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]

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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.