How to teach beginners in quantum theory the POVM concept

[vanhees71]

Fine, I've never realized that you need POVMs to understand a TPC, but I start to get an idea. It's not so much different from Mott's analysis of the cloud-chamber tracks of $\alpha$ particles. Also there you can measure a "position" and reconstruct a "momentum" of the $\alpha$ particle having a magnetic field in place (you could even do it without field when you were able to time-resolve the "formation" of the track accurately enough).

I'm still a bit puzzled what's actually wrong with my simple example. Using Gaussian's is of course an approximation, but that's very common for usual measurements. All the error analysis you learn in the introductory lab are based on it.

 

[A. Neumaier]

I'm still a bit puzzled what's actually wrong with my simple example.

Nothing is wrong per se. But there is the blemish that you create your example in a way that presupposes Born's rule.

Using Gaussian's is of course an approximation, but that's very common for usual measurements. All the error analysis you learn in the introductory lab are based on it.

The Gaussian used in the introductory lab to analyze a sample of measurement results has little to do with the Gaussian used to model a POVM. For in the POVM, all positions close to the nominal position at the center of mass of your pixel, presumably, count with a Gaussian weight towards the pixel's operator, while in a statistical analysis, the discrete measurement results close to the true position count with a Gaussian weight towards the sample average.

 

[vanhees71]

Of course, I presuppose Born's rule, because without it, I've no clue what the physical meaning of the POVM should be. In all books I've seen, it's done using Born's rule, and a PV is explicitly called a special case of a POVM. Finally there seems to be consensus among mathematicians that a POVM is always definable as taking a partial trace over a complete PV measurement. For me, QT is not understandable without Born's probability interpretation.

The 2nd part I don't understand. Do you mean that the width of my Gaussians is not necessarily describing the detector resolution? That's maybe true, but I guess it should be not too far from it. The actual detector resolution could be found by calculating the "pixels' response probability" for a state much better localized than this resolution.

 

[DarMM]

Finally there seems to be consensus among mathematicians that a POVM is always definable as taking a partial trace over a complete PV measurement

Neumark's theorem shows any given POVM on a system is equivalent to a PVM on the system + some ancilla.

However there are two things to note. This merely says that this is a possible way to implement a given POVM, it does not say that is necessarily how a POVM is always implemented. Secondly the passage to such a PVM is blocked in QFT because finite volume states are always mixed.

 

[A. Neumaier]

Of course, I presuppose Born's rule, because without it, I've no clue what the physical meaning of the POVM should be.

The physical meaning is spelled out - without assuming Born's rule - explicitly in post #1 and my detailed example of joint position/momentum/energy measurement in post #35. What is lacking there?

The Gaussian used in the introductory lab to analyze a sample of measurement results has little to do with the Gaussian used to model a POVM. For in the POVM, all positions close to the nominal position at the center of mass of your pixel, presumably, count with a Gaussian weight towards the pixel's operator, while in a statistical analysis, the discrete measurement results close to the true position count with a Gaussian weight towards the sample average.

The 2nd part I don't understand. Do you mean that the width of my Gaussians is not necessarily describing the detector resolution?

No. I just emphsized that the role of the Gaussian is different whether it is used to define an operator in a POVM (done in theoretical work) or the uncertainty in a statistical analysis (done in the lab). Since the POVM should directly interpret what happens in the lab (by Peres' dictum) it should be specified in lab's terms and not by an idealized formula given in terms of theory.

 

[vanhees71]

Well, it was an example, which I thought is close to what's done when position of a particle is measured with a photoplate or CCD cam. If you need more complicated error analysis than the usual Gaussian assumptions, because the errors of distinct outcomes are correlated, then you have to do it for this specific experiment. Your description of the TPC was not concrete enough for me. It was using words but not a concrete definition of the POVM. I've never heard a TPC described as a POVM. I'll ask my experimental colleagues from ALICE, whether they have ever heard about POVMs. I guess they'll think, it's again one of these weird ideas of theorists ;-))).

 

[A. Neumaier]

Your description of the TPC was not concrete enough for me. It was using words but not a concrete definition of the POVM.

