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

[Fra]

Perhaps you touched upon this elsewhere.

Yes, that's what quantum tomography is about.

To accurately determine a momentum vector one also needs more than one measurement.

From the perspective of inference: learning the hamiltonian is as much of a challenge as knowledge of the initial state, and in a real situation the two problems must interfere with each other. I think determining the state is referred to as the state tomography, but determining the hamiltonian is the process tomography?

If one insists that knowledge of the state and the state of the unitary evolution that is applied to the state both qualify as "information", how would one realize the simultaneous process of process and state tomography? Or do you reason as if the hamiltonian and hilbert space is given facts? (not subject to inquiry, subject to similarly critical analysis?)

/Fredrik

 

[A. Neumaier]

Perhaps you touched upon this elsewhere.

From the perspective of inference: learning the hamiltonian is as much of a challenge as knowledge of the initial state, and in a real situation the two problems must interfere with each other. I think determining the state is referred to as the state tomography, but determining the hamiltonian is the process tomography?

Yes, in the special case where the system is conservative; otherwise no hamiltonian exists, only a system of Lindblad generators.

If one insists that knowledge of the state and the state of the unitary evolution that is applied to the state both qualify as "information", how would one realize the simultaneous process of process and state tomography?

This is called self-calibrating tomography. See p.38 of my paper, where I also give some references where further details can be found.

 

[A. Neumaier]

The evaluation of experimental data never uses the quantum formalism. The interpretation of these data of an electron in a Penning trap, however, and thus the "mapping" of measured "beat frequencies" and their mapping to the value of $(g-2)$ is based on the QT formalism.

In the mean time I revised my tomography paper. The new version is better structured and contains a new section on high precision quantum measurements, where the 12 digit accuracy determination of the gyromagnetic ration through the observation and analysis of a single electron in a Penning trap is discussed in some detail.

The standard analysis assumes that the single electron is described by a time-dependent density operator following a differential equation. While in the original papers this involved arguments beyond the traditional (ensemble-based and knowledge-based) interpretations of quantum mechanics, the new tomography-based approach applies without difficulties.

 

[vanhees71]

In the Penning-trap case aren't there the currents of the "mirror charges" in the trap electrodes observed? I'd also expect that what's measured follows within the standard framework is the current due to the motion of the electron within the trap. I'll have a look on your new chapter about this in terms of the POVM formalism.

 

[vanhees71]

Ok, I had a look at it. In Sect. 9.4 you also don't construct a POVM explicitly for this standard Penning-trap setup. There's no formula there :-(.

I also don't understand, what's problematic with the standard treatment in Brown and Gabrielse's RMP article. It's just 1st-order perturbation theory in RPA approximation (in Sect. V.A).

 

[A. Neumaier]

what's problematic with the standard treatment in Brown and Gabrielse's RMP article.

What's problematic, especially for the ensemble interpretation (and Rigo et al. explicitly acknowledge that it goes beyond the standard treatment) is that they use a density operator to describe a single electron rather than an ensemble of electrons. This ensemble is purely imagined (sa, Brown and Gabrielse state explicitly) and has no physical reality. It is needed to derive the formula by which the gyromagnetic ratio is measured.

In the Penning-trap case aren't there the currents of the "mirror charges" in the trap electrodes observed?

Their paper says that they measured two particular frequencies (how doesn't really matter for my paper, but you can find more details by reading their paper yourself), whose quotient gives the gyromagnetic ratio to 12 decimal places.

In Sect. 9.4 you also don't construct a POVM explicitly for this standard Penning-trap setup.

This is because parameter determination such as that of the gyromagnetic ratio, and in fact most of spectroscopy, is not a quantum measurement in the sense of Born's rule nor is it one in in the sense of POVMs. But is uses the objective existence of the density operator and its dissipative dynamics, which are consequences of the detector response principle DRP on which the whole paper is based. That's why I added the material to the paper.

Note that my paper is not primarily about POVMs but about how quantum tomography explains quantum mechanics. Deriving the POVM formalism, including Born's rule where it applies is only a small part of the whole story.

 

[vanhees71]

What's measured are currents, and the necessary amplifiers and the ampere meter do some "time averaging". I thought, it would be possible to describe this with the POVM formalism. So my challenge is standing: How can the abstract POVM formalism be made applicable to describe a real-world experiment in the lab. I've never heard anybody using it to describe a real-world experiment yet.

