A Evaluate this paper on the derivation of the Born rule

[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?

 

[stevendaryl]

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?

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.

 

[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 into the association of states with Hilbert space vectors.

 

[tom.stoer]

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 into the association of states with Hilbert space vectors.

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).

 

[martinbn]

200 pages!

 

[mikeyork]

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).

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.

 

[Prathyush]

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.

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.

200 pages!

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. :)

 

[stevendaryl]

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.

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?

 

[Prathyush]

Well, describing ensembles using density matrices seems to me to be reliant on the Born rule.

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

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.

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?

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)

 

[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.

 

[Prathyush]

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.

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.

So I'm rather mystified why something sensible is expected from approach that is so detached from reality.

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.

 

[zonde]

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.

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.

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.

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.

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.

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.

 

[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.

Amplification is done in detector by classical process. In threshold detector one electron triggers avalanche of other electrons over potential barrier.

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?

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.

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)

 

[A. Neumaier]

I refereed the paper in question here.

⟨A⟩=⟨ψn|A|ψn⟩⟨A⟩=⟨ψn|A|ψn⟩\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.

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.

 

[stevendaryl]

In which way??

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

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.

 

[A. Neumaier]

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.

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.

 

[stevendaryl]

No. Expectation value means the value of a prescribed positive linear functional, nothing more.

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.

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.

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.

 

[A. Neumaier]

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.

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 $\langle A\rangle$!

 

[stevendaryl]

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 $\langle A\rangle$!

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

 

[A. Neumaier]

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

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!

 

[Prathyush]

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

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.

 

[mikeyork]

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!

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?

 

[Prathyush]

You make a silent but incorrect assumption about the interpretation of the statistical mechanics calculus, namely that ⟨A⟩:=⟨ψ|Aψ⟩\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!

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.

 

[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.

 

[A. Neumaier]

I don't quite understand why you are talking about statistical mechanics

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

But it seems to me a little circular to use the ensemble interpretation of QM to derive the Born rule.

describing ensembles using density matrices seems to me to be reliant on the Born rule.

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.

Certainly when we consider the stern gerlach experiment [...]

is a sum over repeated measurements.​

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.

 

[A. Neumaier]

What sort of measurement of a quantum expectation would not involve averaging?

The sort of measurement done in thermodynamics and in ordinary life. One measures a macroscopic system once and gets (if the measurement is done carefully) an accurate value for the measured variable that agrees with the expectation value calculated from statistical mechanics. No repetition is necessary, no averaging is involved. Repeated measurement gives essentially the same value, within the intrinsic (small) uncertainty.

 

[vanhees71]

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.

I'm puzzled how you come to this conclusion. In many-body physics we describe the system's state as any other quantum system by a statistical operator $\hat{\rho}$ and define the expectation value of an observable, represented by a self-adjoint operator $\hat{A}$ by
$$\langle A \rangle_{\rho}=\mathrm{Tr}(\hat{\rho} \hat{A}).$$
This is Born's rule.

What you usually measure for macroscopic systems in a "classical setup" are not arbitrary expectation values of this kind, but relevant "pretty coarse-grained" observables. E.g., for an ideal gas you consider some volume and check quantities like density, pressure, temperature etc. (assuming you have a state in or close to thermal equilibrium). These are quantities describing a large number of particles in a "collective way". The density, e.g., is just counting particle numbers in a volume element containing many particles and then dividing by this volume. The volume is microscopically large, i.e., it' contains many particles, but on the other hand it's macroscopically small, i.e., the properties of the coarse-grained variable don't change much on the scale of the volume. Experience shows that such a "separation of scales" for such coarse-grained variables occurs quite often for macroscopic systems and that makes the success of the thermodynamical approach. You can derive the macroscopic equations like the (Vlasov-)Boltzmann(-Uehling-Uhlenbeck) from this approach (formalizing the "coarse graining procdedure" either by applying the gradient-expansion or some projection method) or further viscous or even ideal hydrodynamics.

