I What is the POVM generalization of Born's rule for quantum measurements?

[A. Neumaier]

TL;DR Summary

Starting from first principles, my new paper gives a self-contained, deductive introduction to quantum measurement and Born's rule, in its generalized form that applies to the results of measurements described by POVMs. It is well-known that a generalization of Born's rule to realistic experiments must be phrased in these terms; Born's rule in its conventional textbook form applies to the small class of projective measurements only.

I just placed my new paper Born's rule and measurement on the arXiv. It contains a self-contained discussion of the POVM generalization of Born's rule for quantum measurements. It is a much extended, polished version of my contributions to the thread How to teach beginners in quantum theory the POVM concept, and relates it to my thermal interpretation of quantum physics.

Your comments are welcome.

 

[vanhees71]

I'll have to read the paper in detail, but I'm already a bit puzzled about the statements concerning the SG experiment. If there's one experiment that can be analyzed in the idealized way described by textbook-projective measurement scenarios, it's this experiment.

Of course you get not simply two lines but a distribution of silver atoms clustered around two lines. Of course this is due to the preparation of the silver beams via the emission from an oven going through apertures and then through a magnetic field. Where in all this one needs POVMs is not clear to me. You can calculate it by solving the time-dependent Schrödinger equation with the appropriate initial condition for the state entering the magnet.

Stern and Gerlach in their analysis didn't even need any quantum mechanics (which in fact didn't really exist at the time in complete form) to obtain, within an uncertainty of measurement of about 10%!) that the beam splits into two (not three as thought by Sommerfeld from Bohr quantization) partial beams (corresponding to two rather than three components of the magnetic moment related to the valence-electron's spin of a silver atom) and that the magnitude of the magnetic moment is $1\mu_{\text{b}}=\frac{e \hbar}{2mc}$ (in Gaussian or Heaviside-Lorentz units).

Of course the real puzzle connected with spin, as we know now, was the explanation of the Zeeman effect, particularly the "anomalous Zeeman effect" which is due to spin rather than orbital angular momentum, but that's another story.

 

[A. Neumaier]

Of course you get not simply two lines but a distribution of silver atoms clustered around two lines. Of course this is due to the preparation of the silver beams via the emission from an oven going through apertures and then through a magnetic field. Where in all this one needs POVMs is not clear to me.

Because the two partial beams overlap, putting a second magnet perpendicular to one of the partial beams (including the overlap region) does not give a zero signal, as it should if the measurement was projective. Thus POVMs are needed. Please read the references given for full details.

the SG experiment. If there's one experiment that can be analyzed in the idealized way described by textbook-projective measurement scenarios, it's this experiment.

It can be done only assuming idealized theoretical assumptions, not realized in the actual historical experiment.

 

[vanhees71]

Ok, there's an overlap region, but this you can make as unimportant as you wish by choosing appropriate setups. The overlap region is never 0 because the magnetic field cannot be in strictly only $z$ direction (because of $\vec{\nabla} \cdot \vec{B}=0$). In this sense you are right, and you need a POVM to give an accurate description of the setup and measurement.

 

[A. Neumaier]

Ok, there's an overlap region, but this you can make as unimportant as you wish by choosing appropriate setups. The overlap region is never 0 because the magnetic field cannot be in strictly only $z$ direction (because of $\vec{\nabla} \cdot \vec{B}=0$). In this sense you are right, and you need a POVM to give an accurate description of the setup and measurement.

Stern and Gerlach reported a significant overlap region. Their actual experiment has no interpretation in terms of Born's rule in textbook form (which wasn't even invented yet).

 

[vanhees71]

As I said, they used classical mechanics with a grain of Bohr to analyse the experiment with an estimated accuracy of about 10-20%. There was no modern QT, no Born's rule and for sure no POVMs ;-)).

 

[A. Neumaier]

As I said, they used classical mechanics with a grain of Bohr to analyse the experiment with an estimated accuracy of about 10-20%. There was no modern QT, no Born's rule and for sure no POVMs ;-)).

Yes, and they concluded their analysis by reporting quantization of direction, not of spin.

But I was talking of the modern interpretation of their original experiment. Like all interpretations of realistic experiments, it needs POVMs.

The relevance of my paper is that it is actually easier to introduce students to POVM measurements than to projective measurements. Both substantially less mathematical machinery is needed and substantially more generality is achieved.

 

[charters]

As an illustration we consider some piece of digital equipment with 3 digit display measuring some physical quantity A using N independent measurements. Suppose the measurement results were 6.57 in 20% of the cases and 6.58 in 80% of the cases. Every engineer or physicist would compute the mean A = 6.578, the variance [...] = 0.0042, and the standard deviation σA = 0.004, concluding that the true value of the quantity A deviates from 6.578 by an error of the order of 0.004N−1/2. Note that, as always when one measures something with a digital device, the error distribution is discrete, not Gaussian. Now we consider the measurement of the Hermitian quantity A ∈ C2× 2 of a 2-state quantum system in the pure up state, using N independent measurements, and suppose that we obtain exactly the same results. Now Born’s statistical interpretation proceeds differently and claims that there is no measurement error. Instead, each measurement result reveals one of the eigenvalues x1 = 6.57 or x2 = 6.58 in an unpredictable fashion.

...

However, Born’s statistical interpretation deviates radically from engineering practice, without any apparent necessity. It does not even conform to the notion of a measurement in the traditional sense since an essential element in the latter’s specification –the reproducibility of the result –is not guaranteed. Shouldn’t we rather proceed as before and draw the same conclusions as the engineer?

I want to suggest there is a tacit assumption motivating the Born approach, which I will call "Setup-Error Independence". SEI is simply the idea that a macroscopic measurement device's error/variance/deviation should be invariant under benign changes such as spacetime rotations and translations or alterations to spatially separated/unconnected components of the experiment. My bathroom scale's margin of error does not change whether I weigh myself or my dog; whether it is facing north or east; or whether my neighbors have their own scales or not.