Another (gedanken) experiment is the measurement of a "trajectory" of a single particle in a cloud chamber. You can put some radioactive material in there and make a movie of the tracks forming, i.e., you can measure directly how the track forms, i.e., a position-momentum joint measurement. The standard quantum description a la Mott is very clear and for me describes the appearance of "trajectories" in this setup satisfactory, but maybe it's interesting to discuss it within the POVM framework too?

 

[A. Neumaier]

So my challenge is standing: How can the abstract POVM formalism be made applicable to describe a real-world experiment in the lab.

The whole of Section 4 of my paper is devoted to real-world experiments that use POVMs rather than projective measurement, with reference to other people's work.

I've never heard anybody using it to describe a real-world experiment yet.

This can only mean that you never bothered to read the associated literature. For example, the quantum information textbook by Nielsen and Chuang is full of POVMs.

Most introductions to quantum mechanics describe only projective measurements, and consequently the general description of measurements given in Postulate 3 may be unfamiliar to many physicists, as may the POVM formalism described in Section 2.2.6. The reason most physicists don’t learn the general measurement formalism is because most physical systems can only be measured in a very coarse manner. In quantum computation and quantum information we aim for an exquisite level of control over the measurements that may be done, and consequently it helps to use a more comprehensive formalism for the description of measurements. [...]

A physicist trained in the use of projective measurements might ask to what end we start with the general formalism, Postulate 3? There are several reasons for doing so. First, mathematically general measurements are in some sense simpler than projective measurements, since they involve fewer restrictions on the measurement operators; there is, for example, no requirement for general measurements analogous to the condition $P_iP_j = \delta_{ij}P_i$ for projective measurements. This simpler structure also gives rise to many useful properties for general measurements that are not possessed by projective measurements. Second, it turns out that there are important problems in quantum computation and quantum information – such as the optimal way to distinguish a set of quantum states – the answer to which involves a general measurement, rather than a projective measurement. A third reason [... is ...] the fact that many important measurements in quantum mechanics are not projective measurements.

 

[A. Neumaier]

the measurement of a "trajectory" of a single particle in a cloud chamber. You can put some radioactive material in there and make a movie of the tracks forming, i.e., you can measure directly how the track forms, i.e., a position-momentum joint measurement. The standard quantum description a la Mott is very clear and for me describes the appearance of "trajectories" in this setup satisfactory, but maybe it's interesting to discuss it within the POVM framework too?

The POVM description of this is similar to that of the joint position-momentum measurement of particle tracks in my Section 4.4, except that the arrangement of wires is replaced by the pixels of the video taken.

 

[Morbert]

-

-
Repeated, nondestructive measurements of the same quantity of the same system should yield the same result each time, I think. This isn't guaranteed for POVMs right? Is there a distinction between asserting a detector imperfectly measures a standard observable built from a projective decomposition, and a detector exactly measuring a quantity built from a POVM like the one above?

 

[A. Neumaier]

Repeated, nondestructive measurements of the same quantity of the same system should yield the same result each time, I think. This isn't guaranteed for POVMs right? Is there a distinction between asserting a detector imperfectly measures a standard observable built from a projective decomposition, and a detector exactly measuring a quantity built from a POVM like the one above?

The distinction is the word 'projective'. Dirac and von Neumann considered only measurements whose repetition yield the same result each time, and were thus lead to the class of projective measurements.

But most measurements in practice are not of this kind, as either each realization of the system can be typically measured only once, or each measurement on the system measures a different state of the system but never the projected one.

 

[A. Neumaier]

or each measurement on the system measures a different state of the system but never the projected one.

The typical example is a faint temporary interaction which changes the system state only slightly (not by projection) but leaves an irreversible record, hence counts as measurement. Particle track detectors are based on this.

 

[Morbert]

As an aside, I think this important character of POVMs (repeated measurements not yielding identical results) is preserved even if we expand the state space to recover a projective measurement a la Luis and Sanchez-Soto (equation 1). The significance of the projector |k><k| in equation 1 is merely that the record of a single measurement will not change no matter how many times the record is reviewed.
[edit]- Then again, maybe the carelessness of the researcher reviewing past records can be modeled with a POVM : )

 

[vanhees71]

But I think, that's a feature of the POVM approach. Almost all real-lab measurements are not projective measurements, and it doesn't make sense to assume a collapse to the projection of the state to the eigenspace of the measured eigenvalue of the self-adjoint operator that represents the measured observable as a generally valid postulate. It rather depends on the kind of interaction between the measured object and the measurement device, what happens to the object in measurement. E.g., if you detect a photon usually it's absorbed in the material. Usually what's used in photon detection is the photo effect, i.e., afterwards there's no photon left to be in any state, and thus you also don't need a state for this photon to describe its behavior. In such cases I guess one could use a POVM to describe the measurement process. However, so far nobody could give a concrete example, how to describe a given measurement procedure with a POVM. I always get the answer that one cannot do that.