There is some community of quantum physicists who likes to try to derive Born's rule from the other postulates, among them Weinberg with his newest textbook, as an attempt to solve an apparent "measurement problem". As far as I understand what they mean by this "problem" is, why a measurement leads to a certain outcome although according to quantum theory, if the system is not prepared such that the measured observable is determined, there are only probabilities to find a certain value, given by Born's rule. The idea behind these attempts seems to be to be able to derive how the certain outcome of the observable's value occurs when measured with these probabilities from the dynamical postulates alone. The problem is of course that you get a circular argument, and Weinberg finally comes to the conclusion that he cannot satisfactorily derive Born's rule from the other postulates. That doesn't mean that not some other clever physicist comes along and finds a convincing set of postulates for QT, from which you can derive Born's rule. Whether or not you find this line of argument convincing or not is subjective. I don't see any merit in such an attempt, except if you find something really new. The paradigmatic example where such a methodology opened a whole new universe (for general relativists quite literally) of mathematical thinking is the attempt to prove the axiom about parallels of Euclidean geometry from the other postulates. It lead to the discovery of non-Euclidean geometries and a plethora of new "geometrical ideas", including the group-theoretical approach a la Klein, which is so vital for modern physics through Noether's theorems and all that.

In the minimal interpretation there is no measurement problem. For the Stern-Gerlach experiment, i.e., measuring a spin component (usually chosen as $\sigma_z$) of a spin-1/2 particle, it's pretty simple to explain everything quantum theoretically, because the system is so simple that you can do so. The experiment consists more or less just of an appropriately chosen static magnetic field, which has a large homogeneous component in $z$ direction and also some inhomogeneity (also along the $z$ direction, but of course one has necessarily another in another direction due to Gauss's Law $\vec{\nabla} \cdot \vec{B}=0$; one simple model for such a field is $\vec{B}=(B_0 + \beta z)\vec{e}_z-\beta y \vec{e}_y$. Then it's quite easy to show that if $\beta |\langle y \rangle| \ll B_0$ that the particle beam splits into two quite well separated partial beams which are almost 100% sorting the two possible states $\sigma_z = \pm 1$. That's how you measure (or even prepare!) nearly pure $\sigma_z$ eigenstates by entangling position and $\sigma_z$. There's nothing mysterious about this, but of course this argument uses the full standard QT postulates, including Born's rule.

 

[akhmeteli]

I refereed the paper in question here.

Thank you for the review. Just some nit-picking:

200 page treatise (from 2011, available as an arXiv preprint only) containing all the details:
A.E. Allahverdyan, R. Balian and T.M. Nieuwenhuizen,
Understanding quantum measurement from the solution of dynamical models,
https://arxiv.org/abs/1107.2138
(201 pages)

This work was published in Phys. Rep. 525 (2013) 1-166

 

[Prathyush]

I largely agree with the analysis done by Vanhees71 in post #27.

The problem is of course that you get a circular argument, and Weinberg finally comes to the conclusion that he cannot satisfactorily derive Born's rule from the other postulates. That doesn't mean that not some other clever physicist comes along and finds a convincing set of postulates for QT, from which you can derive Born's rule.

I don't think any new postulate is required, I think it would only require a careful analysis of what we mean by measurement. As I suggested (admittedly with uncertainty ) on post #21, in the construction of any experiment, we refer to situations that can be described only using macroscopic observables. Born's rule should naturally follow from a clarification on what an ideal experimental apparatus is.

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.

It is possible that Born's rule is ingrained deeply stevendaryl's, but it is also possible that you have not made a sufficiently clear and compelling argument. I too find what you are saying to be in need of clarification.

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.

Perhaps the central point is "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)." which is in direct contrast with the main point of Vanhess71's post. I certainly have to think about this. But It appears that Arnold has a point because when we observe a macroscopic system we certainly do not destroy the entanglements and superposition between various particles, So it cannot be treated as an average of individual measurements over the different particles.

