Insights Quantum Physics via Quantum Tomography: A New Approach to Quantum Mechanics

[vanhees71]

No. The quote describes the experimental findings of the original paper by Stern and Gerlach. Nobody ever thought this would disagree with QM.

You probably never saw a discussion of the real experiment, only its heavily idealized caricature described in introductory textbooks on quantum mechanics!

Which quote are you talking about? On p. 12 of your paper there's the quote by Fröhlich:

The only form of ”interpretion” of a physical theory that I find legiti-
mate and useful is to delineate approximately the ensemble of natural
phenomena the theory is supposed to describe and to construct some-
thing resembling a ”structure-preserving map” from a subset of mathe-
matical symbols used in the theory that are supposed to represent phys-
ical quantities to concrete physical objects and phenomena (or events)
to be described by the theory. Once these items are clarified the theory
is supposed to provide its own ”interpretation”.
J¨urg Fr¨ohlich, 2021 [45, p.238]

with which I fully agree, of course, but that's not referring to the SG experiment.

Of course, when you measure a spin component with the standard ideal SG setup, you don't measure expectation values on a single silver atom but the spin component, which gives either $\hbar/2$ or $-\hbar/2$ as a result with a probability determined by the spin state $\hat{\rho}$ the silver atom is prepared in. When it comes from an oven as in the original experiment, this state is of course $\hat{\rho}=\hat{1}/2$. It's a paradigmatic example, for which a von Neumann filter measurement can be realized.

 

[gentzen]

Which quote are you talking about? On p. 12 of your paper there's the quote by Fröhlich:

This means that you still have v1 of Neumaier's paper. The only quote on p. 12 in v3 is:

The following ’laboratory report’ of the historic Stern-Gerlach experiment stands quite in contrast to the usual textbook ’caricatures’. A beam of silver atoms, produced in a furnace, is directed through an inhomogeneous magnetic field, eventually impinging on a glass plate. [...]
Only visual measurements through a microscope were made. No statistics on the distributions were made, nor did one obtain ’two spots’ as is stated in some texts. The beam was clearly split into distinguishable but not disjoint beams. [...] Strictly speaking, only an unsharp spin observable, hence a POV measure, is obtained.
Busch, Grabowski and Lahti, 1995 [25, Example 1, p.7]

 

[A. Neumaier]

Of course, when you measure a spin component with the standard ideal SG setup, you don't measure expectation values on a single silver atom but the spin component, which gives either $\hbar/2$ or $-\hbar/2$ as a result with a probability determined by the spin state $\hat{\rho}$ the silver atom is prepared in. When it comes from an oven as in the original experiment, this state is of course $\hat{\rho}=\hat{1}/2$. It's a paradigmatic example, for which a von Neumann filter measurement can be realized.

If your interpretation of the measurement results were correct, Stern and Gerlach could have deduced the value of $\hbar$ to infinite precision.

Instead I look at Figure 13 with the actual measurement results of Stern and Gerlach, and see (like Busch et al. in the quote and like everyone who can see) a large number of scattered dots, not two exact numbers involving $\hbar$. Clearly what was measured for each silver atom was position (with a continuous distribution), not spin.

Tu turn these position measurements into a projective spin measurement of $\hbar/2$ or $-\hbar/2$ you need to invoke heavy idealization, including additional theory and uncontrolled approximations.

 

[vanhees71]

Ok, well yes, the original SG experiment did not resolve single-atom results, and it may be well interesting to describe the original SG experiment in terms of the POVM formalism to better understand this formalism. Unfortunately in the quoted book they do this on p. 165ff but in a pretty abstract way instead of (approximately) solving the Schrödinger equation...

 

[A. Neumaier]

Ok, well yes, the original SG experiment did not resolve single-atom results

Can you point to the report of another SG experiment that resolves single silver atoms?

Even then, one only measures atom position and computes from these measurements fairly crude approximations of $\hbar$. It simply isn't a projective (textbook) measurement.

it may be well interesting to describe the original SG experiment in terms of the POVM formalism to better understand this formalism.

You can understand the formalism by reading Sections 2 and 3 of my paper. I did not treat the Stern-Gerlach experiment in detail since a precise description is quite involved. But in Section 3 I discuss in some detail several other measurement situations, which should be enough to get a clear understanding of what the approach means.

