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

[A. Neumaier]

I simply like to understand, what's behind this POVM idea in a physical context rather than a mathematical abstract concept.

The physical idea behind POVMs is just the detector response principle discussed in Section 2.2 of my paper, together with the statement of Theorem 2.1. The proof is relevant only for those who want to understand the mathematical concept behind.

 

[vanhees71]

My problem with your paper that not at a single place you give a concrete description. It's only qualitative. I don't know, how you describe even a Stern-Gerlach experiment with a POVM as long as you don't relate the mathematical operators defining it in theory to the real-world setup in, say, a student lab where this experiment is done.

In the standard description you simply predict the probability for finding the silver atoms on the screen after having gone through the magnet and compare it to what's measured. The agreement is not great but satisfactory.

The high-accuracy version of measuring magnetic moments with a Penning trap, which we also have discussed above, is also described in such a way, but also in this case you didn't derive the POVM from this setup but just describe something with words. It's not better than the standard descriptions of experiments about quantum objects, and indeed you are right, mostly the measurement device is treated with classical physics, but that's not so surprising since measurement devices are macroscopic objects which are well described by classical physics. But that still doesn't answer my question, how to get a concrete POVM. Of course also the other direction would be interesting, i.e., how to design an experiment for a given POVM. But also this I've not seen yet anywhere.

 

[vanhees71]

The physical idea behind POVMs is just the detector response principle discussed in Section 2.2 of my paper, together with the statement of Theorem 2.1. The proof is relevant only for those who want to understand the mathematical concept behind.

The math doesn't seem so much more difficult than the standard QT description. The problem is that it's not clear how to make the connection with equipment in the lab, which is not a problem in the standard description at all.

 

[A. Neumaier]

I don't know, how you describe even a Stern-Gerlach experiment with a POVM

The experiment is described and explained just using shut up and (semiclassical) calculate! Probablilities are not needed, only beam intensities. Using either Born's rule and POVMs would be theoretical overkill.

Stern and Gerlach had neither POVMs nor Born's rule, and their experiment (including variations) can be interpreted without any probability arguments; the only information needed beyond semiclassical models is the fact that with low intensity beams one needs a longer exposure to produce the detailed pattern.

The math doesn't seem so much more difficult than the standard QT description.

In fact the math is both simpler and more general, and hence to be preferred over Born's rule. No eigenvalue stuff is needed.

The problem is that it's not clear how to make the connection with equipment in the lab, which is not a problem in the standard description at all.

One makes the connection with equipment without any reference to either POVM or Born's rule. Instead one refers to known properties of the equipment and the established relations to theory.

 

[vanhees71]

The "eigenvalue stuff" however gives a straight-forward relation between the mathematical abstract object (self-adjoint operator on a Hilbert space) to physical quantities/observables: values you find when measuring the observable accurately. For the POVM I've not seen any such simple relation between physical quantities and the mathematical description.

 

[A. Neumaier]

The "eigenvalue stuff" however gives a straight-forward relation between the mathematical abstract object (self-adjoint operator on a Hilbert space) to physical quantities/observables: values you find when measuring the observable accurately.

Not really.

What is measured in a Stern-Gerlach experiment is simply the silver intensity on the screen. To interpret the latter as a spin measurement you need to invoke the quantum mechanical model of the whole setting - and this in the shut up and calculate version only. Eigenvalues or Born's rule are not involved here at all! So one doesn't expect a use for POVMs either.

 

[vanhees71]

Of course you have to invoke QT to understand the SGE. In fact it was one of the key experiments to convince many physicists about the necessity to give up classical concepts. At the time the underlying model was of course pretty wrong. At best, one would have expected three rather than only two lines (Sommerfeld), but somehow Bohr mumbled something and predicted only two lines. That's why Stern wrote a postcard to Bohr saying "you are right". Concerning the amount of the deflection the two wrong ingredients cancelled, i.e., they assumed orbital angular momenta within the Bohr-Sommerfeld quantization ("old quantum theory") and a gyro-factor 1. Nowdays we know it's spin 1/2 (which was however discovered only in 1926 by Goudsmit and Uhlenbeck after Kramers was persuaded by Pauli not to publish his correct idea ;-)) and a gyro-factor very close to 2.

Of course the eigenvalues of the spin component (magnetic moment, which is proportional to it) in direction of the magnetic field are involved, leading to the prediction of two strips on the screen. In the standard description for an electron moving through an inhomogeneous magnetic field of the right kind it's simply that the magnetic fields leads to an entanglement between position and this spin component, i.e., an Ag-atom beam splits in two pieces which are pretty well separated, and in one beam are with almost 100% probability spin-up and in the other spin-down Ag-atoms. Blocking one beam is an almost perfect preparation for Ag-atoms with determined spin components.