Under the definition of measurement error you are proposing, SEI is not respected by the equipment used in basic quantum experiments.

For example, in the SG experiment, we can prepare our silver atoms with x-up spin, and then measure along the z axis to get the standard "two lip" results. But if we merely rotate the device to align with (or closer to) the x axis, the variance of the results will decrease. Just as rotating my scale 90 degrees doesn't make it more or less error prone, merely rotating an SG device should not change the variance either.

Another example is the MZI. First, in an open MZI, to even treat the distribution of outcomes as the measurement error of a "device", we would have to consider that device to be a non-local macroscopic object, since the two optical sensors can be arbitrarily far apart. This itself is a deviation from standard engineering, economic, and legal definitions, which would consider (and typically provide a manufacturer's warranty on) only the fidelity of each optical sensor separately. Even after allowing for a non locally defined device, it is still the case that just by inserting a (spatially separated) beamsplitter to close the MZI, we again reduce the device's variance, though the change we make seems far too innocuous to alter the function of the optical sensors.

In the Born approach, SEI is preserved as any measurement uncertainty contingent on these benign* tweaks to experimental setups is attributed to the fundamental quantumness of the target system, not the device. This has a significant appeal.

Perhaps SEI and related beliefs about how devices are defined will need to be revisited to make progress on the measurement problem. But nevertheless, I think making this assumption explicit suggests why the TI is not clearly a more conservative or less radical approach to measurement than the Born approach. Rather, as with most issues in quantum foundations, different bullets are being bit, and personal tastes will vary.

*Of course, there is no universal and rigorous definition of what makes a tweak benign, e.g. just wiggling or extending an antenna can improve radio reception, and a camera with a smudged lens takes blurry photos. But these are cases where the tweak is understood to lead to the device receiving a more or less robust signal, which is not our understanding in the SG or MZI.

 

[PeterDonis]

My bathroom scale's margin of error does not change whether I weigh myself or my dog; whether it is facing north or east; or whether my neighbors have their own scales or not.

But it certainly does change if you orient the scale vertically instead of horizontally. So your definition of SEI has a very important qualifier that you left out: it doesn't apply for any variations that affect the mechanism of operation of the measuring device.

in the SG experiment, we can prepare our silver atoms with x-up spin, and then measure along the z axis to get the standard "two lip" results. But if we merely rotate the device to align with (or closer to) the x axis, the variance of the results will decrease

Yes, and that's because the orientation of the SG device is part of its mechanism of operation. You should not expect SEI to hold if you change the SG device's orientation any more than you should expect it to hold if you orient your bathroom scale differently from horizontal.

Another example is the MZI.

This brings in a different set of issues related to nonlocality. I would not advise even trying to tackle those if you don't have a good understanding of the simpler case of a single localized measuring device.

 

[charters]

But it certainly does change if you orient the scale vertically instead of horizontally. So your definition of SEI has a very important qualifier that you left out: it doesn't apply for any variations that affect the mechanism of operation of the measuring device.

Not sure if you stopped reading halfway, but this is precisely the type of objection I was attempting to anticipate in my footnote. I am not advancing a completely universal definition of a "benign tweak", and you can certainly pedantically undermine my argument on these grounds. However, I think:

1) using a scale on its side is clearly not in the scope of the benign tweaks I am thinking about, nor would positioning an SG to not even face the beam be benign.

2) the paper invokes the idea of adherence to standard engineering practice, which is an unrigorous, common sense idea itself. So I am making an argument based on my view of these standard practices (which I believe are widely held), and pointing out what I see as a notable departure. I think, even if it is very hard to formally articulate in a universal way, we all appreciate the difference between a scale facing north or east, versus a scale laying flat or on its side. And I further expect most people will agree the the SG and MZI examples resemble the former - indeed I see this as part of the explanation for why the Born approach has been historically preferred.

I would not advise even trying to tackle those if you don't have a good understanding of the simpler case of a single localized measuring device

Well I believe I do have a good understanding of a single localized device, and my comment discussed the nonlocality issue.

 

[PeterDonis]

this is precisely the type of objection I was attempting to anticipate in my footnote

I don't see that your footnote addresses the issue I raised. It acknowledges that some changes are "benign" and others are not, but it doesn't fix the implied definition of the boundary between the two that you are using.

nor would positioning an SG to not even face the beam be benign

Changing the orientation of the SG at all, even if it continues to face the beam, is not benign, for the same reason that placing the scale vertically is not benign. If you don't understand that, I strongly suggest that you take a step back and think very carefully before making any further claims.

I further expect most people will agree the the SG and MZI examples resemble the former

Not if you are claiming that changing the orientation of a SG device from the z axis to the x axis, while still facing the beam, should be "benign". It obviously isn't. See above.

I believe I do have a good understanding of a single localized device

From what you've said so far, I don't think you have a good understanding of an SG device, because if you did, you would already have grasped the points I made above and in my previous posts.

 

[PeterDonis]

the paper invokes the idea of adherence to standard engineering practice, which is an unrigorous, common sense idea itself. So I am making an argument based on my view of these standard practices

If you are claiming that it is "standard engineering practice" to not care about the specific orientation of an SG device, I strongly suggest that you rethink that position.

 

[charters]

Changing the orientation of the SG at all, even if it continues to face the beam, is not benign, for the same reason that placing the scale vertically is not benign

I disagree. Placing a scale vertical is non-benign because it is now unable to properly receive its input. This is akin to an SG oriented so that the beam does not even feed into its aperture, e.g, the SG just being in the wrong part of the lab.

However, rotating a properly positioned SG along its axis of measurement, while keeping fixed the fact that the beam feeds into its aperture, is benign. This rotation does not alter the device's ability to receive its input, akin to how a scale facing either north or east can equally support an object on its platform.

 

[PeterDonis]

Placing a scale vertical is non-benign because it is now unable to properly receive its input.

