How to teach beginners in quantum theory the POVM concept

[A. Neumaier]

TL;DR Summary

Compared to Born's rule in its traditional squared probability amplitude form, the POVM concept is both more general and more easy to introduce on an elementary level.

[Edit 23.12.2019: A much extended, polished version of my contributions to this thread can be found in my paper Born's rule and measurement (arXiv:1912.09906).]

I'd still not know, how to teach beginners in QT using the POVM concept. [...] I don't think it's possible to introduce POVMs for physicists without using the standard formulation in the usual terms of observables and states.

Well, it is simpler than to introduce in full generality Born's rule.

Everything can be motivated and introduced nicely for a qubit, using polarization of classical light, as in my Insight article on A Classical View of the Qubit. That article concentrated on preparation (i.e., the states) rather than measurement (i.e., the POVMs). One can follow it up with the following discussion of measurement.

In a first course, I'd introduce pure states later than in the Insight article, deriving initially von Neumann's dynamics for the density operator rather than the Schrödinger equation. This would emphasize the idealization involved in the latter. The Schrödinger equation is really needed only much later, as a computational tool.

Having the Hilbert space and the unnormalized density operator for sources, one introduces a detector as a collection of detector elements of which at most one responds at any given time, defining a stochastic process of events. The measurement postulate takes the following simple form:

(DRP) (detector response principle)
A detector element $k$ responds to a stationary source in state $\rho$ with a rate $p_k$ depending linearly on the state $\rho$.

The linearity is well motivated by beam experiments: Changing the intensity amounts to a scalar multiplication of densities, combining two sources to addition. Thus it is easy to check by experiment the linearity of typical instrument responses, and the motivation is complete.

Postulate (DRP) is the only measurement postulate; everything else can be derived from it when the Hilbert space is finite-dimensional.

By linearity, the rates satisfy $p_k=\sum_{i,j} P_{kji}\rho_{ij}$ for suitable complex numbers $P_{kji}$. If the Hilbett space has finite dimension $n$, these coefficients can be found operationally by approximately measuring the rates for at least $n^2$ linearly independent states $\rho$ and solving the resulting linear least squares problem for the coefficients. This is called quantum detection tomography.

Introducing the matrices $P_k$ with $(j,i)$ entries $P_{kji}$, this can be written as
$$p_k=Tr~\rho P_k,$$
thus providing a derivation of the POVM extension of Born's probability formula from very simple first principles. The properties of the matrices can be deduced from the fact that the $p_k$ are rates of a stationary process. Hence they are nonnegative and sum to a constant. Since $p_k$ is real for all states $\rho$, the $P_k$ must be Hermitian. Picking arbitrary pure states $\rho=\psi\psi^*$ shows that $P_k$ is positive semidefinite. Summing the probabilities shows that the sum of the $P_k$ is a multiple of the identity. Requiring this multiple to be 1 is conventional and amounts to a choice of units for the rate in such a way that if the state of the surce is normalized to trace 1, the $p_k$ bdcome probabilities rather than rates. Thus the $P_k$ form a POVM and we have derived everything.

If there are a large number of detector elements, the detection event are usually encoded numerically. The value assigned to the $k$th detection event is pure convention, and can be any number $a_k$, or even a vector when the elements are arranged in a multidimensional array. It is whatever has been written on the scale the pointer points to, or whatever has been programmed to be written by an automatic digital recording device.

The state dependent formula for the expectation of the observable measured that follows from POVM together with the value assignment is $\langle A\rangle=Tr~\rho A$ with the operator (or operator vector) $A=\sum a_kP_k$. We may say, the detector measures an observable represented by the operator (vector) $A$
Note that the same operator $A$ in the expectation can be decomposed in many ways into a linear combinaion of many POVM terms; thus there may be many different POVMs measuring observables corresponding to the same operator $A$.

By picking the values carefully one can choose them to approximate a particular operator $X$ of interest, for example the position operator. This corrsponds to the classical situation of labeling the scale of a meter to optimally match a desired observable.

If the detector can be tuned by adjusting parameters $\theta$ affecting its responses, the $P_k=P_k(\theta)$ depend on these these parameters, giving $A=\sum a_kP_k(\theta)$. Now both the labels $a_k$ and the parameters $\theta$ can be tuned to improve the accuracy with which the desired $X$ is approximated. This is the process called calibration. Constructing detector devices that allow a high quality measurement corresponding to theoreticlly important operators is the challenge of high precision experimental physics.

The derivation just given is simple, intuitive, and complete. It tells everything needed to check and if necessary calibrate arbitrary detectors for their claimed measurement properties.

The derivation is far simpler, far more intuitive, and far more complete than what is needed to introduce students new to quantum physics to Born's rule, with its initially very weird formula for probabilities in a pure state.

Born's rule in expectation form is the very idealized case (realized experimentally only approximately, in very special situations) where the $P_k$ are orthogonal projectors, .e., $P_k^2=P_k=P_k^*$ and $P_jP_k=P_kP_j$ for all $j,k$. In this special case case, and only in this case, the components of $A$ commute and have a joint discrete spectrum, given by the $a_k$. This special case is distinguished in that by relabeling the values $a_k$ to $f(a_k)$, the same detector also measures any function $f(A)$ of $A$.

To get Born's rule in its traditional textbook form, one has to specialize further the state to be a normalized pure state, $\rho=\psi\psi^*$ with $\psi^*\psi=1$, and one finds $p_k=\psi^*P_k\psi$.

 

[DarMM]

How is the state taking the form of a trace class operator motivated prior to this? Or at least a complex matrix?

 

[A. Neumaier]

How is the state taking the form of a trace class operator motivated prior to this? Or at least a complex matrix?