 

[vanhees71]

My point was that for a macroscopic system we don't measure properties of single particles but quite "coarse-grained variables". This we do (perhaps more implicitly) already in classical mechanics. For the purpose of describing, within Newtonian mechanics, the motion of the planets around the Sun we can (in a good approximation) simplify them to "point particles", i.e., instead of describing the entire bodies as complicated systems we simplify them to simply be represented by their center of mass mutually attracted by other centers of mass via Newton's Law of gravity. For the purpose at hand that's a quite satisfactory treatment. The same holds at other layers of approximations going from the "most fundamental theory" (which is also a time-dependent notion; for us nowadays it means the Standard Model of elementary particles and general relativity) to more "effective" descriptions.

The point is that for the observables of macroscopic bodies in almost all cases (particularly of everyday life) the quantum expectation values of these macroscopic bodies are a sufficient description, because the fluctuations (both quantum and "thermal") around these expectation values are small compared to the accuracy which is needed to have a sufficiently detailed description of these macroscopic bodies. This means that the expectation values are as good as the single measurements within the needed accuracy. This doesn't make Born's rule superfluous. To the contrary, you need it to define how to evaluate the expectation values in the first place. Nobody thinks about this using quantum many-body theory, because it's just how the theory is formulated, i.e., Born's rule is an integral part of it, without which you cannot relate the formalism to real-world observations.

As I said before, some people find this still weird after over 90 years of QT. Maybe someone finds another way to formulate quantum theory (or a better more comprehensive other theory, containg QT as an approximation) and then derive Born's rule from these more convincing basis, but as long as we don't have such a convincing new formulation, it's safe to stick with the standard postulates and apply QT in terms of the minimal interpretation to the analysis of real-world experiments.

 

[Prathyush]

My point was that for a macroscopic system we don't measure properties of single particles but quite "coarse-grained variables".

Sorry about misrepresenting your post. I would be interested to see how Arnold has to respond.

This is clearly a question that I haven't thought about in depth, this discussion was extremely fruitful to me because it brought these issues into the forefront.

This is a crude and preliminary analysis, please treat it that way. I think the important point here is to understand in what situations a coarse grained macroscopic description is applicable. The born's rule does not apply when we talk about measuring a macroscopic system, in the sense that when we extract coarse grained information about the macroscopic system, we don't redefine(or recreate ) a new state for the system. Formally we write $$<A> = tr(\rho A)$$ however it does not have the same meaning when we talk about the same equation for a microscopic system. Which is formally defined as a sum over multiple observations.

This sharp differences in meaning, must lie in the fact that our interaction with a macroscopic system does not appreciably change its state, and we can use extrinsic variables for its description. Any precise observation of our macroscopic system will entail a different experimental apparatus, its meaning of an observation must be changed appropriately and a macroscopic description must be changed into a microscopic one.

It requires more analysis, I don't see right now how to proceed from here. More precisely I think this inherent stability when we talk about a macroscopic system could be formally analysed.

 

[vanhees71]

I disagree. If you take the expectation value of, say the total energy or momentum of a macroscopic system, it's just that, the average of these quantities, and it's done with Born's rule. You contradict yourself, because what you wrote down is Born's rule. Given the state $\hat{\rho}$ (positive semidefinite trace-1 self-adjoint operator) then $\langle A \rangle=\mathrm{Tr}(\hat{\rho} \hat{A})$ is the expectation value of $A$, and that's Born's rule.

 

[Prathyush]

I disagree. If you take the expectation value of, say the total energy or momentum of a macroscopic system, it's just that, the average of these quantities, and it's done with Born's rule. You contradict yourself, because what you wrote down is Born's rule

The point I am making is the following, the sense in which we apply Born's rule, to a macroscopic system is in sharp contrast with the sense in which we apply Born's rule to a macroscopic system. I have explained in post#31 is why this different procedures appear to be in sharp contrast.

Just to be clear when we use Born's rule for a microscopic system, we also insist that the state of system be redefined to highlight the new information that was obtained. However no such thing is done for a macroscopic system. It appears that there are 2 completely independent ways in which we are using Born's rule. This fact needs further clarification.