 

[A. Neumaier]

Of course the eigenvalues of the spin component (magnetic moment, which is proportional to it) in direction of the magnetic field are involved, leading to the prediction of two strips on the screen.

But they are involved in the dynamics of the particles, not in their measurement. Thus their appearance is independent of the measurement itself (which only involves the screen) and no eigenvalues.

 

[vanhees71]

The point is that the measurement of the Ag atoms position is in 100% correlation to the spin component due to the preparation of the beam, and that's why the position measurement can be interpreted as a (pretty) precise measurement of the spin component. What has this to do with using a POVM to describe this measurement process?

 

[A. Neumaier]

The point is that the measurement of the Ag atoms position is in 100% correlation to the spin component due to the preparation of the beam, and that's why the position measurement can be interpreted as a (pretty) precise measurement of the spin component. What has this to do with using a POVM to describe this measurement process?

But my point is that the model dynamics already predicts exactly two silver beams at the right spots, with intensities given by the preparation. This only depends on reversible quantum theory - the Schrödinger equation and the definition of intensity analogous to what my quantum tomography paper does for photons in Section 2.1. So POVMs or projection operators don't enter at this stage.

The whole setting is in principle reversible, hence no measurement has taken place. Therefore this part cannot involve Born's rule, which is explicitly about measurement, and has nothing to say in the reversible case.

The subsequent measurement of the silver intensities on the screen is a purely classical process, of the same kind as measuring where and how much paint falls on a screen when sprayed with two faint beams of paint.

Thus the whole experiment nowhere needs quantum measurements for its quantitative understanding! Hence Born's rule is not needed anywhre!

Of course you can mumble 'Born's rule is involved in any measurement'. But this doesn't change anything and adds nothing to the explanation.

 

[vanhees71]

But my point is that the model dynamics already predicts exactly two silver beams at the right spots, with intensities given by the preparation. This only depends on reversible quantum theory - the Schrödinger equation and the definition of intensity analogous to what my quantum tomography paper does for photons in Section 2.1. So POVMs or projection operators don't enter at this stage.

Yes, and I don't understand, where POVMs are needed to explain real-world experiments. That's why I'm asking!

For me all QT does is to predict the probabilities for the outcome of measurements, given a preparation procedure. Here the preparation procedure is to produce a silver-atom beam with an oven and letting this beam go through an inhomogeneous magnetic field, which is designed to split the beam with high accuracy in two partial beams, where the position in a certain region is entangled with the value of the spin component in direction of this magnetic field.

The measurement itself, irreversibly storing the result, is when the silver atoms are absorbed at the photoplate making a spot at the corresponding place. Then you can count the spots and compare with the probability distribution predicted by the calculation. Indeed there's no POVM needed, and the "projection operators" I use are simply to calculate the probability distribution by Born's rule, $P(\vec{x})=\mathrm{Tr} (\hat{\rho} |\vec{x} \rangle \langle x|)=\langle \vec{x} |\hat{\rho} \vec{x} \rangle=\rho(\vec{x},\vec{x})$.

The whole setting is in principle reversible, hence no measurement has taken place. Therefore this part cannot involve Born's rule, which is explicitly about measurement, and has nothing to say in the reversible case.

Of course, before the Ag atom hits the photoplate the procedure is in principle reversible, and no measurement has taken place. Nevertheless, I have to use Born's rule to predict the probability distribution for where the atom will hit the photo plate. Calculations in a Hilbert space are no measurements, of course.

The subsequent measurement of the silver intensities on the screen is a purely classical process, of the same kind as measuring where and how much paint falls on a screen when sprayed with two faint beams of paint.

Exactly. So where do I need a POVM?

Thus the whole experiment nowhere needs quantum measurements for its quantitative understanding! Hence Born's rule is not needed anywhre!

Of course it's needed. What else than the probability distributions should be the testable prediction for the SGE?

Of course you can mumble 'Born's rule is involved in any measurement'. But this doesn't change anything and adds nothing to the explanation.

I don't say Born's rule is involved in any measurement. It's giving the predictions for what will be measured. I still don't understand what's unexplained in this standard description of the SGE and where I need POVMs to predict the outcome of the measurement.

 

[A. Neumaier]

Yes, and I don't understand, where POVMs are needed to explain real-world experiments. That's why I'm asking!

The POVM is not something needed in all contexts. What is always needed are the definitions in Section 2.1 (states and intensities) and 2.2 (detector response principle, DRP). Together with the quantum machinery, they define everything necessary to understand arbitrary measurements!

POVMs are needed only when one wants to know quantitatively what is being measured in a particular setting in the presence of imperfections, since the foundations from the 1930s (based on idealization) show their limitations.

It is then proved in Theoem 2.1 that there is always a POVM hidden behind - independent of whether or not the POVM is being used.