Well, that necessarily the case in the lab, where you only have sources and detectors, but neither operators nor integrals...

In terms of operators, one usually describes joint position-momentum measurements in terms of a POVM built from coherent states. I don't have a good reference ready at the moment, but see, e.g., the introduction of https://arxiv.org/pdf/1805.01012.pdf, where a reference to [10] is given.

 

[vanhees71]

Well, we are discussing how to describe the sources and detectors with a POVM. Then I expect that you describe the POVM for your example you give. A TPC is quite common, and I think I roughly understand how it works (though as a theorist usually I have to trust the experimentalists to understand their device well enough to simply provide the result in physical terms to be confronted with model calculations).

I think, I can vaguely understand the idea with the coherent states. I guess the "von Neumann lattice" is a nice example which can be made concrete as a POVM, though I couldn't "google scholar" yet a reference to that idea. Maybe I try it myself when I find the time...

I'll have a look at the details of the quoted PRL later.

 

[A. Neumaier]

Well, we are discussing how to describe the sources and detectors with a POVM. Then I expect that you describe the POVM for your example you give.

A good reference is

• R.F. Werner, The uncertainty relation for joint measurement of position and momentum, J. Quantum Information Computation 4 (2004), 546-562.

Observables defined by a POVM are defined in Section 2.2 and constructed in Lemma 3 (though not for a TPC).

 

[ftr]

A good reference is

• R.F. Werner, The uncertainty relation for joint measurement of position and momentum, J. Quantum Information Computation 4 (2004), 546-562.

Observables defined by a POVM are defined in Section 2.2 and constructed in Lemma 3 (though not for a TPC).

Is there any concrete experiment that this analysis could be checked by? or at least what would be a good experiment example?

 

[A. Neumaier]

Steinberg's 2014 paper ''Quantum Measurements: a modern view for quantum optics experimentalists''

provides precisely what I'm still lacking in explaining the meaning of POVMs. Now it would be great if somebody could write a paper merging this paper by a practitioning experimenter, providing the physical meaning of the formalism in an intuitive way such that he can work with them as an experimentalist, with the very abstract definitions of mathematical physicists, i.e., something for a phenomenological theoretical physicist like me.

What about the following?

• Lundeen et al., Tomography of quantum detectors, Nature Physics 5 (2009), 27.
• D'Ariano, Maccone & Presti, Quantum Calibration of Measurement Instrumentation. Physical Review Letters 93 (2004), 250407.

 

[thephystudent]

I read this paper a while ago, and found it very enlightening in explaining related concepts on a simple qubit model. POVMs are discussed in one of the first sections, and then it continues about quantum trajectories.

 

[Buzz Bloom]

I was not able to find the definition of the acronym POVM online . I hope someone at PF will post a definition.

 

[PeterDonis]

I was not able to find the definition of the acronym POVM online .

Google is your friend.

 

[Buzz Bloom]

I read this paper a while ago, and found it very enlightening in explaining related concepts on a simple qubit model. POVMs are discussed in one of the first sections, and then it continues about quantum trajectories.

Hi thephystudent:

I just found the definition of POVM in the paper you cited.

positive operator valued measurement​

Unfortunately this techeze is not a definition I am able to understand. And the discussion of this concept in the paper is way over my head, so never mind.

Regards,
Buzz

 

[DarMM]

Hi thephystudent:

I just found the definition of POVM in the paper you cited.

positive operator valued measurement​

Unfortunately this techeze is not a definition I am able to understand. And the discussion of this concept in the paper is way over my head, so never mind.

Regards,
Buzz

Positive operator valued measurement is a very opaque name, don't worry about not being able to figure it out from the name. How familiar are you with textbook projective measurements in quantum theory?

 

[Buzz Bloom]

How familiar are you with textbook projective measurements in quantum theory?

Hi DarMM:

Zero familiar.

I tried to read section 2 of the cited article on Page 8, and I was able to somewhat digest the mathematical definition of

we give a set of positive operators ^{^}ˆE_n which sum to the identity ^{^}1,​

but I do not even know what the identity operator ^{^}1 is.