 

[vanhees71]

I completely disagree with this idea that there are different meanings of the very clear formalism of QT. Born's rule is Born's rule, no matter whether you apply it to a single electron or to a many-body system. I don't know what you mean by the "state of the system is redefined". Do you mean something like a collapse hypothesis? I don't believe in this religion. In almost all measurements the collapse hypothesis is invalid, and where it is valid (a socalled von Neumann filter measurement) it's indeed a preparation procedure (as in my above most simple example of a Stern-Gerlach apparatus).

 

[Prathyush]

I completely disagree with this idea that there are different meanings of the very clear formalism of QT. Born's rule is Born's rule, no matter whether you apply it to a single electron or to a many-body system. I don't know what you mean by the "state of the system is redefined". Do you mean something like a collapse hypothesis? I don't believe in this religion. In almost all measurements the collapse hypothesis is invalid, and where it is valid (a socalled von Neumann filter measurement) it's indeed a preparation procedure (as in my above most simple example of a Stern-Gerlach apparatus).

Now you need to clarify what you mean? Basically yes I am talking about the collapse. When we measure the spin of a system, for any futher measurements we must set newly measured state as the initial state. That is the essence of the born's rule as we learn in textbooks.

Now what don't you believe about this?

 

[A. Neumaier]

define the expectation value of an observable, represented by a self-adjoint operator \hat{A} by

⟨A⟩_ρ=Tr(ρA).​

This is Born's rule.

No. This is not Born's rule. This is just a definition giving a name to a formula. Born's rule is a statement about eigenvalues and the probability of measurement results in an ideal von-Neumann measurement. The above definition has neither a reference to measurement nor to the conditions that make a measurement von-Neumann. SDo how can it be Born's rule? The only connection to Born's rule is that it can be deduced from it in the very special case that the latter is applicable and one averages over a lot of identically prepared measurement results. But the formula holds much generally, and is in fact much more fundamental than Born's rule in that it is very easy to state (no spectral theory needed) and (unlike Born's rule) applies universally.

 

[A. Neumaier]

This work was published in Phys. Rep. 525 (2013) 1-166

Thanks! I updated the reference.

 

[A. Neumaier]

Perhaps the central point is "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)." which is in direct contrast with the main point of vanhees71's post.

Yes, and vanhees71 is wrong about his claims! He just parrots without justification the same stuff I used to believe as well as long as I only read what is repeated in the textbooks since 1932. But when I looked at the true correspondence of what is measured in a macroscopic system and how it is encoded into the statistical thermoddynamics formalism it was very obvious that the correspondence is the one I gave. The observables to which this applies are all those measured in daily engineering practice - in equilibrium they are for a chemical system total energy (represented by the operator $H$ the Hamiltonian), total mass of each component of a chemical mixture, (represented by the operators $m_iN_i$, where $N_i$ is the number operator of chemical component $i$. These are the fundamental operators in quantum mechanics. Each single measurement of any of these agrees with the expectation value to several digits of relative accuracy.

Thus has nothing at all to do with Born's rule which is completely misplaced when applied in a macroscopic context where single measurements are already significant and probabilities are irrelevant.

 

[A. Neumaier]

So it cannot be treated as an average of individual measurements over the different particles.

It cannot be treated as that for the simple reason that the observable measured is not an average over particles but (for mass) a sum over particles, and the energy in an interacting system is not even additive.

 

[A. Neumaier]

This doesn't make Born's rule superfluous. To the contrary, you need it to define how to evaluate the expectation values in the first place.

There is no need for Born's rule to define expectations values. As stevendaryl had already remarked, the latter is just a name given to a mathematical expression borrowed from statistics It need not have any more relation with the name-giving object as the term state vector has with a 3-dimensional arrow that lead to this name..

One can derive all of equilibrium thermodynamics theory from this definition without ever invoking probability concepts or the notion of measurement. Then one can invoke classical 19th century measurement theory to justify that the expectation value can be identified with the measured equilibrium values - since the formulas are just those that had been in use since 1850. Indeed, this is the way thermodynamics is derived in any physics book that bothers to give a derivation.

Nowhere a single trace of Born's rule!

 

[A. Neumaier]

Any precise observation of our macroscopic system will entail a different experimental apparatus

But if the apparatus does its job well and the apparatus in not coupled to a very sensitive system (such as a single quantum spin) the result will not depend on the apparatus, and always agree with the expectation value.

