# B How to teach beginners in quantum theory the POVM concept

#### ftr

Actually this a general question its not about TI per se. I did not know how to ask you since you have disabled the conversation, and I don't know if it was appropriate to open a thread which might look rude towards you. Although As you know I am a fan of TI since I have had similar thoughts based on my own "system" which is based on geometric probability. May be you can reword it and open a thread I would appreciate that.

#### A. Neumaier

Actually this a general question its not about TI per se. [...] I don't know if it was appropriate to open a thread which might look rude towards you
It is appropriate to open a thread rephrasing your questions without mentioning the TI or me.

#### ftr

It is appropriate to open a thread rephrasing your questions without mentioning the TI or me.
Ok thanks.

#### vanhees71

Gold Member
Hi Arnold
Just few questions. Would you tie POVM concept to TI in your "textbook"? If so have you published TI in any peer review journal, what was the response of the mainstream. Also, for TI to become a legitimate interpretation, how does the process must proceed, I mean what kind of authority approval is involved. Thanks.
In the sciences there are no authorities. You send your work to a respected scientific journal, where it gets peer reviewed and then, if found suitable, published. That's it. If it's interesting enough, it will be cited by other researchers, maybe used for further work, maybe criticized.

#### vanhees71

Gold Member
Today, I tried to read something concrete about this POVM business, and the more I read the more I get lost. It's highly abstract. I cannot even make sense of a position measurement, which is the most fundamental measurement you need to begin with.

The standard formalism for the most simple case of a spin-0 particle in one spatial dimension goes as follows. The Hilbert space is constructed from the fundamental observables position $x$ and momentum $p$. The self-adjoint operators fulfill the Heisenberg algebra
$$[\hat{x},\hat{p}]=\hat{1}.$$
From this one constructs the generalized position eigenbasis
$$|x \rangle=\exp(\mathrm{i} x \hat{p}) |x=0 \rangle.$$
The generalized momentum eigenfunction thus is
$$\langle x|p \rangle=\frac{1}{\sqrt{2 \pi}} \exp(\mathrm{i} p x).$$
With the norm chosen such that
$$\langle p|p' \rangle=\delta(p-p').$$
Then we also have
$$\langle x|x' \rangle=\delta(x-x').$$
Now let $\hat{\rho}$ be a state. Then in the standard Born definition of the formalism the probability to find $x$ in and intervale $I=[x_1,x_2]$ is
$$P_I=\int_{x_1}^{x_2} \langle x|\hat{\rho}|x \rangle.$$
Now I tried to find a source, where a POVM position measurement is explained. To no avail.

So I tried to make sense of it myself. Instead of an ideal measurement, represented by the "orthogonal" projectors
$$\hat{P}_x=|x \rangle \langle x|$$
which fulfill
$$\hat{P}_x \hat{P}_y = \delta(x-y) |x \langle \rangle x|,$$
and the generalized matrix elements of these generalized projectors are of course
$$P_x(x_1,x_2)=\langle x_1|\hat{P}_x|x_2 \rangle=\delta(x-x_1) \delta(x-x_2),$$
one introduces some $\hat{\Pi}_i$, which are positive semidefinite self-adjoint operators, which should somehow contain information on the measurment device to define a probability for registration of a particle.
I this should describe a "realistic" position measurement, I guess what this means is that one has a detector with $(N-1)$ discrete pixels at rough positions $x_i$ with some "size" $\Delta x$ (in the 3D world like a photo plate measuring the positions, where particles hit this plane). One may describe this by
$$\hat{\Pi}_i = \frac{1}{N-1} \int_{\mathbb{R}} \mathrm{d} x f(x-x_i) |x \rangle \langle x|,$$
where
$$f(y)=\frac{1}{\Delta x \sqrt{2 \pi}} \exp \left (-\frac{y^2}{2 \Delta x^2} \right)$$
is a normalized Gaussien distribution.