An eigenvalue-free version of Born's rule is then proved - i.e., rigorously derived, not postulated - in Section 3.1. On the other hand, the textbook form of Born's rule (using eigenvalues) cannot be derived since it is valid only under a special assumption - namely that of projective measurements. Then it follows in Section 3.2 as a special case of the general rule.

Why do you insist on the complicated special case (Born's rule for projective measurements) when there is a much simpler and intuitive general rule (the DRP)?

Of course it's needed. What else than the probability distributions should be the testable prediction for the SGE?

Testable predictions are the form and intensities of the silver beams, measured by the mass distribution of the silver on the screen. This is what was measured by Stern and Gerlach, and it is what is measured in a modern student lab reproducing some version of the experiment.

I still don't understand what's unexplained in this standard description of the SGE and where I need POVMs to predict the outcome of the measurement.

The standard textbook description does not explain the odd shape and overlap of the lips of silver reported by Stern and Gerlach. It does not even mention that there is this discrepancy.

POVMs are not needed to understand the measurements, for that the definition of intensity and the DRP are enough. Technically, this is far less complex than Born's rule.

POVMs are needed when you want to describe what really has been measured, including all imperfections of the experiment. This experiment has been analyzed in detail in the reference for the quote on top of p.26 of my paper.

 

[A. Neumaier]

when the silver atoms are absorbed at the photoplate making a spot at the corresponding place.

Stern and Gerlach didn't use a photoplate but a simple glass bottle. The silver arrived and stuck there. Just like when you would use a beam of paint.

 

[vanhees71]

The POVM is not something needed in all contexts. What is always needed are the definitions in Section 2.1 (states and intensities) and 2.2 (detector response principle). Together with the quantum machinery, they define everything necessary to understand arbitrary measurements!

POVMs are needed only when one wants to know quantitatively what is being measured in a particular setting in the presence of imperfections, since the foundations from the 1930s (based on idealization) show their limitations.

It is then proved in Theoem 2.1 that there is always a POVM hidden behind - independent of whether or not the POVM is being used.

I think I understand the math. My problem is to get a concrete POVM for a given physical situation. I'm just trying to find this in the literature and now I'm asking here without any success. Perhaps I suggest the wrong (gedanken) experiments? I consider them the most simple examples one can think of, and indeed I don't need a POVM to describe them. The standard quantum mechanics including Born's rule is sufficient to understand them, but it should as well possible to describe them with POVMs, according to your claim that "there is always a POVM hidden behind.

An eigenvalue-free version of Born's rule is then proved - i.e., rigorously derived, not postulated - in Section 3.1. On the other hand, the textbook form of Born's rule (using eigenvalues) cannot be derived since it is valid only under a special assumption - namely that of projective measurements. Then it follows in Section 3.2 as a special case of the general rule.

Why do you insist on the complicated special case (Born's rule for projective measurements) when there is a much simpler and intuitive general rule (the DRP)?

If it were so simple, why aren't you able to concretely define the POVM for the SGE?

The form and intensities of the silver beams, measured by the mass distribution of the silver on the screen. This is what was measured by Stern and Gerlach, and it is what is measured in a modern student lab reproducing some version of the experiment.

The standard textbook description does not explain the odd shape and overlap of the lips of silver reported by Stern and Gerlach. It does not even mention that there is this discrepancy.

Of course, that's because one doesn't describe the SGE in all detail, because it's complicated. It already starts with using a pure state (Gaussian wave packet) as an initial state instead of a mixture of a beam of silver vapor exiting the oven and then going through some slits to focus it better before entering the magnet. The magnetic field is also simplified such that you can solve the Schrödinger equation exactly etc.

POVMs are not needed to understand the measurements, for that the definition of intensity and the DRP are enough. Technically, this is far less complex than Born's rule.

Born's rule is very simple and transparent compared to the POVM, which seems not to be possible to be constructed for a simple, idealized gedanken experiment as the simplified textbook version of the SGE.

POVMs are needed when you want to describe what really has been measured, including all imperfections of the experiment. This experiment has been analyzed in detail in the reference for the quote on top of p.26 of my paper.

 

[A. Neumaier]

I think I understand the math. My problem is to get a concrete POVM for a given physical situation. I'm just trying to find this in the literature and now I'm asking here without any success.

We discussed the Stern- Gerlach experiment. It didn't need Born's rule, so it doesn't need POVMs.
But both can be used if one likes, and unlike the former, the latter gives a correct account of imperfections.
Look at [33, Example 1, p.7] to see a lengthy discussion in terms of POVMs.

Perhaps I suggest the wrong (gedanken) experiments?

Perhaps optimal quantum discrimination is a POVM concrete enough for you. This particular POVM can be realized using a beam splitter, as described in more generality in Section 4.1 of my paper.

I consider them the most simple examples one can think of, and indeed I don't need a POVM to describe them. The standard quantum mechanics including Born's rule is sufficient to understand them, but it should as well possible to describe them with POVMs, according to your claim that "there is always a POVM hidden behind.