Unfortunately in the quoted book they do this on p. 165ff but in a pretty abstract way instead of (approximately) solving the Schrödinger equation...

Many papers and books using POVMs are quite abstract because they employ a measure-theoretic approach rather than simple quantum measures in the sense of my new paper. This is why my paper is a big step forward towards making the approach more understandable to everyone. Still, you need to do some reading to get the correct picture.

 

[Fra]

It results in different emanating beams, though their properties are the same.

Its an equivalence class only in the same irrelevant sense as in the claim that ''momentum is an equivalence class of preparations of particles in a classical source''. Very different equipment can result in particles with the same momentum.

Using mathematical terminology to make such a simple thing complicated is quite unnecessary.

As I understand your paper, it seems one major improvement in your description, is to make mathematically more explicit, the process of inferring the "quantum state" from REAL interactions - rather than considering and imaginary ensemble that is defined outside the formalism?

Ie. the fictive "equivalence class" is replaced by a real construction - as per quantum tomoghraphy - essentially from a sequence or history of interactions. And this works fine, as long as the sources are as you say "stationary" or does not change until the process of tomopgraphy is completed by margin?

Does this sound like a reasonable?

/Fredrik

 

[A. Neumaier]

As I understand your paper, it seems one major improvement in your description, is to make mathematically more explicit, the process of inferring the "quantum state" from REAL interactions - rather than considering and imaginary ensemble that is defined outside the formalism?

Ie. the fictive "equivalence class" is replaced by a real construction - as per quantum tomography - essentially from a sequence or history of interactions. And this works fine, as long as the sources are as you say "stationary" or does not change until the process of tomography is completed by margin?

The real advance is to make the quantum state a property of the source - i.e., of a macroscopic object.

This makes all talk about fictitious stuff (like ensembles, equivalence classes, multiple worlds, or presumable changes of states of knowledge) obsolete, without a change in the operational content of quantum mechanics.

 

[vanhees71]

I've started to read the paper, and I first had the impression, it's now much closer to the real use of QT as a physical theory as done by physicists since 1926, than the previous papers on your "thermal interpretation", but now it seems again, I'm completely misunderstanding its intended meaning. Obviously I misunderstood what you mean by "source". For me a "source" is just some device which "prepares quantum systems", and this can be also a single "particle" or even a single "photon" and not only macroscopic systems.

 

[Fra]

The real advance is to make the quantum state a property of the source - i.e., of a macroscopic object.

This makes all talk about fictitious stuff (like ensembles, equivalence classes, multiple worlds, or presumable changes of states of knowledge) obsolete, without a change in the operational content of quantum mechanics.

Given your ambitions, I guess this makes good sense. It's just that it does not solve the mysteries, at least not for me.

I still see your perspective as as limiting case of a general (yet unknown) theory.

As a macroscopic object is essentially just a part of the classical reality, this to me seems quite close to Bohrs angle to the CI (in contrast to Heisenbergs). You still need a CONTEXT, and this context is classical reality. This is of course a good thing from the perspective of human science... and as the context is the good old classical reality, it becomes more trivial as except for relativity, the observer-observer interaction is more trivial.

But it's a bad thing if you think there is explanatory power to be found by considering the logic of interacting observers. And I am not sure how it helps with fine tuning problems or unification quests. As I understand it, it isn't the ambition either. Then it's fine. I think part of the confusion, is different expectations, which I think what you wrote yourself in post 15 as well.

/Fredrik

 

[A. Neumaier]

I still see your perspective as as limiting case of a general (yet unknown) theory.

I prefer to frame the known in an optimally rational way, rather than to speculate about the unknown.

As a macroscopic object is essentially just a part of the classical reality, this to me seems quite close to Bohrs angle to the CI (in contrast to Heisenbergs). You still need a CONTEXT, and this context is classical reality.

I define classical reality in Section 7.2 of my paper as that part of quantum reality that can be deduced from it in the form of a local equilibrium description. Thus the context is quantum physics itself.

But it's a bad thing if you think there is explanatory power to be found by considering the logic of interacting observers.

Everything in my paper is observer-independent. It doesn't matter who observes, excpt that poor observations lead to poor approximations of the state.

