B How to teach beginners in quantum theory the POVM concept

[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.