This is a theorem, not just a claim. You can easily convince yourself of its validity.

If it were so simple, why aren't you able to concretely define the POVM for the SGE?

Because for the idealized textbook version the POVM is projective and for more realistic version it is no longer that simple (but given in the above reference). I don't want to copy long explanations from books that you can easily read yourself.

Born's rule is very simple and transparent compared to the POVM,

The definition of intensity and the DRP is very simple and transparent (and more general) compared to Born's rule. It is enough to specify the testable predictions of the textbook version of the Stern-Gerlach experiment. And it leads directly to POVMs (though in toy examples like this the POVM turns out to be projective due to the idealizations involved).

Make your pick by Occam's razor!

 

[A. Neumaier]

We discussed the Stern- Gerlach experiment. It didn't need Born's rule, so it doesn't need POVMs.
But both can be used if one likes, and unlike the former, the latter gives a correct account of imperfections.
Look at [33, Example 1, p.7] to see a lengthy discussion in terms of POVMs.

In the SGE the "pointer" is the particle's position, right? If you accept this, it's the most simple example for a measurement describable completely by quantum dynamics (sic!), i.e., the motion of a neutral particle with a magnetic moment through an inhomogeneous magnetic field!

You yourself conceded that Born's rule is not involved, dynamics does everything!

 

[vanhees71]

Born's rule is always involved. It simply states that $|\psi^2(t,\vec{x})|^2$ is the probability distribution for detecting a particle at $\vec{x}$ at time $t$. That implies that the intensity of the traces of the silver atoms on the plate is a measure for this probability distribution when using a beam of silver atoms (i.e., an ensemble ;-)).

 

[A. Neumaier]

Born's rule is always involved. It simply states that $|\psi^2(t,\vec{x})|^2$ is the probability distribution for detecting a particle at $\vec{x}$ at time $t$. That implies that the intensity of the traces of the silver atoms on the plate is a measure for this probability distribution when using a beam of silver atoms (i.e., an ensemble ;-)).

Not on the most fundamental level.

The true foundations of quantum mechanics is relativistic quantum field theory in the Heisenberg picture. There a beam of silver is given by the 1-point function $\langle j(x)\rangle$ of the silver 4-current $j(x)$ being peaked in a neighborhood of the beam. Silver is transported along the beam as described (in principle) exactly by the Kadanoff-Baym equations. In its coarse-grained approximation it is described by hydromechanics. The amount of silver deposited at some position is the integral of the intensity $I(x)=\langle j_0(x)\rangle$ of the silver beam at that position. This gives a quantitatively valid explanation of what happens when a silver beam hits a glass bottle. Together with standard semiclassical dynamical reasoning, this fully explains the original Stern-Gerlach experiment.

This is completely analogous to the treatment of polarized light in Section 2.1 of my tomography paper, where the current is modeled by the 4 components of the relativistic Pauli vector $\sigma$. The spacetime dependence is suppressed in this simple qubit setting; otherwise (as discussed in detail in the book by Mandel and Wolf) again a field theoretic current would figure.

Thus on the most fundamental level, a beam of silver is - just like a beam of light - a field, not an ensemble of silver atoms.

Describing a beam of silver (or a beam of light) by an ensemble of individual particles following each other along the beam, so that Born's rule is applicable, requires additional, theoretically murky semiclassical approximations. Already the particle concept in relativistic quantum field theory is somewhat dubious, as you well know. That something is wrong with your recipe can also be seen from the fact that your recipe cannot be applied to the detection of photons, which follow experimentally exactly the same pattern, although one can not even define the wave function that your recipe requires.

Thus the approach pursued in my paper is much to be preferred. In addition, it has the advantage of being simpler to motivate and to work with!

 

[gentzen]

The true foundations of quantum mechanics is relativistic quantum field theory in the Heisenberg picture.

Can you give a reference (and explicitly quote of the relevant text) to some paper (or book, by you, or by somebody else) for this assertion? I couldn't find it in Quantum tomography explains quantum mechanics. The best I could find in Foundations of quantum physics II. The thermal interpretation was section "4.2 Dynamics in quantum field theory"

Since the traditional Schrödinger picture breaks manifest Poincaré invariance, relativistic QFT is almost always treated in the Heisenberg picture.

But even this minor endorsement is weakened by section "1 Introduction"

We introduce the Ehrenfest picture of quantum mechanics, the abstract mathematical framework used throughout.

A lukewarm endorsement in Foundations of quantum physics III. Measurement was section "4.6 Conservative mixed quantum-classical dynamics"

New in quantum-classical systems – compared to pure quantum dynamics – is that in the Heisenberg picture, the Heisenberg state occurs explicitly in the differential equation for the dynamics. But it does not take part in the dynamics, as it should be in any good Heisenberg picture. The state dependence of the dynamics is not a problem for practical applications since the Heisenberg state is fixed anyway by the experimental setting.