 

[A. Neumaier]

I've started to read the paper, and I first had the impression, it's now much closer to the real use of QT as a physical theory as done by physicists since 1926,

It describes the real use of QT as a physical theory as done by physicists since 1970, when the more comprehensive view of measurement that goes beyond von Neumann's was introduced.

but now it seems again, I'm completely misunderstanding its intended meaning. Obviously I misunderstood what you mean by "source". For me a "source" is just some device which "prepares quantum systems", and this can be also a single "particle" or even a single "photon" and not only macroscopic systems.

Yes. You probably need to read the whole to see the differences and understand the new point of view.

From the perspective of the present considerations, quantum particles appear to be ghosts in the beams. This explains their spooky properties in the quantum physics literature!

Experiments in physics laboratories use controllable sources, filters and detectors to explore the nature of the microscopic world. All these are macroscopic objects, the only things that can be observed. The microscopic aspecs are not obsved but inferred; they define inferables, not observables. On p.13 of my paper I quote:

If you visit a real laboratory, you will never find there Hermitian operators. All you can see are emitters (lasers, ion guns, synchrotrons and the like) and detectors. The experimenter controls the emission process and observes detection events. [...] Quantum mechanics tells us that whatever comes from the emitter is represented by a state ρ (a positive operator, usually normalized to 1). [...] Traditional concepts such as ”measuring Hermitian operators”, that were borrowed or adapted from classical physics, are not appropriate in the quantum world. In the latter, as explained above, we have emitters and detectors.

A quantum source is simply a piece of equipment that has some measurable effect on detectors placed at some distance from them. How this effect comes about is not observed but described by theoretical models whose consequences can be checked against experimental results. The techniques and results described in my paper are agnostic about the models (Section 7.1), they are just about the observable (and hence macroscopic) aspects.

The present approach works independent of the nature or even the presence of a mediating substance: What is measured are properties of the source, and this has a well-define macroscopic existence. We never needed to make an assumption on the nature of the medium passed from the source to the detector. Thus the present approach is indifferent to the microscopic cause of detection events. It does not matter at all whether one regards such an event as caused by a quantum field or by the arrival of a particle. In particular, a microscopic interpretation of the single detection events as arrival of particles is not needed, not even an ontological statement about the nature of what arrives. Nor would these serve a constructive purpose.

This is the reason why the state is assigned to the source and not to something postulated as being transmitted. More precisely, the measured property (a quantum expectation) is assigned to the particular location at which the measurement is done, which leads naturally to a quantum field picture (Section 5.4).

Suppose that we have a detector that is sensitive only to quantum beams entering a tiny region in space, which we call the detector’s tip. We assume that we can move the detector such that its tip is at an arbitrary point x in the medium, and we consider a fixed source, extended by layers of the medium so that x is at the boundary of the extended source. The measurement performed in this constellation is a property of the source. The results clearly depend only on what happens at x, hence they may count as a measurement of a property of whatever occupies the space at x. Thus we are entitled to consider it as a local property of the world at x at the time during which the measurement was performed.

 

[vanhees71]

Given your ambitions, I guess this makes good sense. It's just that it does not solve the mysteries, at least not for me.

I still see your perspective as as limiting case of a general (yet unknown) theory.

It may well be that there is a more comprehensive theory then contemporary quantum theory. Who knows, what the solution of the problem to describe the gravitational interaction quantum mechanically and what this might imply for the description of spacetime, but there's no hint for such a new theory. Empirically there are no phenomena which are not described with our standard theories, as incomplete they might be.

As a macroscopic object is essentially just a part of the classical reality, this to me seems quite close to Bohrs angle to the CI (in contrast to Heisenbergs). You still need a CONTEXT, and this context is classical reality. This is of course a good thing from the perspective of human science... and as the context is the good old classical reality, it becomes more trivial as except for relativity, the observer-observer interaction is more trivial.

I think the contrary makes more sense. Macroscopic objects as we observe them are only consistently describable with quantum theory, given the atomistic structure. There'd not even be stable atoms as bound states of charged particles let alone molecules and condensed matter within classical physics. The "classical reality" is emergent, understood via the classical physical laws as effective description of the dynamical behavior of macroscopic coarse-grained observables.