Regards,
Buzz

 

[DarMM]

How familiar are you with the mathematics of quantum theory then? Have you only encountered it in "solving equations" form or have you no experience with the mathematics.

 

[Mentz114]

[]

So let us consider momentum measurement at LHC. It can be used to check whether momentum and energy are indeed what is claimed to be prepared, but it is of real interest in measuring momenta and energies of secondary decay products (where one doesn't know beforehand what is prepared). The discussion will also shed light on position measurement.

Instead of a fully realistic momentum measurement, let us consider a somewhat simplified but still reasonably realistic momentum measurement in a Time Projection Chamber (I don't know precisely what the LHC is using, but this doesn't matter as only the basic principle is to be illustrated). The beam passes a number of wires arranged in $L$ layers of $w$ wires each and generates current signals, ideally exactly one signal per layer. From these signals, time stamps and positions are being computed by a least squares process (via the Kalman filter), assuming the track (of a charged particle in a magnetic field) is a helix (due to ionization energy loss in the chamber). From the classical tracks reconstructed by least squares, the momentum is computed in a classical way. (In the description in Section 5.2 of https://arxiv.org/pdf/nucl-ex/0301015.pdf, only 2 Layers are present, so one uses linear tracks. The LHC uses more layers and a helical track finder, see http://inspirehep.net/record/1643724/files/pdf.pdf)

Note that we measure both position and momentum, which is not covered by Born's rule.

But it is described by a POVM with an operator for each of the $w^L$ possible signal patterns. The value assignment is done by a nontrivial computer program for the least squares analysis and produces a 7-dimensional phase space vector (including the energy). The operators exist by my general analysis in post #1, and can probably be approximately described in mathematical terms. But this is not essential for the principle itself, which - as you wanted - should be given in laboratory terms only.

The TPC is new to me and this is very informative. The measurements in this case do actually give good estimates of the positions and momenta but they don't look like projectors in the way I understand projection. Is that right ?

 

[A. Neumaier]

The TPC is new to me and this is very informative. The measurements in this case do actually give good estimates of the positions and momenta but they don't look like projectors in the way I understand projection. Is that right ?

It cannot be projections to common eigenspaces of the observables measured (as Born's rule would require it) since there are no such common eigenspaces.

 

[Mentz114]

Thank you. I'm reading the two accounts of the CERN track detectors and I hope that will throw some light on this.

 

[A. Neumaier]

I cannot even make sense of a position measurement, which is the most fundamental measurement you need to begin with. [...] Now I tried to find a source, where a POVM position measurement is explained.

Fine, I've never realized that you need POVMs to understand a TPC, but I start to get an idea.

One needs a POVM whenever one simultaneously measures observables corresponding to noncommuting operators, since this cannot be handled by Born's rule.

An idealized joint measurement of position and momentum was described by a coherent state POVM (with infinitely many projectors $|\alpha\rangle\langle\alpha|$ to all possible coherent states $|\alpha\rangle$, where $\alpha$ is a complex phase space variable) in:

• Arthurs & Kelly, BSTJ briefs: On the simultaneous measurement of a pair of conjugate observables. The Bell System Technical Journal, 44 (1965), 725-729.

(See also https://arxiv.org/abs/quant-ph/0008108 and https://arxiv.org/abs/1305.0410.)

By discretizing this using an arbitrary partition of unity (i.e., a collection of smooth nonnegative functions $e_k(\alpha)$ summing to 1) , these can be grouped into finitely many positive operators $P_k:=\pi^{-1}\int d\alpha e_k(\alpha)|\alpha\rangle\langle\alpha|$ corresponding to finite resolution measurements, making it look more realistic. This would be suitable as a simple analytic for presentation in a course.

But to check whether an actual joint measurement of position and momentum fits this construction for some particular partition of unity would be a matter of quantum tomography!

 

[vanhees71]

Great! So subsituting coherent states instead of using my "smeared position measures" makes my idea of a position POVM correct for a joint (weak) measurement of position and momentum.