You are quibbling. [Edit: See an expanded response in post #15.]

rotating a properly positioned SG along its axis of measurement, while keeping fixed the fact that the beam feeds into its aperture, is benign.

No, it isn't, because changing the orientation of the SG device changes which observable it is measuring.

If you continue to post along these lines, you will receive a warning and a thread ban.

 

[PeterDonis]

Placing a scale vertical is non-benign because it is now unable to properly receive its input.

No, it isn't. You could put the scale vertical and lie on the floor next to it and put your feet on it. It would register zero, and that reading would be correct, because you are not experiencing any proper acceleration or feeling any weight in the direction the scale's spring is oriented. In other words, placing the scale vertical means it is measuring a different observable than it is when it is horizontal, just as a SG device oriented in the x direction is measuring a different observable than one oriented in the z direction.

 

[charters]

You are just misreading what I am saying.

I am not talking about observed experimental outcomes or theory. Of course, rotating the SG changes the observed pattern, and of course rotating the SG measure a different Pauli operator in theory.

The issue here is to what we *attribute* this empirical observation and theoretical statement. Is it due to the nature of the measured quantum mechnical systems, as in the conventional approach? Or is it due to the nature of macroscopic measuring devices, as in the TI?

The claim in the paper is that the variance observed in quantum experiments can be understood as due to devices having inherent margins of error, and that this explanation holds closer to the normal treatment of measurement errors in engineering and device manufacture. My point is: the way that these margins of error vary across setups in quantum experiments is fairly different from conventional assumptions about devices, their margins of error, and how their error rates vary across setups. If we attribute the variance to the devices as the TI suggests, we are now thinking about devices and their margins of error in a substantially different way than we do in classical physics or in the existing quantum interpretations.

I am not saying this is necessarily a bad thing, just that it is not obvious to me this new account has the clear advantage of being more conservative or intuitive.

If you continue to post along these lines, you will receive a warning and a thread ban.

I have said nothing rude nor argued for a personal crackpot theory. It is poor form to choose to debate a point (and interpret me uncharitably) while simultaneously exercising your moderator powers, which should be used in a viewpoint neutral manner.

 

[PeterDonis]

I am not talking about observed experimental outcomes or theory.

Then I fail to see how anything you say is relevant to this thread, since it is about experimental outcomes and how theory models them. But surely you don't actually mean what you say here; your very next sentence is:

Of course, rotating the SG changes the observed pattern, and of course rotating the SG measure a different Pauli operator in theory.

Which certainly seems to be about an experimental outcome and theory.

I think you need to think more carefully about what you are saying.

The issue here is to what we *attribute* this empirical observation and theoretical statement.

Um, to the fact that we observe a change in the outcome and the theory accounts for it by saying we are measuring a different observable, so naturally we get a different outcome?

I have said nothing rude or argued for a personal crackpot theory.

Can you give a reference (textbook or peer-reviewed paper) for your idea of "setup-error independence"? If not, it's personal theory.

In any event, you don't have to say anything rude (which you haven't) or argue for a personal theory to get a misinformation warning or a thread ban; the scope of both is broader than that.

However, you are not getting either at this point, because you have now raised a question which appears, at least to me, to be a reasonable one for @A. Neumaier to respond to:

The claim in the paper is that the variance observed in quantum experiments can be understood as due to devices having inherent margins of error, and that this explanation holds closer to the normal treatment of measurement errors in engineering and device manufacture. My point is: the way that these margins of error vary across setups in quantum experiments is fairly different from conventional assumptions about devices, their margins of error, and how their error rates vary across setups. If we attribute the variance to the devices as the TI suggests, we are now thinking about devices and their margins of error in a substantially different way than we do in classical physics or in the existing quantum interpretations.

I am not saying this is necessarily a bad thing, just that it is not obvious to me this new account has the clear advantage of being more conservative or intuitive.

I don't have an opinion offhand about this one way or the other (@A. Neumaier is in a much better position to respond to it than I am), but I can say that your claims about "SEI" (which, as I noted above, is your personal theory unless you can give references for it) do not appear to me to be either necessary or relevant to raising the issue you are raising in the above quote. You could have just said what you said in the above quote in your first post in this thread, and I would have had no issue with it.

 

[vanhees71]

Yes, and they concluded their analysis by reporting quantization of direction, not of spin.

But I was talking of the modern interpretation of their original experiment. Like all interpretations of realistic experiments, it needs POVMs.

The relevance of my paper is that it is actually easier to introduce students to POVM measurements than to projective measurements. Both substantially less mathematical machinery is needed and substantially more generality is achieved.

I still have problems with the approach to claim that all that's measured are "q expectation values", which is simply not true. Also that I'm not allowed to use the usual statistical interpretation of these expectation values within your statistical interpretation, is an obstacle to use it as a didactical approach. I have nothing against POVMs, but I've not the impression that they are used much in analysis of the usual experiments as in high-energy physics. Maybe they are more necessarily needed in quantum information theory.

 

[A. Neumaier]

I still have problems with the approach to claim that all that's measured are "q expectation values", which is simply not true. Also that I'm not allowed to use the usual statistical interpretation of these expectation values within your statistical interpretation, is an obstacle to use it as a didactical approach. I have nothing against POVMs, but I've not the impression that they are used much in analysis of the usual experiments as in high-energy physics. Maybe they are more necessarily needed in quantum information theory.

Most of the paper is independent of the thermal interpretation, and just gives an exposition of POVMs without starting with the conventional assumptions.

But everything disussed fits much more easily with the thermal interpretation than with Born's eigenvalue interpretation. For the measurement of angular momentum, the eigenvalue interpretation requires that half multiples of $\hbar$ are measured - this is clearly the case only approximately. Thus the numbers in Born's interpretation are not measurement results but theoretical values only (part of the interpetation-free shut up and calculate part of QM), and Born's rule would say nothing about the relations to true measurement.