But it's a bad thing if you think there is explanatory power to be found by considering the logic of interacting observers. And I am not sure how it helps with fine tuning problems or unification quests. As I understand it, it isn't the ambition either. Then it's fine. I think part of the confusion, is different expectations, which I think what you wrote yourself in post 15 as well.

/Fredrik

 

[vanhees71]

It describes the real use of QT as a physical theory as done by physicists since 1970, when the more comprehensive view of measurement that goes beyond von Neumann's was introduced.

Yes. You probably need to read the whole to see the differences and understand the new point of view.

Experiments in physics laboratories use controllable sources, filters and detectors to explore the nature of the microscopic world. All these are macroscopic objects, the only things that can be observed. The microscopic aspecs are not obsved but inferred; they define inferables, not observables. On p.13 of my paper I quote:

A quantum source is simply a piece of equipment that has some measurable effect on detectors placed at some distance from them. How this effect comes about is not observed but described by theoretical models whose consequences can be checked against experimental results. The techniques and results described in my paper are agnostic about the models (Section 7.1), they are just about the observable (and hence macoscopic) aspects.

This is the reason why the state is assigned to the source and not to something postulated as being transmitted. More precisely, the measured property (a quantum expecation) is assigned to the particular location at which the measurement is done, which leads naturally to a quantum field picture (Section 5.4).

In this way it makes a lot of sense again, but as I said before, today the experimentalists are able to prepare single-quantum (particles, atoms, molecules, photons) states and observe them. The observations are of course always via macroscopic measurement devices.

 

[A. Neumaier]

In this way it makes a lot of sense again, but as I said before, today the experimentalists are able to prepare single-quantum (particles, atoms, molecules, photons) states and observe them. The observations are of course always via macroscopic measurement devices.

In this case the traditional interpretation in terms of Born's rule is vacuous since probabilities are meaningful only in an ensemble context but individual quantum systems do not form an ensemble.

Instead, these experiments are traditionally interpreted in the fashion of classical stochastic processes, which have a trajectory interpretation for individual realizations, so that individual systems can be discussed. For example, this explains observed quantum jumps in atoms in an ion trap subject to external fields; see, e.g.,

• Plenio, M. B., & Knight, P. L. (1998). The quantum-jump approach to dissipative dynamics in quantum optics. Reviews of Modern Physics, 70(1), 101.

In the context of the present paper, one has to distinguish several cases of tiny quantum systems.

(A) A tiny quantum system mounted on a macroscopic objects.

1. If mounted for subsequent view in a scanning tunneling microscope, the tiny system acts as a stationary quantum source of the kind discussed in my paper, and its state can be determned by quantum tomography.
2. If mounted on an ion trap, the tiny quantum system can be manipulated by applying external classical field. This turns it into an instationary quantum system, for which standard quantum tomography is limited to short times within which the system cn be regarded as stationary. This means that its state can be measured only to limited accuracy - only very limited information can be reliably collected. This is discussed in Section 4.5 of my paper.
3. Nonstationary quantum tomography has hardly been studies, so it is too early to tell in detail which kind of limitations this imposes on what can be achieved experimentally.
4. For example, in the quantum jump experiments, observations interpretable in terms of the system state are restricted to observing a noisy piecewise constant time series that shows that apart from very short times of transitions (jumps) the system is stationary - being in one of to eigenstates of the appropriate Hamiltonian. Thus, per time step, one only gets one bit of information about the changing density operator.

(B) A tiny quantum system freely moving in a homogeneous macroscopic medium.

1. Single massive particles in flight can be observed in a bubble chamber or, in the context of modern particle accelerators, in a time projection chamber. The latter case is discussed in Section 3.4 of my paper; the former can be done in a similar way.
2. Single photons in flight cannot be observed; at best one can observe their death. It is impossible to do quantum tomography on them. Thus it is experimentally impossible to measure their state. Hence assigning a state to them is very questionable. Instead, a highly nonstationary photon state must be assigned to the cavity producing the photon, and case (A) applies.

 

[vanhees71]

This we discussed repeatedly. One can do statistics using a single particle in, e.g., a Penning trap, as described here:

https://doi.org/10.1088/0031-8949/1988/T22/016

but isn't this indeed a paradigmatic example for your formulation?

Also nondestructive photon measurements are done, but also the standard photon detection of course measures properties of single photons like energy, momentum, and polarization, or what else do you think the photon measurements in all the accelerators in HEP and heavy-ion physics provide?