And again here, the introduction to the section weakens the endorsement

Since the differences between classical mechanics and quantum mechanics disappear in the Ehrenfest picture in favor of the common structure of a classical Hamiltonian dynamics, we can use this framework to mix classical mechanics and quantum mechanics.

The older Classical and quantum mechanics via Lie algebras contains similar statements in section "19.2 Quantum-classical dynamics", but less lukewarm:

By design, in the Heisenberg picture, the state does not take part in the dynamics. What is new, however, compared to pure quantum dynamics is that the Heisenberg state occurs explicitly in the differential equation. In practical applications, the Heisenberg state is fixed by the experimental setting; hence this state dependence of the dynamics is harmless. However, because the dynamics depends on the Heisenberg state, calculating results by splitting a density at time t = 0 into a mixture of pure states no longer makes sense. One gets different evolutions of the operators in different pure states, and there is no reason why their combination should at the end give the correct dynamics of the original density. (And indeed, this will usually fail.) This splitting is already artificial in pure quantum mechanics since there is no natural way to tell of which pure states a mixed state is composed of. But there the splitting happens to be valid and useful as a calculational tool since the dynamics in the Heisenberg picture is state independent.

Back then, the Ehrenfest picture was not yet used to weaken the endorsement.

So for me, the question arises whether you now decided against the Ehrenfest picture in favor of the Heisenberg picture. But why? Or do you just use the Heisenberg picture as a substitute, because you expect that very few physicists would be familiar with the Ehrenfest picture, especially when it comes to relativistic quantum field theory?

 

[A. Neumaier]

Can you give a reference (and explicitly quote of the relevant text) to some paper (or book, by you, or by somebody else) for this assertion?
[...]
So for me, the question arises whether you now decided against the Ehrenfest picture in favor of the Heisenberg picture. But why?

I didn't decide against the Ehrenfest picture.

I am speaking here from a pragmatic point of view. Quantum field theory in the form of the standard model is known to be the basis of all our quantum physics. It is phrased almost exclusively in the Heisenberg picture, without any wave functions present. In my post I intended to refer implicitly to this, currently most fundamental, setting.

QFT can be reformulated in the Schrödinger picture via wave functionals, but this is hardly ever used in the literature.

All that quantum field theory calculates are N-point functions and stuff derived from these (like scattering cross sections and transport coefficients). These are the stuff that belong to the Ehrenfest picture. Thus the latter is the unifying umbrella.

But (unlike the Heisenberg and Schrödinger picture) the designation 'Ehrenfest picture' is little used in the literature. Since in discussions it is better to refer to well-known things where these convey the same information as less known concepts, I used the Heisenberg picture with which every QFT specialist is thorougly familiar.

 

[vanhees71]

Not on the most fundamental level.

The true foundations of quantum mechanics is relativistic quantum field theory in the Heisenberg picture. There a beam of silver is given by the 1-point function $\langle j(x)\rangle$ of the silver 4-current $j(x)$ being peaked in a neighborhood of the beam. Silver is transported along the beam as described (in principle) exactly by the Kadanoff-Baym equations. In its coarse-grained approximation it is described by hydromechanics. The amount of silver deposited at some position is the integral of the intensity $I(x)=\langle j_0(x)\rangle$ of the silver beam at that position. This gives a quantitatively valid explanation of what happens when a silver beam hits a glass bottle. Together with standard semiclassical dynamical reasoning, this fully explains the original Stern-Gerlach experiment.

Sure, but all your words are just using Born's rule. I know that you deny that your expectation-value brackets have a different than the usual straight-forward meaning of Born's rule, but I don't see what's the merit should be not to accept Born's rule (of course in its general form for general states, i.e., also for mixed states).

This is completely analogous to the treatment of polarized light in Section 2.1 of my tomography paper, where the current is modeled by the 4 components of the relativistic Pauli vector $\sigma$. The spacetime dependence is suppressed in this simple qubit setting; otherwise (as discussed in detail in the book by Mandel and Wolf) again a field theoretic current would figure.

Thus on the most fundamental level, a beam of silver is - just like a beam of light - a field, not an ensemble of silver atoms.

Describing a beam of silver (or a beam of light) by an ensemble of individual particles following each other along the beam, so that Born's rule is applicable, requires additional, theoretically murky semiclassical approximations. Already the particle concept in relativistic quantum field theory is somewhat dubious, as you well know. That something is wrong with your recipe can also be seen from the fact that your recipe cannot be applied to the detection of photons, which follow experimentally exactly the same pattern, although one can not even define the wave function that your recipe requires.

Thus the approach pursued in my paper is much to be preferred. In addition, it has the advantage of being simpler to motivate and to work with!