Reproducibilty is the main issue, essential for metrology - even in quantum metrology.
Reproducible are probabilities, which are q-expectations, and macroscopic field values, which again are q-expectations. Heisenberg began 1925 modern QM by insisting on that the theory should model the observable stuff only. Heisenberg's observables were transition probabilities, i.e., q-expectations - not the quantities dubbed observables in textbook expositions.

Individual observations are not reproducible hence qualify only as noisy measurements, reproducing the mean value (again a q-expectation) within a few standard deviations.

 

[vanhees71]

The outcome of the SG measurement was not the expectation value of the magnetic moment, which is 0, but $1 \mu_{\text{B}}$ with an accuracy of about 10%.

 

[A. Neumaier]

Also that I'm not allowed to use the usual statistical interpretation of these expectation values within your statistical interpretation, is an obstacle to use it as a didactical approach.

Did you read my paper? I carefully distinguish between the two. The q-expectations $\langle A\rangle$ are theoretical numbers figuring in the formal part of QM. The statistical expectation values $E(a_k)$ are experimental quantities computed (in the limit of an arbitrarily large number of repetitions) from the actual outputs $a_k$ of a detector. In case of real-valued outputs, they are related by $E(a_k)=\langle A\rangle=\bar A$ for some operator $A$, typically an approximation of the operator corresponding to the observable one wishes to measure. But the corresponding variances only satisfy $E((a_k-\bar A)^2)\ge\langle (A-\bar A)^2\rangle$, in real experiments typically with strict inequality. Thus statistical expectations and q-expectations are mathematically different objects, with different properties.

Also, if they weren't it would not even make sense to interpret Kennard's uncertainty inequality in the traditional textbook - i.e., Heisenberg's - way as a limit of joint approximate measurability of position and momentum. Joint approximate measurement would even be an impossibility in Born's interpretation without POVMs. But whenever we measure momentum, we measure it of an object with a roughly observable position. Hence such measurements are obviously possible.

 

[vanhees71]

In standard QT the usual uncertainty relation you quote in your paper has not (!) the meaning of a joint approximate measurability of position and momentum but it describes a statistical property of the joint preparability of position and momentum.

In standard QT to verify it experimentally for a given state (preparation procedure!) you have to repeat the preparation and then measure the position accurately and then you repeat the preparation and then meausure position accurately. In the ideal case of accurate measurements you get the qm. standard deviations for position and momentum, satisfying the uncertainty relation.

Of course there are strictly speaking no mathematically accurate measurements of position or momentum. So to be more precise you have to say that you have to measure position and momentum with a much higher resoluation than the expected QM standard deviations of these quantities to get a sufficiently accurate verification of the quantum mechanical predictions (the same holds for the statistical error, which you have to bring down by a sufficient number of repetitions of the preparation and measurement procedure).

The joint (then necessarily inaccurate!) measurement of position and momentum is a far more complicated problem and needs specific analysis of the specific measurement done. In my understanding it's in such usually quite complicated cases the POVM approach is necessary, and thus it's for sure not an approach for the introductory QM lecture.

 

[A. Neumaier]

I want to suggest there is a tacit assumption motivating the Born approach, which I will call "Setup-Error Independence". SEI is simply the idea that a macroscopic measurement device's error/variance/deviation should be invariant under benign changes such as spacetime rotations and translations or alterations to spatially separated/unconnected components of the experiment. [...]

Under the definition of measurement error you are proposing, SEI is not respected by the equipment used in basic quantum experiments.

For example, in the SG experiment, we can prepare our silver atoms with x-up spin, and then measure along the z axis to get the standard "two lip" results. But if we merely rotate the device to align with (or closer to) the x axis, the variance of the results will decrease. Just as rotating my scale 90 degrees doesn't make it more or less error prone, merely rotating an SG device should not change the variance either.

SEI has never been a requirement of metrology; I haven't seen it mentioned in the metrology textbook by Rabinovich quoted in my paper.

One has the same lack of SEI in the classical measurement of a cigar as in the quantum measurement of angular momentum.

Seen from far apart, the position of the cigar is in the upper left corner of the table, with a measurement error of roughly 10 cm, and the intrinsic angular momentum of a silver atom is zero with a measurement error of roughly $\hbar$. Measuring positionor angular momentum in some direction closely enough, the result and the uncertainty strongly depends on the direction chosen and on the state of the of the object measured - the shape determined by the detailed state of the cigar and of the ellipsoid determined by the detailed state of the silver atom.

 

[A. Neumaier]

In standard QT the usual uncertainty relation you quote in your paper has not (!) the meaning of a joint approximate measurability of position and momentum but it describes a statistical property of the joint preparability of position and momentum.

Mathematiclly speaking you are correct, but Heisenberg considered it to be the former, and the Heisenberg microscope is traditional textbook illustration material.

Of course there are strictly speaking no mathematically accurate measurements of position or momentum. So to be more precise you have to say that you have to measure position and momentum with a much higher resolution than the expected QM standard deviations

As you see, even you need to distinguish between the statistical concept (resolution) and the q-expectation version (QM standard deviation), since they disageee in your situation. My paper just makes this very explicit, instead of sweeping it under the carpet.

The joint (then necessarily inaccurate!) measurement of position and momentum is a far more complicated problem and needs specific analysis of the specific measurement done. In my understanding it's in such usually quite complicated cases the POVM approach is necessary, and thus it's for sure not an approach for the introductory QM lecture.

My paper introduces all these issues in a clear and elementary way suitable for an introductory course, and in a way that prevents a lot of later confusion!

 

[vanhees71]

Mathematiclly speaking you are correct, but Heisenberg considered it to be the former, and the Heisenberg microscope is traditional textbook illustration material.

Yes, and Heisenberg was wrong and was corrected immediately by Bohr at the time.

As you see, even you need to distinguish between the statistical concept (resolution) and the q-expectation version (QM standard deviation), since they disageee in your situation. My paper just makes this very explicit, instead of sweeping it under the carpet.