 

[A. Neumaier]

One can do statistics using a single particle in, e.g., a Penning trap, as described here:
https://doi.org/10.1088/0031-8949/1988/T22/016
but isn't this indeed a paradigmatic example for your formulation?

One can do statistics with any collection of measurement results.
But in the case you mention, where the data come from a single particle, the statistics is not governed by Born's rule. Each data point is obtained at a different time, and at each time the particle is in a different state affected in an unspecified way by the previous measurement. So how could you calculate the statistics from Born's rule?

Instead, the statistics is treated in the way I discussed in case (A).

Also nondestructive photon measurements are done,

If the nondestructive single photon measurements result in a time series, the situation for this photon is the same as for the particle in the Penning trap.

but also the standard photon detection of course measures properties of single photons like energy, momentum, and polarization, or what else do you think the photon measurements in all the accelerators in HEP and heavy-ion physics provide?

I didn't know that accelerators measure momentum and polarization of individual photons. Could you provide me with a reference where I can read details? Then I'll be able to show you how it matches the description in my paper.

 

[vanhees71]

I wouldn't call results of experiments with single electrons, protons, ions etc. in Penning traps which are among the most precise ever "of limited precision" ;-). The tgeoretical description uses standard quantum theory based on Born's rule (see Dehmelt's above quoted review).

Detectors measure particles and photons of coarse. Real and virtual photons (dileptons) are among the most interesting signals in pp, pA, and heavy-ion collisions at CERN for some decades.

 

[Fra]

I prefer to frame the known in an optimally rational way, rather than to speculate about the unknown.

This is certainly respectable position, and I think your exposition is good from this position.

With what "is known" I think you effectively refer to human science. But if we even here consider and obsererver: What is the real difference between what an observers knows, and what it THINKS it knows? And does it make difference to the observes betting strategy? (action)

It doesn't matter who observes, excpt that poor observations lead to poor approximations of the state.

Just to contrast: In an interacting agent view, "poor approximations" should lead to "poor actions", "poor betting" which should be observable by other agents. My take on this is different. As I see it, the inside agent/observer has no access to an external judgement of what is a good or bad approximation. The agent just has to act on available evidence. What is right or wrong, poor or precise, should be irrelevant from the perspective of the agents betting. So the causal mechanism is independent of wether the information is "right". Information as well as desinformation will provoke a response which is depedent of the subjective information . But these ideas are IMO part of the non-equilibrium parts. Ie. an agent that is consisently "wrong", will soon be put out of business in the overall game. So instead of thinking of a ordinary equilibration, I see it as a evolution (as state spaces also evolve, there is no objective entropic flow). During this process, agents has two choices, learn (improve their predictions) or face destruction (deleted from the population pool). In this process lies also the emergence of symmetries. I seek and struggle with the mathematical or rather, algorithmic description on this. This is admittedly more speculative though. But I have the opinion that "speculation" and revision from feedback is at the heart of true learning. And I take this seriously also applied to the "observer". Even a measurement, can be seen as a "speculation" considering the choice of WHICH measurement to perform (in order for say maximal information gain).

Can we infer Vanhees secret speculations about he hopes to find(that you don't say loud as scientists should not be biased;), from the way the chooses to construct the next measurement or experiment?

This is why I can not help viewing, current QM as a limiting case of a very massive dominant agent, which is for all practical purposes classical and provides the background. I see that both the "preparation", and "detectors" are constructed from the agent itself. The information gained, is between the "action"(which can be seen as a "preparation") and the "backreaction of the environment", which I abstractly see as the correspondence of a general measurement of a real inside observer.

I feel that, trying to "close" or "polish" the limit theory case has a big value in itself, but it also risks polish away the open ends that are the clues to progress, and I prefers the open ends as clues forward.

/Fredrik

 

[A. Neumaier]

With what "is known" I think you effectively refer to human science. But if we even here consider and obsererver: What is the real difference between what an observers knows, and what it THINKS it knows? And does it make difference to the observes betting strategy? (action)

Science has no single betting strategy. Each scientist makes choices of his or her own preference, but published is only what passed the rules of scientific discourse, which rules out most poor judgment on the individual's side. What schience knows is an approximation to what it thinks it knows, and this approximation is quite good, otherwise resulting technology based on it would not work and not sell.