Now I'm still puzzled what this has to do with the TPC, which measures a momentum, right?

 

[A. Neumaier]

Great! So substituting coherent states instead of using my "smeared position measures" makes my idea of a position POVM correct for a joint (weak) measurement of position and momentum.

correct as an idealized model, yes. (''weak'' is not the correct label. A weak measurement is something different.)

Now I'm still puzzled what this has to do with the TPC, which measures a momentum, right?

The TPC measures position and velocity (or momentum if the mass is known) since it measures a whole path. Though one can of course ignore part of the information gathered. The momentum and energy loss are the quantities of interest for scattering, but for secondary decays one also needs the decay position.

 

[A. Neumaier]

in all the nice mathematical sources you quoted not a single one gives a clear physical description of a POVM measurement of position or the "fuzzy common measurement of position and momentum" (as I'd translate what seems to be intended by the very abstract formulations of the POVM formalism I've seen so far).

There is a nice book by Busch, Grabowski and Lahti 2001, extensively discussing POVMs
for realistic measurements. Their starting example is:

The following 'laboratory report' of the historic Stern-Gerlach experiment stands quite in contrast to the usual textbook 'caricatures'. A beam of silver atoms, produced in a furnace, is directed through an inhomogeneous magnetic field, eventually impinging on a glass plate. The run time in the original experiment was 8 hours. Comparison was made with a similar experiment with the magnet turned off, run time 4.5 hours. The result of the magnet-off case was a single bar of silver on the glass approximately 1.1 mm long, 0.06-0.1 mm wide. In the magnet-on case, a pair-of-lips shape appeared on the glass 1.1 mm long, one lip 0.11 mm wide, the other 0.20 mm wide, the maximum gap between the upper and lower lips being approximately the order of magnitude of the width of the lips. Both lips appeared deflected relative to the position of the bar.

Only visual measurements through a microscope were made. No statistics on the distributions were made, nor did one obtain 'two spots' as is stated in some texts. The beam was clearly split into distinguishable but not disjoint beams; yet this was considered to be enough to justify the conclusion that some property had been demonstrated. Gerlach and Stern viewed this property as 'space quantisation in a magnetic field.'

[...] For the moment a simplified description shall suffice to show that, strictly speaking, only an unsharp spin observable, hence a POV measure, is obtained.

 

[vanhees71]

Seems to be a great book! I've even (legal) access to it, though it's published in 1995. Is this the book you have in mind? At least it's about the right topic and also discusses the SGE in some detail.

https://rd.springer.com/book/10.1007/978-3-540-49239-9

 

[A. Neumaier]

Seems to be a great book! I've even (legal) access to it, though it's published in 1995. Is this the book you have in mind? At least it's about the right topic and also discusses the SGE in some detail.

https://rd.springer.com/book/10.1007/978-3-540-49239-9

Yes, this is the book I had quoted from. There is also another, quite recent book with two authors the same:

• P. Busch, P. Lahti, J. Pellonpää and K. Ylinen, Quantum Measurement, Springer, Berlin 2016.

with a number of chapters on realistic POVM measurements, but the other book is much more elementary.

 

[A. Neumaier]

At least it's about the right topic and also discusses the SGE in some detail.

Here is another book of interest:

• W.M. de Muynck, Foundations of Quantum Mechanics, an Empiricist Approach, Kluwer, Dordrecht 2002. (note: the online copy linked to is slightly different from the published version)

Chapter 7 discusses a number of realistic examples. In the introduction to the chapter (p.258 of the online version) he writes:

The examples discussed in sections 7.2 through 7.5 show that a generalization of the formalism is necessary for describing even the most common methods of quantum mechanical measurement, like the detection of photons using a detector that is not 100% efficient. This also holds true for such experiments as the double slit experiment, being a paradigm of standard quantum mechanics. It will not be surprising, then, that an analysis of this experiment based on the standard formalism can hardly be a reliable one, and that conclusions based on such an analysis should be considered with some reservation.

With ''generalization of the formalism'' he means the POVM formalism generalizing the traditional textbook formalism which only features projective measurements.