No, I follow the distinction in terms of the standard meaning according to which the state describes the "preparation" of a system (where preparation has a wide meaning, e.g., it can also be simply a rough information about a macroscopic status like measuring the temperature and density of a gas leading to a description in terms of the (grand-)canonical equilibrium stat. op.). The outcome of measurements of any given observable is then described propabilistically, i.e., the knowledge of the state allows to predict the probabilities for the outcome of measurements. In the original sense of Born's rule in terms of what in the context of measurement theory is called an exact measurement; in the ideal case it's even a projective measurement, but that's not necessary, as e.g., usually photons are absorbed in a precise measurement.

In my opinion to understand and define the much more complicated notion of a POVM you need to rely on these physical (sic!) foundations of QT.

My paper introduces all these issues in a clear and elementary way suitable for an introductory course, and in a way that prevents a lot of later confusion!

In your paper you just give an axiomatic mathematical scheme. This is something you can understand in a very specialized lecture after the students have heard at least the two standard course lectures (QM I and II).

 

[A. Neumaier]

In your paper you just give an axiomatic mathematical scheme. This is something you can understand in a very specialized lecture after the students have heard at least the two standard course lectures (QM I and II).

The axiomatic mathematical scheme in my paper simply replaces Born's rule in the traditional axiomatic scheme figuring in the standard courses on QM. My paper is selfcontainedd and leads to all important facts about quantum measurement (including those not obtainable by Born's interpretation). In particular, no prior knowledge of quantum mechanics is needed.

In my opinion to understand and define the much more complicated notion of a POVM you need to rely on these physical (sic!) foundations of QT.

Please substantiate your claim by pointing out the places in my paper where these physical foundations of QT that you allude to are relied upon.

 

[vanhees71]

Your paper is fine as a scientific paper (though I still don't believe that it's physically sensible to claim that what's measured are q-expectation values or approximations thereoff instead of observables as in the standard formulation of QT). The only thing I argue against is the claim that this can be used as an approach to teach students in the introductory QM lecture.

 

[A. Neumaier]

Your paper is fine as a scientific paper (though I still don't believe that it's physically sensible to claim that what's measured are q-expectation values or approximations thereoff instead of observables as in the standard formulation of QT). The only thing I argue against is the claim that this can be used as an approach to teach students in the introductory QM lecture.

For the purpose of an introductory course on QM, one can stop after Section 3.2 of my paper. Then one remains completely within the tradition and still has a fully elementary introduction to POVMs including the special case of projective measurement. What would be too difficult for an introductory course??

The relation with the thermal interpretation is just a suggestive afterthought, for those who are not committed to taking the tradition for the ultimate truth. Voluntary reading for the students.

 

[vanhees71]

In math you start with a bunch of axioms and definition and derive theorems and lemmas, though I'm not so sure whether this is really good math teaching either. Too much Bourbakism may teach you a lot of ready mathematical structures in a very systematic way, but does it teach you to do math and create new math yourself? I doubt it.

In physics you have the additional "difficulty" to describe real-world observations, also as a theoretical physicist. I've no clue, how you should understand an abstract concept with some operations with vectors and operators in a Hilbert space without having the idea about its connection to the stuff in the lab and beyond.

Also I still do not understand, why you insist on giving up the standard treatment of observables as described by self-adjoint operators in Hilbert space with their spectrum giving the possible values they can take and the standard Born rule, no matter, whether you first introduce pure states only as usually done or already much earlier the somewhat more complicated idea of a general state as a statistical operator; I must admit I have not found a really good way to teach the stat-op concept except the standard one by introducing the pure states first and then the stat. op. as a "mixture" (which is highly artificial though, because I'm not aware that any standard experiment provides a mixture in such a way).

The reason for my rejection of the idea to treat q-expecation values as the observable quantitities is that it seems not to be correct to me. Maybe I still misunderstand your concept, but taken at phase value I think it would imply that in the Stern-Gerlach experiment you would expect one broad spot/line around 0 (the expectation value $9$ of $s_z$ in the usual setup with a silver-atom beam from an oven), which is just somewhat broadend with magnetic field compared to the control measurement with 0 magnetic field rather than the two discrete lines the real experiment shows. This was really a breakthrough at the time, because it really showed that the idea of "quantization of direction" ("Richtungsquantelung") was in principle correct though the possibility of spin 1/2 wasn't dicovered yet nor the Lande-Faktor of about 2 was clearly understood (though known from the correction of the wrong Einstein-de Haas measurement (1915) by Barnett's measurement using the reverse effect (also 1915)).

 

[A. Neumaier]

In math you start with a bunch of axioms and definitions and derive theorems and lemmas,

In your introduction to quantum physics, you start with a bunch of unmotivated notation, definitions and postulates:

Wir wollen diese Formulierung der Quantenmechanik axiomatisch an die Spitze stellen.

What you present is much worse than what is in my paper. For example, compare Sections 1.1-1.2 of my paper with p.13 (second half) to p.15 (first half) of your introductory lecture notes Prinzipien der Quantentheorie. You throw lots of (for the uninitiated) formal gibberish at the student that becomes intelligible physically only much later. You use the advanced notion of selfadjointness without even giving a definition. That (2.2) is well-defined is a lemma (though not called so) that you formally prove. A few pages later you also prove a formally stated theorem (Satz 4), with a nontrivial technical proof extending over two full pages. Why isn't this too much Bourbakism? I claim that my axioms, definitions and theorems are much more elementary than yours!

In your very recent lecture notes for teachers you proceed similarly, presenting at the start three abstract postulates motivated by perfectly polarized light (similar to my motivation, but mine works with the more realistic case of partially polarized light) :

Diese sehr abstrakten Postulate werden wir I am Folgenden noch genauer analysieren. Es ist erfahrungsgemäß recht schwierig, die physikalische Bedeutung des Formalismusses zu verstehen, aber man gewöhnt sich mit der Zeit an diese „quantenphysikalische Denkweise“, die radikal mit den gewohnten Begriffen der klassischen Physik bricht.