 

[A. Neumaier]

In almost all measurements the collapse hypothesis is invalid,

And in almost all measurements Born's rule is violated as most measurements are not von-Neumann measurements. All POVM measurements violate Born's rule.

 

[vanhees71]

Now you need to clarify what you mean? Basically yes I am talking about the collapse. When we measure the spin of a system, for any futher measurements we must set newly measured state as the initial state. That is the essence of the born's rule as we learn in textbooks.

Now what don't you believe about this?

The collapse hypothesis is not part of the formal postulates and fortunately not necessary. It causes more problems than it is in any way useful to understand quantum theory.

What happens with the measured system is a question that cannot be part of the general formalism for the simple reason that it depends on how your measurement apparatus is constructed. E.g., if you measure a photon's polarization by checking whether it runs through a polarization foil or not, you absorb it (either by the foil or by the detector telling you that it has gone through the polarizer). In the end you don't have a photon left at all. On the other hand, it's indeed a filter measurement in the literal sense, i.e., you can be (almost) sure that any photon that comes through the polarizer is polarized in the corresponding direction.

 

[vanhees71]

And in almost all measurements Born's rule is violated as most measurements are not von-Neumann measurements. All POVM measurements violate Born's rule.

As far as I know, the definition of POVM measurements relies also on standard quantum theory, and thus on Born's rule (I've read about them in A. Peres, Quantum Theory: Concepts and Methods). It just generalizes "complete measurements" by "incomplete ones". It's not outside the standard rules of quantum theory.

 

[Prathyush]

Arnold may I recommend using 1 or 2 posts per reply as it will be helpful to the reader, and using the multi quote function for this purpose.

On the other hand, it's indeed a filter measurement in the literal sense, i.e., you can be (almost) sure that any photon that comes through the polarizer is polarized in the corresponding direction.

I understand this particular situation is a filter measurement. But that does not appear to be a relevant point.

One can construct experimental apparatus that won't destroy the particle that we intend to measure. Consider the double slit experiment the Born's rule applies in the sense that if we observe which slit the particle when though for making future prediction about it, we need to use this new information to calculate probabilities that is the essence, I don't understand how there can be a disagreement about this. Sure in some situations we destroy the particle, sometime we change the state of the particle but that can be understood from from first principles based on how the apparatus was constructed.

Born's rule is $$p_i = <\psi|A_i|\psi>$$ where Ai is the the projection of the operator onto the nth state. In a suitably constructed apparatus, one can say the final state after measurement is precisely the eigenvalue of the measurement operator. This is what we understand by an ideal measurement. If we have a mixed state it can be written in terms of density matrices etc.
$$<A> = Tr(A \rho )$$ is a straight forward consequence of this rule.

Would you disagree with the above?

Also please tell how to include latex into the main text without going to a new line each time. I used the 2 dollar symbols to add latex.

 

[vanhees71]

There is no need for Born's rule to define expectations values. As stevendaryl had already remarked, the latter is just a name given to a mathematical expression borrowed from statistics It need not have any more relation with the name-giving object as the term state vector has with a 3-dimensional arrow that lead to this name..

One can derive all of equilibrium thermodynamics theory from this definition without ever invoking probability concepts or the notion of measurement. Then one can invoke classical 19th century measurement theory to justify that the expectation value can be identified with the measured equilibrium values - since the formulas are just those that had been in use since 1850. Indeed, this is the way thermodynamics is derived in any physics book that bothers to give a derivation.

Nowhere a single trace of Born's rule!

Well, usual statistics textbook start with equilibrium distributions and define, e.g., the grand canonical operator
$$\hat{\rho}=\frac{1}{Z} \exp[-\beta (\hat{H}-\mu \hat{Q})],$$
to evaluate expectation values using Born's rule, leading to the Fermi- and Bose-distribution functions.

Classical statistical mechanics is also based on probability concepts since Boltzmann & Co. I don't know, which textbooks you have in mind!

