# A Evaluate this paper on the derivation of the Born rule

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1. May 24, 2017

### Prathyush

I have encountered this paper "Curie Wiess model of the quantum measurement process". https://arxiv.org/abs/cond-mat/0203460

Another work by the same authors is "Understanding quantum measurement from the solution of dynamical models" https://arxiv.org/abs/1107.2138

I am still evaluating the papers. I find the general lessons implied to be interesting and probably compelling.

In the context of the model studied is this paper accurate? What do you think about the overarching viewpoint presented by the authors?

2. May 24, 2017

### stevendaryl

Staff Emeritus
I think it's great that they made such a careful analysis of a measurement process. But it seems to me a little circular to use the ensemble interpretation of QM to derive the Born rule.

3. May 24, 2017

### mikeyork

The difficulty that people generally perceive in the Born rule arises because of the historical association of the abstract idea of probability with relative frequency. The reason for the association is obvious because classically relative frequency is the natural practical and historical way we have to measure probability. So they assume that "probability" must be non-negative and, in particular, equate it to the normalized relative frequency.

But a moment's thought should tell us that any mathematical encoding that tells us how to compute the relative frequency can serve as a theoretical probability.

When it comes to QM, the scalar product of a "final" (i.e. outcome) state with (i.e. conditional on) the initial state is a natural encoding of probability because it is largest (i.e. most likely) when the final state vector is closest (in the usual scalar product of vectors sense) to the initial state vector.

Of course there remains a little mystery as to why the squared modulus is the correct way to compute the relative frequency, although it does seem to be the most straightforward.

But my point is that that little mystery is the entire content of the Born rule, once we have a Hilbert space because the association of theoretical probability with the scalar product is built in to the association of states with Hilbert space vectors.

Last edited: May 24, 2017
4. May 25, 2017

### tom.stoer

Certainly not.

If I use a Hilbert space setting to analyze for example the multipole expansion in classical electrodynamics, the mathematics is the same as in quantum mechanics, but there is no probability at all. The scalar product has a different meaning, and this this meaning does not come with the mathematical framework. The meaning is due to our interpretation of the mathematical framework, and this interpretation is different in classical electrodynamics and in quantum mechanics (and it is different in the various different interpretations of quantum mechanics).

5. May 25, 2017

### martinbn

200 pages!

6. May 25, 2017

### mikeyork

I didn't say that the scalar product has that meaning in any Hilbert space. I said it comes from the association of physical states with state vectors.

Any function p serves as a probability encoding as long as there is some single-valued function f(p) that predicts the asymptotic relative frequency. In QM, the scalar product serves this purpose in a very natural way (with f(p) monotonic in |p|) because of the meaning we attach to state vectors. It is the same as saying that big changes are less likely than small changes -- a fairly obvious, perhaps even tautological, notion. The content of the Born rule lies with the simplicity of f(p) = |p|^2.

7. May 25, 2017

### Prathyush

Why do you think the reasoning may be circular? I am in stage where I am still evaluating the arguments presented in the paper, assuming they are correct, here is what I think about the general context. We understand exactly how our measurement apparatus was constructed, it is reasonable to suppose that we use a statistical ensemble describe it. At a microscopic level we understand how degrees of freedom of the apparatus interact with system. We want to understand the physics that describes the macroscopic changes that the apparatus sees when it interacts with the system. The paper appears to precisely do this, and arrives with Born's rule as a consequence of this analysis. I see no circularity in this reasoning.

The original paper is 5 pages. You may find "lectures on dynamical models for quantum measurements" useful https://arxiv.org/abs/1406.5178 Which is in lies in between at 45 pages. :)

8. May 25, 2017

### stevendaryl

Staff Emeritus
Well, describing ensembles using density matrices seems to me to be reliant on the Born rule. The point of a density matrix is that if $\rho$ is a density matrix, and $A$ is some operator, then the expectation value for $A$ is given by:

$\langle A \rangle = tr(\rho A)$

Now write $\rho$ in terms of a complete orthonormal basis $|\psi_n\rangle$:

$\rho = \sum_n p_n |\psi_n\rangle \langle \psi_n |$

Then the above definition of $\langle A \rangle$ is equivalent to:

$\langle A \rangle = \sum_n p_n \langle \psi_n | A | \psi_n \rangle$

That says that for a pure state $|\psi_n\rangle$,

$\langle A \rangle = \langle \psi_n | A | \psi_n \rangle$

That definition for the expectation value of an operator when evaluated in a pure state seems to me equivalent to the Born rule.

Actually, now that I think about it, the above expression for expectation values is equivalent to the Born rule if we make the additional assumption that a measurement always produces an eigenvalue of the operator corresponding to the observable being measured. So maybe the point is that by considering the composite system (system being measured + measuring device), the fact that you always get an eigenvalue is derivable?