Again you use the notion of selfadjointness without explaining its meaning. I never need this very advanced notion. On p.14 you state and prove two lemmas (without calling them lemmas, but this makes no difference). Then you have several pages with lots of formal computations to establish basic properties. Why isn't all this too much Bourbakism? Compare your exposition with what I need in my paper!

though I'm not so sure whether this is really good math teaching either. Too much Bourbakism may teach you a lot of ready mathematical structures in a very systematic way, but does it teach you to do math and create new math yourself? I doubt it.

Where in my paper is there any Bourbakism? Unlike Bourbaki I motivate in a physical way all concepts and results introduced. That I label some results 'Theorem' and their derivations 'Proof' is just a matter of style - I believe it helps to be clear. One could as well delete the word 'Theorem' and replace the word 'Proof' by 'Indeed', without any difference in the contents conveyed.

In physics you have the additional "difficulty" to describe real-world observations, also as a theoretical physicist. I've no clue, how you should understand an abstract concept with some operations with vectors and operators in a Hilbert space without having the idea about its connection to the stuff in the lab and beyond.

I have no clue where I introduced abstract concepts without having connected them to the stuff in the lab and beyond. In addition to the motivating polatization experiments, my paper contains a whole chapter with physical examples!

Also I still do not understand, why you insist on giving up the standard treatment of observables as described by self-adjoint operators in Hilbert space with their spectrum giving the possible values they can take and the standard Born rule

Because

• lots of observable items (spectral line widths, decay rates, reaction cross sections, coupling constants) are not described by self-adjoint operators in Hilbert space, and
• the standard Born rule is almost universally only approximately valid, and in many cases not at all.

I must admit I have not found a really good way to teach the stat-op concept except the standard one by introducing the pure states first and then the stat. op. as a "mixture" (which is highly artificial though, because I'm not aware that any standard experiment provides a mixture in such a way).

Yes, indeed.

Compare this with the introduction of the density operator in my paper. Everything is very naurally motivated from classical polarization. The Stokes vector does not generalize but the polarization tensor does, and has simple composition rules. Hence it is the natural object of study.

The reason for my rejection of the idea to treat q-expecation values as the observable quantitities is that it seems not to be correct to me. Maybe I still misunderstand your concept

Yes, you make an additional assumption that I do not make, and that causes you trouble:

but taken at face value I think it would imply that in the Stern-Gerlach experiment you would expect one broad spot/line around 0 (the expectation value

This would be the case if I had claimed that the errors are necessarily Gaussian distributed, which you seem to assume. But it is known that measurement errors may have very different statistics, depending on what one measures.

Instead I claim that the errors have a bimodal distribution, which is indeed what one observes. The expectation value of such a distribution need not be close to one of the peaks.

 

[vanhees71]

I admit that I haven't found a satisfactory introduction to QM 1 either, but in your paper you simply define q-expectation values using the most general notion of a state as a statistical operator. I don't believe that this is simpler for the beginners of QT than the more standard approach starting with pure states.

The approach with polarized light, using just plane waves, is the most simple I can think of. It's also already known from the elctrodynamics lecture. To use the complete Stokes-vector formalism seems also a bit overdone for an introductory lecture.

That's not a criticism against your paper per se, but I doubt that this approach starting from the most general and sophisticated case is useful for the introductory course.

 

[A. Neumaier]

in your paper you simply define q-expectation values using the most general notion of a state as a statistical operator. I don't believe that this is simpler for the beginners of QT than the more standard approach starting with pure states.

It is not postulated but derived by a simple linearity argument, and has a very intuitive form. Whereas Born's original formula is quadratic and strange at first sight - Born even got it wrong initially!

In addition, Born's approach needs the whole spectral machinery including multiplicities to produce the expectation formula, quite a messy process.

 

[A. Neumaier]

The approach with polarized light, using just plane waves, is the most simple I can think of. It's also already known from the elctrodynamics lecture. To use the complete Stokes-vector formalism seems also a bit overdone for an introductory lecture.

In an introductory lecture in quantum mechanics, I'd regard it to be a bit overdone using as motivation a very special case of the Maxwell equations, which govern a completely different field. The latter are appropiate as motivation when one is going to describe QED, which merges the two fields!

Just as one commonly states the results of the Stern-Gerlach experiment without prior fundamental theory, the simple laws for partially polarized light can also be introduced in a purely phenomenological way. When asking about the physical description of ordinary light, students should not be left in the dark!

Unpolarized light is much more common than completely polarized light. The laws that describe both polarized and unpolarized light are easly stated; they were known since 1852, even before Maxwell's equations were formulated. Just as Stern-Gerlach was known before its explanation through spin and the Schrödinger equation.

 

[A. Neumaier]

I admit that I haven't found a satisfactory introduction to QM 1 either,

You should not criticize my work on the basis of standards that you are not able to follow yourself.

What I proposed in my paper is a detailed, fully working and well motivated alternative to the traditional approach to quantum mechanics. My proposal may not be perfect but it is as good as the latter, and has the advantages of greater simplicity anf greater generality.

 

[RUTA]

The discussion about how one should best introduce QM is interesting. With applications to nuclear physics, solid state physics, quantum optics, quantum information/computing, particle physics, and foundations to name a few, one could start almost anywhere. There are three of us teaching intro courses in these areas for engineers at my small undergrad institution and there is very little overlap in content between the courses, even though all the courses analyze actual experiments and/or applications.

 

[vanhees71]

You should not criticize my work on the basis of standards that you are not able to follow yourself.

What I proposed in my paper is a detailed, fully working and well motivated alternative to the traditional approach to quantum mechanics. My proposal may not be perfect but it is as good as the latter, and has the advantages of greater simplicity anf greater generality.

You asked for comments. I'm sorry if you take it as undue criticism.

 

[A. Neumaier]

You asked for comments. I'm sorry if you take it as undue criticism.