My challenge stands: The most simple example for a unsharp joint measurement of position and momentum of a particle seems to be the example that is described (imho fully satisfactory) by Mott in the famous paper about the tracks of a charged particle in a cloud chamber. One can extend this indeed to an observation of approximate positions and momenta by simply taking a movie and measuring the position of the end of the track as a function of time and then deduce both a position and the momentum of the particle along this track. Shouldn't it be possible to describe this (gedanken) experiment in terms of a POVM?

 

[Morbert]

But I think, that's a feature of the POVM approach. Almost all real-lab measurements are not projective measurements, and it doesn't make sense to assume a collapse to the projection of the state to the eigenspace of the measured eigenvalue of the self-adjoint operator that represents the measured observable as a generally valid postulate. It rather depends on the kind of interaction between the measured object and the measurement device, what happens to the object in measurement. E.g., if you detect a photon usually it's absorbed in the material. Usually what's used in photon detection is the photo effect, i.e., afterwards there's no photon left to be in any state, and thus you also don't need a state for this photon to describe its behavior. In such cases I guess one could use a POVM to describe the measurement process. However, so far nobody could give a concrete example, how to describe a given measurement procedure with a POVM. I always get the answer that one cannot do that.

I'm going to naively apply consistent histories to try and construct one.
Say you want to destructively measure the linear polarisation of a photon (system $s$), which is registered by a detector (system $D$) at time $t_1$. Unitary evolution would look like $$U\frac{1}{\sqrt{2}}(|V\rangle+|H\rangle)|D_0\rangle = \frac{1}{\sqrt{2}}(|D_V\rangle+|D_H\rangle)$$ We write down the histories $$C_H=|H\rangle\langle H|(t_1-\delta t) \otimes D_H(t_1)$$$$C_V=|V\rangle\langle V|(t_1-\delta t) \otimes D_V(t_1)$$
We can construct the POVM with the two members $$P_V=\mathrm{tr}_D \rho_D C_VC_V^\dagger$$$$P_H=\mathrm{tr}_D \rho_D C_HC_H^\dagger$$

 

[vanhees71]

How can unitary transformation go out of the Hilbert space? A unitary operator is an isomorphism on a fixed Hilbert space and not between two different Hilbert spaces as you write in your 1st formula.

 

[Morbert]

How can unitary transformation go out of the Hilbert space? A unitary operator is an isomorphism on a fixed Hilbert space and not between two different Hilbert spaces as you write in your 1st formula.

I am assuming $|D_H\rangle,|D_V\rangle$ are members of $\mathcal{H}_\mathrm{phonon}\otimes\mathcal{H}_\mathrm{detector}$ but not $\mathcal{H}_\mathrm{detector}$, such that e.g. $U^\dagger|D_H\rangle = |H\rangle|D_0\rangle$ but maybe this is too naive

 

[A. Neumaier]

I always get the answer that one cannot do that.

That's not true. You always got the answer that it can be done only if you specify the full measuring process from the interaction of the measured system till the measurement results. And then you lost patience or interest, and didn't follow up on my answers.

The most simple example for a unsharp joint measurement of position and momentum of a particle seems to be the example that is described (imho fully satisfactory) by Mott in the famous paper about the tracks of a charged particle in a cloud chamber.

Mott does not perform a single measurement in his analysis. So how can one extract a POVM from his discussion if nothing is measured? The POVM would depend on details about how the cloud chamber track is observed to actually get the results of the measurement.

One can extend this indeed to an observation of approximate positions and momenta by simply taking a movie and measuring the position of the end of the track as a function of time and then deduce both a position and the momentum of the particle along this track. Shouldn't it be possible to describe this (gedanken) experiment in terms of a POVM?

Now that you defined a recipe for getting a position and momentum that can be carried out experimentally, this is indeed possible. Your recipe leads to a POVM in essentially the same way as my analysis in Section 4.4 of v4 of my quantum tomography paper, except that the grid of wires is replaced by a grid of pixels encoding the video. That this POVM is complicated comes from the fact that extracting a position and a momentum from a video is complicated.

