare what I've said to what he says there.

I haven't read von Neumann either. I have just encountered similar mathematical expressions in a bunch of different places, and they often mention von Neumann as the source. I think chapter 9 of Ballentine is a good place to get introduced to this stuff.

 

[Demystifier]

I think chapter 9 of Ballentine is a good place to get introduced to this stuff.

It's good, but for some reason (unclear to me) Ballentine never mentions decoherence. For example, after Eq. (9.13) he says:
"It is clear that the nondiagonal terms in (9.13) cannot vanish so as to reduce the state to the form of (9.12) ..."
At this point nothing would be more natural than to mention that decoherence provides a FAPP mechanism to arrive at (9.12). Yet, for some reason, Ballentine does not say anything of that sort. Do you have any idea what his reasons might be?

 

[peteratcam]

If you're concerned about what exactly the H looks like, I'm sure there are lots of toy models in the literature, but I haven't studied any of them (except for a quick look at section II in that paper...I didn't try to understand all the details).

Yes, I guess that's exactly what I'm concerned about. So I found
A. Peres, Phys Rev D. 39 3943
which at least describes the model apparatus a bit more fully. However, it is not a finite-state system, so isn't as easy to play around with it on Matlab or Octave.

I think chapter 9 of Ballentine is a good place to get introduced to this stuff

I'm glad you mentioned that. This from Ch 9 of Ballentine:
"We then introduce an interaction that produces a unique correspondence between the value of the dynamical variable and the indicator. The properties of the interaction are specified implicitly by defining the effect of the time development operator U, and since only the properties of U enter the analysis, there is no point in discussing the interaction in any more detail."
So you can see why I wanted to find out what it should look like. Admittedly, Problem 9.4 in Ballentine describes basically the same model for the apparatus as in the PRD by Peres.

So is what I've described useful at all? Since the resolution of my apparatus is arbitrary (since N can be chosen at will in the model), perhaps it provides a useful way of looking at information flow into an N-state system - I don't know.
For those who are interested, I've attached a Matlab program which implements it.