Well, it is the way you phrase your comments. For example you wrote:

In your paper you just give an axiomatic mathematical scheme. This is something you can understand in a very specialized lecture after the students have heard at least the two standard course lectures (QM I and II).

But in two different introductory courses you also just gave an axiomatic mathematical scheme! This double standard is a bit unfair, even when only regarded as a comment.

 

[vanhees71]

The discussion about how one should best introduce QM is interesting. With applications to nuclear physics, solid state physics, quantum optics, quantum information/computing, particle physics, and foundations to name a few, one could start almost anywhere. There are three of us teaching intro courses in these areas for engineers at my small undergrad institution and there is very little overlap in content between the courses, even though all the courses analyze actual experiments and/or applications.

The problem is to get the radically new physical thinking as compared to classical physics which (apparently) is straight-forward from the very first moment. I don't think that just to put the "trace formula" for expectation values on top of a axiomatic is good for the introductory course lecture. It's very good for a treatment in an advanced lecture on quantum foundations, quantum information, or something similar.

The criticism about my "photon-polarization first" approach is also well justified. I made this choice, because the lecture about electrodynamics, which is part of my course on theoretical physics for high-school teachers, ends with a quite complete treatment of electromagnetic waves, including a discussion of plane-wave solutions, polarization and also of course diffraction on slits and gratings.

Then my idea was that it is best to start with the most simple case of a 2D quantum-mechanical Hilbert space, as is done in many textbooks I like using spin 1/2 and the Stern-Gerlach experiment. This is however in my opinion also not so good for an introductory QM lecture, i.e., one where the students never have heard about spin before, i.e., and that's why I start rather with the polarization of electromagnetic waves and then argue heuristically (and I don't think that you can introduce QM at this level in any sensible way purely deductively without a big portion of heuristics) what happens if you use single-photon sources (without of course being able to explain the subtle and in fact important difference between "dimmed laser light" and real one-photon Fock states). I see the advantage in the fact that you have all the material of electromagnetism lecture at hand, particularly the fact that the most intuitive observable of light, which is intensity (i.e., "brightness" as a qualitative entity), is an adequate (temporal) average over the energy density of the electromagnetic field, i.e., for a plane wave something $\proportional |\vec{E}^2|$.

Then you have with the most simple example of using polaroid foil(s) doing experiments with polarized light (which is nowadays really no problem to do even in the introductory expermental-physics lecture, since one has lasers at hand to produce nicely coherent light; even with "natural light" it's no big deal to get polarized light using a polaroid filter) all the arguments at hand to motivate the QT formalism:

-polarization states as preparation procedure: Just take a polaroid to provide a source of linearly polarized light in a given direction

-self-adjoint operators as representants of polarization states: Just use another polaroid and describe it as projection operarator

-the superposition principle and complete orthonormal systems: You work within a framework of linear optics, where the em. field equations are linear equations, and you can find new solutions by superposition and even all solutions in terms of complete sets of orthonormal function systems (a concept also known from the E&M lecture, where this is treated in the context of Fourier series and Fourier integrals).

Of course there are also some drawbacks of this method, because it introduces photons at the introductory stage, and there's some danger that the students get the wrong idea as if one could treat photons in a kind of 1st-quantization approach, which of course then follows directly after the introduction with photon polarization, where the Hilbert space for one particle is introduced in the usual heuristics a la Schrödinger (in a very simplified form, not taking recourse to the Hamilton-Jacobi theory of classical mechanics).

 

[vanhees71]

Well, it is the way you phrase your comments. For example you wrote:

But in two different introductory courses you also just gave an axiomatic mathematical scheme! This double standard is a bit unfair, even when only regarded as a comment.

The only manuscript for an introductory lecture is the manuscript for teachers students, and with Theorie III I'm not really satisfied, but this is far from an axiomatic treatment. It's very heuristic and hand-waving. It's not even complete, as indeed in an axiomatic treatment you have to start with statistical operators, and I even think that for that purpose your axiomatics is pretty elegant (though I still don't believe that one should identify the q-expectation values with observables, for the already mentioned reason).

 

[RUTA]

The problem is to get the radically new physical thinking as compared to classical physics which (apparently) is straight-forward from the very first moment. I don't think that just to put the "trace formula" for expectation values on top of a axiomatic is good for the introductory course lecture. It's very good for a treatment in an advanced lecture on quantum foundations, quantum information, or something similar.

The criticism about my "photon-polarization first" approach is also well justified. I made this choice, because the lecture about electrodynamics, which is part of my course on theoretical physics for high-school teachers, ends with a quite complete treatment of electromagnetic waves, including a discussion of plane-wave solutions, polarization and also of course diffraction on slits and gratings.

Then my idea was that it is best to start with the most simple case of a 2D quantum-mechanical Hilbert space, as is done in many textbooks I like using spin 1/2 and the Stern-Gerlach experiment. This is however in my opinion also not so good for an introductory QM lecture, i.e., one where the students never have heard about spin before, i.e., and that's why I start rather with the polarization of electromagnetic waves and then argue heuristically (and I don't think that you can introduce QM at this level in any sensible way purely deductively without a big portion of heuristics) what happens if you use single-photon sources (without of course being able to explain the subtle and in fact important difference between "dimmed laser light" and real one-photon Fock states). I see the advantage in the fact that you have all the material of electromagnetism lecture at hand, particularly the fact that the most intuitive observable of light, which is intensity (i.e., "brightness" as a qualitative entity), is an adequate (temporal) average over the energy density of the electromagnetic field, i.e., for a plane wave something $\proportional |\vec{E}^2|$.

Then you have with the most simple example of using polaroid foil(s) doing experiments with polarized light (which is nowadays really no problem to do even in the introductory expermental-physics lecture, since one has lasers at hand to produce nicely coherent light; even with "natural light" it's no big deal to get polarized light using a polaroid filter) all the arguments at hand to motivate the QT formalism:

-polarization states as preparation procedure: Just take a polaroid to provide a source of linearly polarized light in a given direction

-self-adjoint operators as representants of polarization states: Just use another polaroid and describe it as projection operarator

-the superposition principle and complete orthonormal systems: You work within a framework of linear optics, where the em. field equations are linear equations, and you can find new solutions by superposition and even all solutions in terms of complete sets of orthonormal function systems (a concept also known from the E&M lecture, where this is treated in the context of Fourier series and Fourier integrals).