 

[A. Neumaier]

I am assuming $|D_H\rangle,|D_V\rangle$ are members of $\mathcal{H}_\mathrm{phonon}\otimes\mathcal{H}_\mathrm{detector}$ but not $\mathcal{H}_\mathrm{detector}$, such that e.g. $U^\dagger|D_H\rangle = |H\rangle|D_0\rangle$ but maybe this is too naive

This is not too naive but too sloppy to be correct. In fact you define
$|D_H\rangle :=U(|H\rangle|D_0\rangle)$ and similarly $|D_V\rangle$, and your formula follows.

 

[vanhees71]

That's not true. You always got the answer that it can be done only if you specify the full measuring process from the interaction of the measured system till the measurement results. And then you lost patience or interest, and didn't follow up on my answers.

You never gave a concrete answer. That's the problem.

Mott does not perform a single measurement in his analysis. So how can one extract a POVM from his discussion if nothing is measured? The POVM would depend on details about how the cloud chamber track is observed to actually get the results of the measurement.Now that you defined a recipe for getting a position and momentum that can be carried out experimentally, this is indeed possible. Your recipe leads to a POVM in essentially the same way as my analysis in Section 4.4 of v4 of my quantum tomography paper, except that the grid of wires is replaced by a grid of pixels encoding the video. That this POVM is complicated comes from the fact that extracting a position and a momentum from a video is complicated.

Were is the concrete POVM given in this paper?

 

[A. Neumaier]

You never gave a concrete answer. That's the problem.

Where is the concrete POVM given in this paper?

The very concrete answer is in the last paragraph of Section 4.4 on p.32. I had several times referred to it. The POVM consists of the quantum measure together with a reindexing of the matrices by the measurement results.

 

[vanhees71]

There are many words, no concrete construction of the POVM.

 

[A. Neumaier]

There are many words, no concrete construction of the POVM.

The words define in the first sentence a unique quantum measure. The relabeling needed to get the POVM is described in the remainder, and can be exactly described by the computer programs used to analyze the video (in your version of the experiment), summing the contributions that lead to the same label. Since you described the analysis of the video in words only, I cannot do better.

Thus the construction of the POVM is as concrete as your gedanken experiment.

 

[vanhees71]

We obviously have a different understanding what "concrete" means. I don't know, how experimental particle physicists program their computers, but I'm pretty sure, it's not based on the POVM paradigm of quantum measurement theory. I don't blame you, but I simply like to understand, what's behind this POVM idea in a physical context rather than a mathematical abstract concept.

It seems to be very difficult to construct it for even a so much simpler setup as a measurement of "particle tracks" with a cloud chamber...

 

[A. Neumaier]

We obviously have a different understanding what "concrete" means. I don't know, how experimental particle physicists program their computers, but I'm pretty sure, it's not based on the POVM paradigm of quantum measurement theory.

That's not necessary to apply the POVM principles. They also don't base it on the projective paradigm of quantum measurement theory that you favor.

Instead they rely on the claims of manufacturers or peers how the equipment works. Almost every experimental reasoning is completely classical, together with a little semiclassical quantum mechanics for some crucial points.

I don't blame you, but I simply like to understand, what's behind this POVM idea in a physical context rather than a mathematical abstract concept.

It seems to be very difficult to construct it for even a so much simpler setup as a measurement of "particle tracks" with a cloud chamber...

This is because your question is not adapted to how POVMs are actually used.

In practice, people (i) either have a given setup and want to calibrate it; so they do quantum tomography to find the POVM. Or (ii) they want to realize a given POVM with a suitable experiment. The latter is particularly relevant for potential applications in quantum computing.

In my paper, (i) is described in full detail and full generality in Section 2.2, and (ii) is described for a multiphoton setting in Section 4.1, and in more detail in the papers by Leonhardt cited there.

The exact POVM of concrete experiments is as complex as the experimental setting itself. But when one measures something one is not interested in these details unless ones want to construct a more accurate detector. Thus one usually idealizes the description to the bare minimum.

One can do the same for POVMs. For a joint measurement of position and momentum this is done in Section 4.2. The formula there is physically motivated and as simple as one can want it; the experimental realization is given in the paper cited there. For the partition of unity one can take any collection of hat functions describing the smearing, divided by their sum.

 

[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).