If you do the SGE with single silver atoms, there's no other way to get from the formalism to the observations than Born's rule, and it's one of the very few examples, where no semiclassical approximations are needed. You can solve the Schrödinger equation in this case exactly assuming a simplified magnetic field or use numerics.

 

[gentzen]

Sure, but all your words are just using Born's rule. I know that you deny that your expectation-value brackets have a different than the usual straight-forward meaning of Born's rule, but I don't see what's the merit should be not to accept Born's rule (of course in its general form for general states, i.e., also for mixed states).

From my point of view

• the thermal interpretation is a "Copenhagen like interpretation",
• which is more careful when it comes to treating arbitrary self-adjoint operators as observables.
• If you start like this, then it also makes sense to restrict the "definite measurement result" via Born's rule to fewer self-adjoint operators, and allow the "well defined measurable average results" for more self-adjoint operators.

And with fewer "definite measurement result" via Born's rule observables, it also makes sense to be similarly careful when it comes to preparation procedures, or quantum operations more generally.

Back to the question: The merit to not fully accept Born's rule for arbirary observables is that you can ask the question what is actually physical true in specific experimental setups.

Disclaimer: This is my own opinion, and these are my own words. A. Neumaier's opinions and words will certainly be different.

 

[vanhees71]

For me Born's rule holds for all states and all observables. The state (a "preparation procedure") is described by the statistical operator $\hat{\rho}$. A "complete measurement" is described by measuring a complete set of independent compatible observables $A_i$, represented by self-adjoint operators $\hat{A}_i$ with eigenvalues $a_{ij}$ and a CONS $|a_{1j},a_{2j},\ldots, a_{dj} \rangle$. Then the probability get any possible outcome of such a complete measurement is
$$P(a_{1j},\ldots,a_{dj}) = \langle a_{1j},a_{2j},\ldots, a_{dj}|\hat{\rho}|a_{1j},a_{2j},\ldots, a_{dj} \rangle.$$
The usual qualifications apply if you have operators which have a continuous (parts of their) spectrum.

 

[gentzen]

For me Born's rule holds for all states and all observables.

The risk with this approach is that observables themselves will no longer provide much physical structure, and you will be tempted to ascribe (physical) meaning directly to some preferred basis of the Hilbert space. Which is of course often totally harmless. Sometimes this might cause some undesired gauge fixing, but undesired gauge fixing might even happen in case (physical) meaning is only ascribed via specific observables. I guess only the "function of ..." part of the thermal interpretation can fully avoid this, and even then only if you are really careful.

 

[vanhees71]

Why should this imply any preferred basis? It's independent of the choice of basis (i.e., the choice of a complete set of compatible observables).

 

[A. Neumaier]

Sure, but all your words are just using Born's rule.

No; you just project your understanding of the foundations of quantum physics into my words.

None of my words or underlying concepts uses Born's rule, unless you empty it from all connections to measurements.

Indeed I mention measurement nowhere. Everything mentioned is macroscopic nonequilibrium thermodynamics together with the purely formal definition $\langle A\rangle:=\mathrm{Tr} \rho A$ (which is pure math, not physics), instantiated by taking for $A$ the components of a current to give the physical meaning it has in nonequilibrium thermodynamics. Calling a mathematical definition Born's rule is not appropriate.

I know that you deny that your expectation-value brackets have a different than the usual straight-forward meaning of Born's rule, but I don't see what's the merit should be not to accept Born's rule (of course in its general form for general states, i.e., also for mixed states).

The advantage is that eigenvalues play no role, and that nonprojective measurements are covered without any additional effort. Thus my approach is both simpler and more general than working with Born's rule, and the explanatory value is higher.

A "complete measurement" is described by measuring a complete set of independent compatible observables

Such a complete measurement cannot be done for most quantum systems (except for those with very few degrees of freedom). My approach does not need such fictions.

If you do the SGE with single silver atoms,

Most physics students in the lab will not do SGE with single silver atoms, but with continuous beams of silver. Therefore I only discussed the standard Stern-Gerlach experiment, as performed by them. This involves no ensemble but a silver field in the form of a dispersed beam.

If you do the SGE with single silver atoms, there's no other way to get from the formalism to the observations than Born's rule.

Really? I get the same result not from Born's rule but from the detector response principle DRP, without using eigenvalues or projections.

Maybe you will call the DRP Born's rule to save your view. Then we agree, except for the terminology.