The matrix elements are
$$\Pi_i(y_1,y_2)=\frac{1}{N-1} f(y_1-x_i) \delta(y_1-y_2).$$
These are all positive semidefinite, because for any square-integrable wave function one has
$$\langle \psi|hat{\Pi}_i|\psi \rangle=\int_{\mathbb{R}} \mathrm{d} y_1 \int_{\mathbb{R}} \mathrm{d} y_2 \psi^*(y_1) \Pi_i(y1,y_2) \psi(y_2) = \frac{1}{N-1} \int_{\mathbb{R}} \mathrm{d} y_1 f(y_1-x_i) |\psi(y_1)|^2 \geq 0.$$
Then to make the POVM complete, one has to write
$$\hat{\Pi}_N = \hat{1}-\sum_{i=1}^{N-1} \hat{\Pi}_i, \quad \Pi_{N}(y_1,y_2)=\delta(y_1-y_2) \left [1-\frac{1}{N-1}\sum_i f(y_1-x_i) \right].$$
That's also positive semidefinite:
$$\langle \psi|\hat{\Pi}_N|\psi \rangle = \int_{\mathbb{R}} \mathrm{d} y_1 |\psi(y_1)|^2 \left [1-\frac{1}{N-1}\sum_{i=1}^{N-1} f(y_1-x_i) \right]=1-\frac{1}{N-1} \sum_{i} \int_{\mathbb{R}} \mathrm{d} y_1 |\psi(y_1)|^2 f(y_1-x_i) \geq 0.$$
Then if the particle is prepared in the state $\hat{\rho}$ the probability that pixel $i \in \{1,\ldots,N-1 \}$ registers the particle is
$$P_i=\mathrm{Tr} (\hat{\rho} \hat{\Pi}_i )=\int_{\mathbb{R}} \mathrm{d} y \int_{\mathbb{R}} \mathrm{d} y_1 \rho(y,y_1) \Pi_i(y_1,y)=\frac{1}{N-1} \int_{\mathbb{R}} \mathrm{d} y \rho(y,y_1) f(y-x_i).$$
The probability that the particle is not registered at all then is
$$P_N=1-\sum_{i=1}^{N-1} P_i.$$
Is this a valid description of a POVM for position measurements? If so, how can one derive this without taking the usual Born formulation of QT for granted?

#### ftr

In the sciences there are no authorities. You send your work to a respected scientific journal, where it gets peer reviewed and then, if found suitable, published. That's it. If it's interesting enough, it will be cited by other researchers, maybe used for further work, maybe criticized.
The bolded is the authority isn't it?

#### vanhees71

Gold Member
Well, it's a community of peers. It's the argument that counts, now who has made it.

#### A. Neumaier

Maybe look at joint position and momentum measurements in the references at the top of p. 2 of arXiv:1307.5733. Will respond in more detail when I have more time.

#### 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).
This is because joint position and momentum measurements hold no challenge, hence few take an interest in them. The POVMs of research interest are in quantum optics and quantum cryptography. There the quadratures are the analogues of position and momentum, and joint measurements of these are of interest. See, e.g.,

#### A. Neumaier

Today, I tried to read something concrete about this POVM business, and the more I read the more I get lost. It's highly abstract. 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. To no avail.

So I tried to make sense of it myself. [...]
Is this a valid description of a POVM for position measurements?
It is, but far too mathematical (and far too much assuming) for a physical description of what goes on in a real position measurement. As you had stressed, in the lab there are only sources and detectors, no Gaussians. Thus the former, not the latter must figure in the explanation.
In my example the LHC is a "preparation machine" for (unpolarized) proton beams with a quite well-defined momentum and energy. [...] On the physical operational level an observable is (an equivalence class of) a measurement procedure. In my example you can define any kind of observable on "colliding proton beams".
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.

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#### vanhees71

Gold Member
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

Gold Member
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

Gold Member
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

Gold Member
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

Gold Member
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

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?

#### 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

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

#### PeterDonis

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

#### Buzz Bloom

Gold Member
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

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