9. May 25, 2017

### Prathyush

I think I understand what you are saying but I am not entirely sure. I will paraphrase you, to be sure.
When we define a state
$$|\psi> = c_1 |\psi_1> + c |\psi_2>$$
There is an implicit use of the born rule because by definition a state vector is equivalent to the probabilities obtained when an ensemble of measurements are performed.
Yes that is true, but that is not what we are trying to investigate here. As soon as we introduce operators into quantum mechanics, in a sense one can say that Born's rule comes for free.

What we are(or I am) interested in investigating here is if can one understand the use of the Born's rule when we study the interaction between the experimental apparatus and the system.

Yes it is an expectation that measurement will produce an eigenvalue of the operator corresponding to be observable being measured. It has be shown by using a context. It may be a very difficult problem to show this in a sufficiently general context and would require a lot of mathematical sophistication.(even if the conclusion we arrived at is wrong it would require thorough investigation). What the authors have done is used a reasonable caricature of a real life situation and have show this. (assuming the derivation works through)

Last edited: May 25, 2017
10. May 26, 2017

### zonde

It is very popular approach to represent quantum measurement as interaction with single device that can leave "measurement device" in different states depending on something. This paper in OP follows the same approach.
But I can not see any correspondence to real experimental setups. In all real setups that I know about there is some sort of manipulation device that does not produce any readings and detector that simply amplifies energy deposited in detector.
So I'm rather mystified why something sensible is expected from approach that is so detached from reality.

11. May 26, 2017

### Prathyush

What you have written here is sufficiently vague, perhaps a clarification is needed. What is this manipulation device you are talking about? What are the principles behind the construction of an amplification apparatus? How do we understand amplification in terms of the behaviour of microscopic degrees of freedom.

What I am interested in doing is to understand the interaction apparatus and the system using formal mathematical tools. For this one has to make precise notions such as what it means to record information, amplification, macroscopic states etc. I am looking for a simple system which is sufficiently complex to display a measurement like phenomenon and provide an explanation of these concepts.

One can study other more complicated like cloud chambers or photo multiplier tubes but these systems are too complex for me to analyze. The essential details of a real life example can be captured using simple models. Once we can understand a caricature of a real life example by changing the details we can understand more complicated situations. To me this model looks like a reasonable caricature.

12. May 26, 2017

### zonde

In that paper spin of particle is tested. So the manipulation device would be SG apparatus. If we are testing ion in a trap, manipulations are performed with radiation pulses of different wavelength. If we test beam of photons, we use filters and (polarization) beam splitters.
Amplification is done in detector by classical process. In threshold detector one electron triggers avalanche of other electrons over potential barrier.
Mathematical tools are required when you want quantitative details of the model. But before you start working out quantitative details you need qualitative understanding of the model. I don't see how math can help you there except for getting some rough estimates.
What is caricature of photographic plate type device in this model? Photographic plate does not make different spots on it depending on electrons spin.
The caricature of detector is device that takes whatever particle as input and produces "click" in the output by classical process.

13. May 26, 2017

### Prathyush

I largely agree with the qualitative assessments you have made about the various apparatus(barring ion traps which I am unfamiliar with). I will examine details you have provided.

I don't understand in what sense you are using the word "classical process"? You have used this term twice, and it requires clarification. Certainly whatever this process you are referring to is, It can be understood in terms of microscopic details using an appropriate statistical ensemble and Hamiltonian dynamics right?

This model is in no way a caricature of the photographic plate in any direct way. I don't yet know what a good caricature of a photographic plate is, certainly analyzing Ag and Br atoms is too complex. Now that you asked this question this model is not a caricature of any measurement apparatus that I am aware of in any direct way.

However It is a caricature in the sense that it has some salient features shared by other measurement apparatus. Consider a cloud chamber for instance, we understand that upon the interaction of the charged particles with the water molecules a phase transition happens.(It turns from transparent to cloudy). This models also shares the property that the measurement apparatus is prepared in a metastable state and when it interacts with the state, it changes into one of its 2 ground states(ferromagnetic states). So the information about the state of system is amplified into something that is macroscopically observable. So in a very loose sense I say this is a caricature of the measurement phenomenon.(perhaps not any specific apparatus)

14. May 26, 2017

### A. Neumaier

I refereed the paper in question here.

In which way??

The formula you stated is just a simple mathematically valid statement about certain formulas in a Hilbert space.

Whereas Born's rule claims that the formula has a relation with theoretically ill-defined, complex physical processes (measurement) and philosophically loaded notions (probability). It is these two properties that make the incorporation of Born's rule into any foundation questionable.

15. May 26, 2017

### stevendaryl

Staff Emeritus
No, it's not. "Expectation value" for a measurable quantity means the limiting average of the quantity over many measurements (sometimes it means measurements of different members of an ensemble, or possible many measurements of the same system over time). That's not derivable from the mathematics of Hilbert space.

16. May 26, 2017

### A. Neumaier

No. Expectation value means the value of a prescribed positive linear functional, nothing more. It is defined for arbitrary random variables, independent of the possibility or not of measuring them.

The use made of it in the statistical mechanics of a grand canonical ensemble, say, confoirms to this. One measures a single time a single number, for example the mass of all particles together, and gets a number very well predicted by $\langle M\rangle$ figuring in thermodynamics as derived from statistical mechanics.