In any case, introducing the DRP is much more intuitive than the introduction of Born's rule in your statistical physics lecture notes:

So let’s begin with some formalism concerning the mathematical structure of quantum mechanics as it is formulated in Dirac’s famous book.
[...]
If |o, j〉 is a complete set of orthonormal eigenvectors of O to the eigenvalue o, theprobability to find the value o when measuring the observable O is given by
$$P_ψ(o) =\sum_j |〈o, j |ψ〉|^2. ~~~~~~~~~~~~~~~~~~(2.1.3)$$

which is full of nonintuitive formal baggage that falls from heaven without any motivation. As only reference you give Dirac's famous book; I have the third edition from 1947. There he introduces eigenvectors on p.29, without any motivation, and states Born's rule in (45) on p.47, with formal guesswork as only motivation, and in a very awkward way, where one cannot recognize how it is related to your formulation. A more digestible version comes later in (51) on p.73,
but this is equivalent to yours only in the case of nondegenerate eigenvalues. The name Born's rule is nowhere mentioned in the book - so little importance does Dirac give to it!

Conclusion: In the foundations favored by you the students first have to swallow ugly toads, just based the promise that it will ultimately result in a consistent quantum theory later...

The DRP, in contrast, needs no eigenvalues at all, no separate consideration of degenerate cases, not even self-adjoint operators (themselves nontrivial to define but needed for the spectral resolution). The little stuff needed is simple and easy to motivate from Stokes' treatment of polarization in 1852.

it's one of the very few examples, where no semiclassical approximations are needed. You can solve the Schrödinger equation in this case exactly assuming a simplified magnetic field or use numerics.

The primary semiclassical approximation needed is the one that goes from quantum field theory to an ensemble of a sequence of single atoms moving along a beam and arriving at the bottle.

I don't know of a single paper explaining in detail how this transition in conceptual language can be justified from the QFT formalism. Maybe you can help me here with a reference?

 

[vanhees71]

We had this discussion over and over again. I don't think that we find a consensus here, because I don't see the connection between your formalism and the physics it's proposing to describe. I guess that's why Born's standard probabilistic interpretation has never been substituted by anything else by the majority of physicists.

I also don't understand your problem with the standard description of the SGE in QM. An example is

https://arxiv.org/abs/quant-ph/0409206

 

[A. Neumaier]

We had this discussion over and over again.

Each time I discuss it with you I add some new perspective, otherwise the discussion would not be interesting for me. At least I learn through the exchange (not about your view but more about mine), while you seem to be stuck in tradition.

I don't see the connection between your formalism and the physics it's proposing to describe.

My paper contains as much physics as Dirac's famous book! That you don't see it can only mean that you don't read it with open eyes.

I also don't understand your problem with the standard description of the SGE in QM.

On the quantum mechanical level there is no problem with it.

But a fundamental description would have to come from relativistic quantum field theory, where there are no ensembles. One cannot repeatedly prepare a quantum field extending over all of spacetime.

I don't know of a single paper explaining how the transition in conceptual language from a single quantum field to an ensemble of particles can be justified from the QFT formalism. Maybe you can help me here with a reference?

An example is

https://arxiv.org/abs/quant-ph/0409206

Sections 2-4 are pure theory without contact to experiment (i.e., before hitting the glass bottle); nothing to complain given their assumptions.
Section 5 interprets the final theoretical result as probability without explaining why they are allowed to do this.

The DRP gives this interpretation of the final theoretical result. On the other hand, Born's rule for projective measurements does not do the job: In Section VI, the authors of the paper write:

Thus, we can conclude that the Stern-Gerlach experiment is not, even in principle, and ideal experiment, which would “project” the internal state into the eigenvalues of the measurement operator. [...]
Our calculations indicate that the Stern-Gerlach experiment is not an ideal measuring apparatus, in the sense of [5].

[5] is von Neumann's 1932 book (in its 1955 English translation), where he describes measurement exclusively in terms of projection operators.

Thus projective measurements (and hence Born's rule) cannot be applied without making the additional semiclassical approximations proved invalid in the paper.

 

[gentzen]

Why should this imply any preferred basis? It's independent of the choice of basis (i.e., the choice of a complete set of compatible observables).

No, it doesn't imply a preferred basis. It is just tempting to go down that road, if you throw away the structure of the observables.

Let me expand my thoughts: If you say that any self-adjoint operator is an observable, and then use that "freedom" to use exactly one specific self-adjoint operator to describe the observable corresponding to the measurement in the experiment that you investigate, then the only other relevant self-adjoint operator seems to be the Hamiltonian of your system.

But how will you now ascribe physical meaning to your Hamiltonian and your selected observable? A tempting "cheap way" is to ascribe physical meaning directly to some preferred basis of your Hilbert space.

 

[vanhees71]

Each time I discuss it with you I add some new perspective, otherwise the discussion would be not interesting for me. At least I learn through the exchange (not your view but more about mine), while you seem to be stuck in tradition.My paper contains as much physics as Dirac's famous book! That you don't see it can only mean that you don't read it with open eyes.On the quantum mechanical level there is no problem with it.

But a fundamental description would have to come from relativistic quantum field theory, where there are no ensembles. One cannot repeatedly prepare a quantum field extending over all of spacetime.