 

[A. Neumaier]

I wouldn't call results of experiments with single electrons, protons, ions etc. in Penning traps which are among the most precise ever "of limited precision".

I didn't call these results "of limited precision" but said that they determine the state to limited precision only. The state in these experiments is a continuous stochastic function $\rho(t)$ of time with $d^2-1$ independent real components, where $d$ is the dimension of the Hilbert space. Experiments resulting in $N$ measured numbers can determine this function only to limited precision. By the law of large numbers, the error is something like $O((dN^{1/2})^{-1})$.

What is usually done is to simply assume a Lindblad equation (which ignores the fluctuating part of the noise due to the environment) for a truncated version of $\rho$ with very small $d$. Then one estimates from it and the experimental results a very few parameters or quantum expectations.

This is very far from an accurate state determination...

The theoretical description uses standard quantum theory based on Born's rule (see Dehmelt's above quoted review).

Since Born's rule is a statement about the probability distribution of results for an ensemble of identically prepared systems, it is logically impossible to obtain from it conclusions about a single of these systems. A probability distribution almost never determines an individual result.

I'll read the review once I can access it and then comment on your claim that it derives statements about a single particle from Born's rule.

Detectors measure particles and photons of coarse. Real and virtual photons (dileptons) are among the most interesting signals in pp, pA, and heavy-ion collisions at CERN for some decades.

Then it should be easy for you to point to a page of a standard reference describing how the measurement of photon momentum and polarization in collision experiments is done, in sufficient detail that one can infer the assumptions and approximations made. I am not an expert on collision experiments and would appreciate your input.

 

[DrDu]

That's a nice article. However I somehow miss an explanation, what actually is meant with "quantum tomography" and one has to revert to the arxiv preprint to get an explanation. Given the title of the insights article, maybe you could add some words on what is meant with quantum tomography.

 

[vanhees71]

I didn't call these results "of limited precision" but said that they determine the state to limited precision only. The state in these experiments is a continuous stochastic function $\rho(t)$ of time with $d^2-1$ independent real components, where $d$ is the dimension of the Hilbert space. Experiments resulting in $N$ measured numbers can determine this function only to limited precision. By the law of large numbers, the error is something like $O((dN^{1/2})^{-1})$.

What is usually done is to simply assume a Lindblad equation (which ignores the fluctuating part of the noise due to the environment) for a truncated version of $\rho$ with very small $d$. Then one estimates from it and the experimental results a very few parameters or quantum expectations.

This is very far from an accurate state determination...

Since Born's rule is a statement about the probability distribution of results for an ensemble of identically prepared systems, it is logically impossible to obtain from it conclusions about a single of these systems. A probability distribution almost never determines an individual result.

I'll read the review once I can access it and then comment on your claim that it derives statements about a single particle from Born's rule.

Then it should be easy for you to point to a page of a standard reference describing how the measurement of photon momentum and polarization in collision experiments is done, in sufficient detail that one can infer the assumptions and approximations made. I am not an expert on collision experiments and would appreciate your inpu.

In Dehmelt's paper it is describe, how various quantities using single electrons/ions in a Penning trap are measured. I still don't understand, why you think there cannot be statistics collected using a single quantum. I can also get statics of throwing a single coin again and again to check whether it's a fair one or not. I just do the "random experiment" again and again using the same quantum and collect statistics and evaluate confidence levels and all that. Another review paper, which may be more to the point, because it covers both theory and experiment, is

https://doi.org/10.1103/RevModPhys.58.233

I also think that very rarely one does full state determinations. What's done are preparations and subsequent measurements of observables of interest.

I'm also not an experimental physicist and far from knowing any details, how the current CERN experiments (ATLAS, CMS, and ALICE) measure electrons and photons. I use their results to compare to theoretical models, which are based on standard many-body QFT and simulations of the fireball created in heavy-ion collisions. All this is based on standard quantum theory and thus after all on Born's rule. Here you can look at some papers by the ALICE collaboration as one example for what's measured concerning photons created in pp, pA, and AA collisions (pT spectra, elliptic flow, etc.). Concerning polarization measurements (particularly for dileptons) that's a pretty new topic, and of course an even greater challenge than the spectra measured for decades now. After all these are "rare probes".