Of course there are also some drawbacks of this method, because it introduces photons at the introductory stage, and there's some danger that the students get the wrong idea as if one could treat photons in a kind of 1st-quantization approach, which of course then follows directly after the introduction with photon polarization, where the Hilbert space for one particle is introduced in the usual heuristics a la Schrödinger (in a very simplified form, not taking recourse to the Hamilton-Jacobi theory of classical mechanics).

There is nothing wrong with "heuristics," indeed we must use intuitively understood, but strictly undefined, concepts from the outset in physics, e.g., normal force, string tension, friction. Arguably, since we still do not have a unified axiomatic basis for all of physics, heuristics are unavoidable. Your approach is pedagogically sound because it introduces the new topic starting with what the student already knows. Serway and Jewett's intro physics text introduces the wave function using $u \propto \vec{E}^2$ from E&M and $E = hf$ assuming monochromatic radiation, then asking how many photons exist per unit volume. It's a reasonable (and heuristic) way to introduce the probability amplitude.

 

[Morbert]

As an aside: I think a straightforward link can be made between POVMs associated with detectors referenced in OPs paper, and the typical von Neumann notion of properties/measurement results as projectors. Given the POVM $\{P_k\}$ associated with measurement outcomes $\{k\}$, we can construct a spectral decomposition for each $P_k$ $$P_k = \sum_j\lambda_{jk}\xi_{jk}$$ (where $\lambda_{jk}$ are scalars and $\xi_{jk}$ are projectors) such that if we observe measurement outcome $k$, we can infer property $\xi_{ik}$ with probability $$\lambda_{ik}\frac{\mathbf{Tr}\left[\rho\xi_{ik}\right]}{\mathbf{Tr}\left[\rho P_k\right]}$$
where $\rho$ is the preparation as usual.

 

[A. Neumaier]

As an aside: I think a straightforward link can be made between POVMs associated with detectors referenced in OPs paper, and the typical von Neumann notion of properties/measurement results as projectors. Given the POVM $\{P_k\}$ associated with measurement outcomes $\{k\}$, we can construct a spectral decomposition for each $P_k$ $$P_k = \sum_j\lambda_{jk}\xi_{jk}$$ (where $\lambda_{jk}$ are scalars and $\xi_{jk}$ are projectors) such that if we observe measurement outcome $k$, we can infer property $\xi_{ik}$ with probability $$\lambda_{ik}\frac{\mathbf{Tr}\left[\rho\xi_{ik}\right]}{\mathbf{Tr}\left[\rho P_k\right]}$$
where $\rho$ is the preparation as usual.

But what should be the meaning of this property?

 

[A. Neumaier]

Yes, and Heisenberg was wrong and was corrected immediately by Bohr at the time.

You got the history completely wrong - please read (or reread) some of the old papers! Bohr criticised details of Heisenberg's argument, not the general principle; see this post. In Bohr's paper

• N. Bohr, The Quantum Postulate and the Recent Development of Atomic Theory, Nature Suppl., April 14, 1928, 580-590. (German version: Das Quantenpostulat und die neuere Entwicklung der Atomistik, Naturwissenschaften 16(1928), 245-257.)

he says explicitly on p.583:

The product of the least inaccuracies with which the positional co-ordinate and the component of momentum in a definite direction can be ascertained is therefore just given by formula (2).

Thus Bohr reaffirms Heisenberg's statement that you had wrongly claimed Bohr had corrected. Such joint measurements of position and momentum with limited accuracy cannot be described in terms of Born's statistical interpretation.

That Heisenberg's three views of the uncertainty relation (limits on preparation, unavoidable disturbance through measurement, and uncertainty of joint measurements) are all valid according to the modern understanding is discussed in detail in the references in this post from another thread.

 

[Morbert]

But what should be the meaning of this property?

I don't think it would necessarily have a classical interpretation, other than perhaps the implication that there is in principle a perfect apparatus that can resolve the alternatives $\{\xi_{jk},I-\xi_{jk}\}$. Questions about the nature of quantum properties in general likely lead us into the usual candy shop of QM interpretations.

 

[*now*]

You got the history completely wrong - please read (or reread) some of the old papers! Bohr criticised details of Heisenberg's argument, not the general principle; see this post. In Bohr's paper

• N. Bohr, The Quantum Postulate and the Recent Development of Atomic Theory, Nature Suppl., April 14, 1928, 580-590. (German version: Das Quantenpostulat und die neuere Entwicklung der Atomistik, Naturwissenschaften 16(1928), 245-257.)

he says explicitly on p.583:

Thus Bohr reaffirms Heisenberg's statement that you had wrongly claimed Bohr had corrected. Such joint measurements of position and momentum with limited accuracy cannot be described in terms of Born's statistical interpretation.

That Heisenberg's three views of the uncertainty relation (limits on preparation, unavoidable disturbance through measurement, and uncertainty of joint measurements) are all valid according to the modern understanding is discussed in detail in the references in this post from another thread.

What is your view of the Bohr Rosenfeld arguments and so forth?

 

[A. Neumaier]

What is your view of the Bohr Rosenfeld arguments and so forth?

Which of these arguments relate to the present thread?

 

[*now*]

To start, Zur Frage der Messbarkeit der elektromagnetischen Feldgrössen, 1933.

 

[A. Neumaier]

To start, Zur Frage der Messbarkeit der elektromagnetischen Feldgrössen, 1933.

The thermal interpretation is consistent with the observation that only smeared (q-expectations of) fields without significant spatial or temporal high frequency oscillations can be measured reliably.

 

[*now*]

I see, thank you.