 

[vanhees71]

Arnold may I recommend using 1 or 2 posts per reply as it will be helpful to the reader, and using the multi quote function for this purpose.
I understand this particular situation is a filter measurement. But that does not appear to be a relevant point.

One can construct experimental apparatus that don't destroy the particle that we intend to measure. Consider the double slit experiment the Born's rule applies in the sense that if we observe which slit the particle when thought for making future prediction about it, we need to use this new information to calculate probabilities that is the essence, I don't understand how there can be a disagreement about this. Sure in some situations we destroy the particle, sometime we change the state of the particle but that can be understood from from first principles based on how the apparatus was constructed.

Born's rule is $$p_i = <\psi|A_i|\psi>$$ where Ai is the the projection of the operator onto the nth state. In a suitably constructed apparatus, one can say the final state after measurement is precisely the eigenvalue of the measurement operator. This is what we understand by an ideal measurement. If we have a mixed state it can be written in terms of density matrices etc.
$$<A> = Tr(A \rho )$$ is a straight forward consequence of this rule.

Would you disagree with the above?

Also please tell how to include latex into the main text without going to a new like each time. I used the 2 dollar symbols to add latex.

Just to clarify my postings from before: To simplify the discussion, I usually don't dinguish pure and mixed states. For me all states are defined by a statistical operator (positive semidefinite self-adjoint operator with trace 1). A state is by definition pure if it is represented by a projection operator $\hat{\rho}_{\text{pure}}=|\psi \rangle \langle \psi|$ with some normalized vector $|\psi \rangle$. This is much simpler than to talk about unit rays in Hilbert space!

To get inline LaTeX just enclose them by two # instead of two $ signs.

 

[Prathyush]

Just to clarify my postings from before: To simplify the discussion, I usually don't dinguish pure and mixed states. For me all states are defined by a statistical operator (positive semidefinite self-adjoint operator with trace 1). A state is by definition pure if it is represented by a projection operator $\hat{\rho}_{\text{pure}}=|\psi \rangle \langle \psi|$ with some normalized vector $|\psi \rangle$. This is much simpler than to talk about unit rays in Hilbert space!

To get inline LaTeX just enclose them by two # instead of two $ signs.

Sure I will use density matrices as you prefer. Born's rule in this language is, probability $p_i = Tr (\rho A_i)$ where $A_i$ is the projection of the operator that we intend to measure. In a suitably constructed apparatus, the final state is $\rho_i = |\psi_i><\psi_i|$ where $|\psi_i>$ is the eigenvector corresponding to the measurement performed.
Is there a disagreement here?

 

[vanhees71]

I'm not familiar with your terminology. So let me give you mine in a nutshell. Then we can see, whether we understand the same thing when talk about (quantum theory, which I doubt ;-)).

(a) The state of the system is given by a positive semidefinite self-adjoint operator on a (separable) Hilbert space with trace 1, $\hat{\rho}$ (statistical operator). A state is pure iff there exists a normalized vector $|\psi \rangle$ such that $\hat{\rho}=|\psi \rangle \langle \psi |$.

(b) An observable $A$ is represented by a self-adjoint operator $\hat{A}$.

(c) Possible outcomes of precise (complete) measurements of $A$ are the eigenvalues of $\hat{A}$. In the following I use a complete orthonormalized set of eigenvectors of $\hat{A}$, which I denote with $|a,\beta \rangle$, where $a$ is a possible eigenvalue:
$$\hat{A} |a,\beta \rangle=a |a,\beta \rangle, \quad \langle a,\beta|a',\beta' \rangle=\delta_{aa'} \delta_{\beta \beta'}.$$
The label $\beta$ are one or more variables to label the different orthogonal eigenstates to the same eigenvalue. For simplicity I only consider the case that we have discrete spectra of the operators (if you have variables with continuous spectra it becomes only a bit more complicated since you have to use distributions and integrals instead of sums). The eigenvectors form a complete orthonormalized set of vectors,
$$\sum_{a,\beta} |a,\beta \rangle \langle a,\beta|=\hat{1}.$$