 

[A. Neumaier]

I still don't understand, why you think there cannot be statistics collected using a single quantum.

I don't think that, and I explicitly said this. The point is that this statistics is not statistics about an ensemble of identically prepared systems hence has nothing to do with what Born's rule is about.

I can also get statistics of throwing a single coin again and again to check whether it's a fair one or not.

In this case the system identically prepared is the throw, not the coin. The coin is a system described by a rigid body, with a 12D phase space state $z(t)$, in contact with an environment that randomizes its motion through its collision with the table. The throw is what you can read off when the coin is finally at rest.

The state of the coin is complicated and cannot be identically prepared (otherwise it would fall identically and produce identical throws);. But the state of the throw is simple - just a binary variable, and the throwing setup prepares its state identically. Each throuw is different - only the coin is the same; that's why one gets an ensemble.

This is quite different from a quantum particle in a trap, unless (as in a throw) you reset before each measuement the state of the particle in the trap. But then the observation bevomes uninteresting. The interesting thing is to observe the particle's time dependence. Here the state changes continuously, as with the coin and not as with the throw.

 

[A. Neumaier]

All this is based on standard quantum theory and thus after all on Born's rule.

The 'thus' is not warranted.

Quantum field theory is completely independent of Born's rule. It is about computing $N$-point functions of interest.

Weinberg's QFT book (Vol.1) mentions Born's rule exactly twice - once in its review of quantum mechanics, and once where the probabilistic interpretation of the scatttering amplitude is derived. In the latter he assumes an ensemble of identically prepared particles to give a probabilistic meaning in terms of the statistics of collision experiment.

Nothing at all about single systems!

 

[vanhees71]

I don't think that we reach consensus about this issue. For me Born's rule is one of the fundamental postulates of QT (including QFT). You calculate the correlation functions (Green's functions) in QFT to get statistical information about observables like cross sections. How these correlation functions are related to the statistics of measurement outcomes is derived based on the fundamental postulates of QT, including Born's rule. Of course, that's what Weinberg and any other book on QFT does. A cross-section measurement consists of course always of collecting statistics over very many collision events using not the same particles again and again.

You use yourself Born's rule all the time since everything is based on taking averages of all kinds defined by $\langle A \rangle=\mathrm{Tr} \hat{\rho} \hat{A}$ (if you use normalized $\hat{\rho}$'s).

 

[A. Neumaier]

Another review paper, which may be more to the point, because it covers both theory and experiment, is
https://doi.org/10.1103/RevModPhys.58.233
[...] you can look at some papers by the ALICE collaboration as one example for what's measured concerning photons created in pp, pA, and AA collisions (pT spectra, elliptic flow, etc.). Concerning polarization measurements (particularly for dileptons) that's a pretty new topic,

Are the papers where I can read about ALICE measurements and about polarization measurements in the above review?

 

[vanhees71]

I was referring to the measurements on single particles in a trap, not on ALICE photon measurements. There are tons of papers about "direct photons":

https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=find title photons and cn alice

Polarization measurements for dileptons or photons are very rare today. There's a polarization measurement by the NA60 collaboration on di-muons:

https://arxiv.org/abs/0812.3100

 

[A. Neumaier]

That's a nice article. However I somehow miss an explanation, what actually is meant with "quantum tomography" and one has to revert to the arxiv preprint to get an explanation. Given the title of the insights article, maybe you could add some words on what is meant with quantum tomography.

Thanks. I added to the Insight article a link to Wikipedia and an explaining paragraph.

 

[A. Neumaier]

I was referring to the measurements on single particles in a trap, not on ALICE photon measurements. There are tons of papers about "direct photons":

https://inspirehep.net/literature?sort=mostrecent&size=25&page=1&q=find title photons and cn alice

Polarization measurements for dileptons or photons are very rare today. There's a polarization measurement by the NA60 collaboration on di-muons:

https://arxiv.org/abs/0812.3100

Thanks for the pointers. Will reply in more detial after having read more. I expect that it will mean that the instancs of case (B) are not so different from those of case (A) in my earlier classification of single-particle measurements.

 

[vanhees71]

I still do not understand why you say that the content of the review papers by Dehmelt and Brown contain anything denying the validity of Born's rule. For me it's used all the time!