17. May 26, 2017

### stevendaryl

Staff Emeritus
That's quibbling over definitions. My point is that the identification of $\langle \psi|A|\psi\rangle$ with the average value of $A$ over many measurements is equivalent to the Born rule, or at least is very closely related. What you call the terms is not very interesting.

That doesn't make any sense. If $\langle M \rangle$ is just a mathematical expression denoting the value of a positive linear functional, then it's not a prediction, at all. To relate $\langle M \rangle$ to measurements requires a physical hypothesis.

18. May 26, 2017

### A. Neumaier

But in statistical thermodynamics (which is the most successful application of the expectation calculus of statistical mechanics) one never makes this identification!

Instead one identifies the expectation value of the mass, say, with the single, macroscopic value of the mass! Each single measurement agrees in this way with the predictions of statistical mechanics! Probabilities arise only when a system is so small that uncertainties become significant, and many repetitions are needed to reduce the uncertainty by taking an average over many measurements. But microscopically this is then $K^{-1}\sum_{k=1}^K\langle A_k\rangle$ where $A_k$ is the observable in the $k$th measurement, and not $\langle A\rangle$!

19. May 26, 2017

### stevendaryl

Staff Emeritus
I'm not getting anything other than irritation from this discussion. I don't understand what point you're making.

20. May 26, 2017

### A. Neumaier

You make a silent but incorrect assumption about the interpretation of the statistical mechanics calculus, namely that $\langle A\rangle:=\langle\psi|A\psi\rangle$ is always interpreted as the average value of A over many measurements. But this interpretation is almost never valid. You can see this by comparing what is actually measured in thermodynamics, and how the thermodynamic formulas are derived from statistical mechanics. In each case you'll find that the extensive macroscopic variables (of a single thermodynamic system such as a single brick of gold) are encoded in statistical thermodynamics as expectation values of the corresponding microscopic operators. No repeated measurements, no averages anywhere!

21. May 26, 2017

### Prathyush

I will try and clarify what Arnold is saying(please correct me if I am wrong). I am not making any assumptions at this moment about its possible correctness. The sense in which we use Born's rule currently imply the existence of ideal measurement apparatus that work. However if one has to carefully analyze this situation in depth, i.e if we want to understand exactly how an ideal measurement is made by construction, we must refer to arrangements that must be described using macroscopic observables. And measurements made by these "ideal apparatus" ultimately refer to macroscopic changes of the apparatus. For instance what an electron gun or a cloud chamber is can be described using language that only refers to macroscopic observables.

Last edited: May 26, 2017
22. May 26, 2017

### mikeyork

That is a really interesting point you make. However, can you clarify how expectation in this statistical mechanics sense relates physically to expectation in QM in a way that does not imply averaging? I understand the mathematical point you make; it is the physical interpretation I have trouble with. The QM interpretation in the sense of averaging is clear. What sort of measurement of a quantum expectation would not involve averaging?

23. May 26, 2017

### Prathyush

Now I don't quite understand why you are talking about statistical mechanics, and how it can address stevendaryl's question about the use of Born rule in quantum mechanics. Certainly when we consider the stern gerlach experiment $$<\sigma_z> = p_{up}+p_{down}$$ is a sum over repeated measurements.

Last edited: May 26, 2017
24. May 26, 2017

### vanhees71

Well, what's measured on macroscopic systems usually are indeed very "coarse-grained observables", i.e., e.g., the position and momentum of a macroscopic body is something like the center-of-mass (relativistically center-of-energy) position and the macroscopic momentum the total momentum of the many-body system. Very often (particularly in situations when the system is close to an equilibrium state) in such cases the measured values are close to the expectation value, because the quantum (and even the thermal!) fluctuations of such macroscopic ("collective") observables (i.e., the corresponding standard deviations) are small compared to the macroscopically relevant scales. That's why macroscopic systems very often behave "classically" concerning the macroscopic observables.

I haven't looked on the 200p detailed treatise and I cannot say, whether the authors really prove Born's rule from only the dynamical laws of QT. It's hard to conceive, how this should be done without using the probability interpretation of the quantum formalism, which basically is Born's rule. It's quite probable that the authors somehow use Born's rule to prove it, but as I said, I cannot claim that, because I haven't the time to study their thesis carefully.

25. May 27, 2017

### A. Neumaier

Because the methods used are those of statistical mechanics. All is about the behavior of a macroscopic system (which can be analyzed only through statistical mechanics) interacting with a microscopic one. And because st
is based on a misinterpretation of how statistical mechanics relates to observation. I tried to correct this misinterpretation, but it is so deeply ingrained in stevendaryl's thinking that the information I provided only confuses him.
Yes, but s single spin is a tiny system, whereas statistical mechanics is about the behavior of macroscopic systems, in our case of the measurement device. Measuring a macroscopic system is not governed by Born's rule but by the identification of the expectation value with the measured value (within the intrinsic uncertainty). Born's rule is valid only for complete measurements of very small quantum systems such as a Stern-Gerlach spin.