(d) If the system is prepared in a state $\hat{\rho}$ the probability to find the value $a$, when observable $A$ is measured precisely on this system is given by Born's rule,
$$P(a)=\sum_{\beta} \langle a,\beta|\hat{\rho}|a,\beta \rangle.$$

The expectation value of $A$ is given by
$$\langle A \rangle=\sum_{a} a P(a) = \sum_{a,\beta} a \langle a,\beta |\hat{\rho}|a,\beta \rangle = \sum_{a,\beta} \langle a,\beta| \hat{\rho} \hat{A} a,\beta \rangle=\mathrm{Tr} (\hat{\rho} \hat{A}).$$

The probabilities $P(a)$ can indeed also be formulated with the projection operators to the different eigen spaces of $\hat{A}$, because with
$$\hat{P}_a=\sum_{\beta} |a,\beta \rangle \langle a,\beta|$$
obviously we have
$$P(a)=\text{Tr} (\hat{P}_a \hat{\rho}).$$

There is no need to know in which state the system is after measurement. We don't need to complicate this discussion by bringing up the collapse hypothesis, which is in my opinion completely flawed and not commonly assumed anywhere in practitioning QT.

We also don't need to complicate things by thinking about more general incomplete measurements here. In my understanding the socalled "measurement problem" is to somehow explain, why the outcome of a precise measurement is always one and only one eigenvalue of the associated operator $\hat{A}$. For me that's an empty question. What I've written down are the condensed postulates of QT as a theory to describe what's observed in nature by measuring observables (as is also classical physics by the way). The only thing that counts is, whether this theory describes the real-world experiments and observations in nature, and indeed it does with a breathtaking accuracy. So there is no "measurement problem", because the formalism describes everything we have observed so far. There's not more to be expected from a physical theory. The QT we learn today has been formed in the 1st quarter of the 20th century from a careful analysis of observations of the behavior of matter, particularly atomic and subatomic physics, and that's why it works so well (including also the understanding of the "classical" behavior of the macroscopic matter surrounding us with many of its quantitative properties through statistical many-body quantum physics).

You can load QT (as any mathematical model of reality) with some philosophical (not to call it esoterical) questions like, why we always measure eigenvalues of self-adjoint operators, but physics doesn't answer why a mathematical model works, it just tries to find through an interplay between measurements and mathematical reasoning such models that describe nature (or even more carefully formulated what we observe/measure in nature) as good as possible.

 

[Prathyush]

I'm not familiar with your terminology. So let me give you mine in a nutshell. Then we can see, whether we understand the same thing when talk about (quantum theory, which I doubt ;-)).

Fair enough. I use the same formalism.

First a minor point.

The probabilities P(a) can indeed also be formulated with the projection operators to the different eigenspaces of $\hat{A}$, because with $$\hat{P}_a=\sum_{\beta} |a,\beta \rangle \langle a,\beta|$$

I usually don't include the $\beta$ into the definition of the projection operator or born rule, for instance I would write $\hat{P}_a= |a,\beta \rangle \langle a,\beta|$ and other formulas would change appropriately. Ofcourse it has to be used based on the context.

There is no need to know in which state the system is after measurement. We don't need to complicate this discussion by bringing up the collapse hypothesis, which is in my opinion completely flawed and not commonly assumed anywhere in practitioning QT.

In my understanding the so called "measurement problem" is to somehow explain, why the outcome of a precise measurement is always one and only one eigenvalue of the associated operator $\hat{A}$.

This is something you are suggesting that is orthogonal to what most textbooks write. I can understand where you are coming from when you say what happens to the system after measurement is irrelevant, I will critically analyze this statement in a later post.

For now however Consider a 2 slit experiment(for instance feynman's description of it), If we measure which slit the particles went through, the for all future measurements we have to use this information. This is the reason why most textbooks include the collapse postulate. When a measurement is performed this information must be reflected in the state of the particle atleast in this particular context. Ofcourse you can say once the measurement is performed we can move both the particles to the same place(or change it however you want), so the collapse posulate is not a general rule. I will analyze this "rule" carefully based on your response.

We can talk about macroscopic and microscopic descriptions once we resolve the collapse stuff.