I don't know of a single paper explaining how the transition in conceptual language from a single quantum field to an ensemble of particles can be justified from the QFT formalism. Maybe you can help me here with a reference?Sections 2-4 are pure theory without contact to experiment (i.e., before hitting the glass bottle); nothing to complain given their assumptions.
Section 5 interprets the final theoretical result as probability without explaining why they are allowed to do this.

The DRP gives this interpretation of the final theoretical result. On the other hand, Born's rule for projective measurements does not do the job: In Section VI, the authors of the paper write:[5] is von Neumann's 1932 book (in its 1955 English translation), where he describes measurement exclusively in terms of projection operators.

Thus projective measurements (and hence Born's rule) cannot be applied without making the additional semiclassical approximations proved invalid in the paper.

Dirac describes clearly the physical meaning including the probabilistic interpretation of the quantum state a la Born. You use the same symbols and forbid to interpret them in this standard way but don't give any clear physical interpretation I should use instead. That's why I'm stuck.

Within QFT you can as well prepare single Ag atoms as you can in QM. QFT is also used since it's conception to describe scattering cross sections, and that's also due to the standard interpretation of the quantum state in terms of Born's rule. It's explained in any QFT textbook, e.g., Weinberg, QT of Fields vol. 1.

The point of the paper by Potel is a complete quantum description of the SGE, and there's a probability for spin flips and that's why the SGE is not strictly an ideal von Neumann projection measurement. The point is that it's a complete quantum description in terms of the standard interpretation a la Born.

 

[A. Neumaier]

The point of the paper by Potel is a complete quantum description of the SGE, and there's a probability for spin flips and that's why the SGE is not strictly an ideal von Neumann projection measurement. The point is that it's a complete quantum description in terms of the standard interpretation a la Born.

You are asserting the opposite of the conclusion of the paper.

It is not a description in terms of the standard interpretation a la Born, since Born's rule is only about ideal von Neumann projective measurements (proof: p.20 of your statistical mechanics lecture notes), and Potel et al. proved that the SGE is not of this kind.

But it is a complete quantum description in terms of the interpretation by my detector response principle!

 

[vanhees71]

No, Born's rule gives the probability to find an Ag atom with a given spin component. It has nothing to do whether you get ideal entanglement between spin and position as stated in the simplified approximations in textbooks or the more accurate calculations in Potel's paper.

I still don't understand, what you think to achieve giving up the simple and well-established (as well as very successful) standard foundations of QT. I think that the minimal interpretation without collapse is the only interpretation you need.

You may introduce POVMs, but only if they can be made concrete for the analysis of real-world experiments. I don't see, where they are ever used in real-world applications of QT yet.

 

[A. Neumaier]

I still don't understand, what you think to achieve giving up the simple [...]

I achieve true simplicity. I find Born's rule far from simple, and very restrictive.

But it is impossible to communicate this to you. You block off all m< attempts with the mantra Born's rule explains all measurements., together with bogus arguments to justify it.

The problem is that it's not clear how to make the connection with equipment in the lab, which is not a problem in the standard description at all.

It is not a problem in my setting either. Together with the dynamics implies by the experimental setting, the DRP does it, with much less theoretical baggage than Born's rule:

(DRP): Detector response principle. A detection element $k$ responds to an incident
stationary source with density operator $ρ$ with a nonnegative mean rate $p_k$ depending linearly on ρ. The mean rates sum to the intensity of the source. Each $p_k$ is positive for at least
one density operator ρ

 

[A. Neumaier]

Within QFT you can as well prepare single Ag atoms as you can in QM. QFT is also used since it's conception to describe scattering cross sections, and that's also due to the standard interpretation of the quantum state in terms of Born's rule. It's explained in any QFT textbook, e.g., Weinberg, QT of Fields vol. 1.

One can model a single silver atom in this way. But how do you model an ensemble of 100 silver atoms moving at well separated times along a beam by quantum field theory? You cannot prepare multiple instances of a field extending over all of spacetime. The only use! of Born's rule in Weinberg's Vol. 1 (namely where he interprets the scattering amplitutes) doesn't address this issue - scattering has nothing to do with this question!

I don't know of a single paper explaining how the transition in conceptual language from a single quantum field to an ensemble of particles can be justified from the QFT formalism.

You also seem to know no place where this is done.

 

[Fra]

But a fundamental description would have to come from relativistic quantum field theory, where there are no ensembles. One cannot repeatedly prepare a quantum field extending over all of spacetime.

I don't know of a single paper explaining how the transition in conceptual language from a single quantum field to an ensemble of particles can be justified from the QFT formalism. Maybe you can help me here with a reference?

I agree these are some important points. Even if we don't agree on the solution, agreeing on what is a problem big enough to beg for a solution, and what we can sweep under the rug for a far future is a good start.

/Fredrik