For the qubit, in the way stated in the insight article cited. It leads very naturally to complex positive semidefinite Hermitian operators. Then by natural generalization to arbitrary Hilbert spaces.

 

[DarMM]

For the qubit, in the way stated in the insight article cited. It leads very naturally to complex positive semidefinite Hermitian operators. Then by natural generalization to arbitrary Hilbert spaces.

The insight article is great. It's the natural generalization I'm curious about. How do we get out the tensor product structure for multiple copies and the insight article only deals with photons, what kind of exposition are you imagining here to justify dealing with all physical systems in a formalism introduced for beams of light.

 

[A. Neumaier]

The insight article is great. It's the natural generalization I'm curious about.

From the Insight article one learns in a well-motivated way that a complex Hilbert space of dimension 2 models the simplest quantum phenomena with a positive definite Hermitian $\rho$ describing the state of an arbitrary source, the trace of $\rho$ as the intensity of the source, and certain Hermitian operators representing key quantities.

Probably having mastered this let's every beginning student of quantum physics accept the generalization. It is enough to say that in nearly 100 years of experimental work it was established beyond reasonable doubt that not only photon polarization but an arbitrary quantum system is describable in terms of an arbitrary complex Hilbert space, with a positive definite Hermitian $\rho$ with finite trace describing the state of an arbitrary source, the trace of $\rho$ defining the macroscopic intensity of the source, and certain Hermitian operators (with details depending on the quantum system) define key quantities.

Of course this is not a proof, but when creating foundations for a beginners course one may refer to authorities, and only make plausibility arguments that are easily grasped.

How do we get out the tensor product structure for multiple copies

This is a question quite different from the one you had asked in post #2.

After the qubit I'd introduce the anharmonic oscilator, the second simplest system of fundamental importance. This shows that finite-dimensional Hilbert spaces are not enough and infinite dimensions (i.e., functional analysis) is needed. (This is important even for people interested only in quantum information theory. They need to know that real systems oscillate and need an infinite-dimensional Hilbert space. Otherwise they are prone to draw misleading inferences from their limited $N$-qubit point of view on quantum mechanics.)

Here a lot of elementary phenomena (related to boundary conditions, bound states and scattering states, tunneling) can be discussed. It is only here that the Schrödinger equation starts to become important. I'd motivate it through the considerations in Sections 2.2 and 2.3 of Part I of my foundational series of papers.

Then I'd introduce Ehrenfest's theorem and the classical limit, as in Part IV of my foundational series of papers. This establishes the close connection to classical mechanics, including the fact that the
Poisson bracket is the classical limit of the scaled commutator.

From this one can easily see that coupled harmonic oscillators require a tensor product of Hilbert spaces. Again, it is enough to say that this generalizes to arbitrary composite systems.

After that one can introduce annihilation and creation operators and bosonic Fock spaces over an $N$-dimensional Hlbert space. This leads to the notion of indistinguishability of $N$ oscillators . One can then play with the consruction and look at fermionic Fock spaces. This is the Hilbert space of $N$ qubits, and leads to quantum infomation theory.

Then one can raise curiosity about Fock spaces over infinite-dimensional Hlbert spaces and relate it to quantum fields and systems of arbitrarily many free particles.

This concludes the foundation.

The next step would be to discuss approximation methods.

[If anyone is interested to work out a chapter of an introductory course along these lines I'd be happy to cooperate - offline!]

 

[DarMM]

Then one can easily see that coupled harmonic oscillators require a tensor product of Hilbert spaces

Just curious about this. Recent work suggests it is Relativity that forces the tensor product structure. I'm wondering how you get at the tensor product of the two Hilbert spaces being the correct one to describe the product system as opposed to some other Hilbert space. What feature are you using?

I don't mean rigorously establish or similar, I'm only asking in terms of pedagogy.

 

[A. Neumaier]

Just curious about this. Recent work suggests it is Relativity that forces the tensor product structure. I'm wondering how you get at the tensor product of the two Hilbert spaces being the correct one to describe the product system as opposed to some other Hilbert space. What feature are you using?

I don't mean rigorously establish or similar, I'm only asking in terms of pedagogy.

The same as Dirac 1926, namely classical correspondence. The Poisson algbra of a composite system is also the tenspr product of the Poisson algebra of its constituents. I find this very compelling.

 

[vanhees71]

Summary: Compared to Born's rule, the POVM concept is both more general and more easy to introduce on an elementary level.

Well, it is simpler than to introduce in full generality Born's rule.

Everything can be motivated and introduced nicely for a qubit, using polarization of classical light, as in my Insight article on A Classical View of the Qubit. That article concentrated on preparation (i.e., the states) rather than measurement (i.e., the POVMs). One can follow it up with the following discussion of measurement.

In a first course, I'd introduce pure states later than in the Insight article, deriving initially von Neumann's dynamics for the density operator rather than the Schrödinger equation. This would emphasize the idealization involved in the latter. The Schrödinger equation is really needed only much later, as a computational tool.

Having the Hilbert space and the unnormalized density operator for sources, one introduces a detector as a collection of detector elements of which at most one responds at any given time, defining a stochastic process of events. The measurement postulate takes the following simple form:

(DRP) (detector response principle)
A detector element $k$ responds to a stationary source in state $\rho$ with a rate $p_k$ depending linearly on the state $\rho$.

The linearity is well motivated by beam experiments: Changing the intensity amounts to a scalar multiplication of densities, combining two sources to addition. Thus it is easy to check by experiment the linearity of typical instrument responses, and the motivation is complete.

Postulate (DRP) is the only measurement postulate; everything else can be derived from it when the Hilbert space is finite-dimensional.

By linearity, the rates satisfy $p_k=\sum_{i,j} P_{kji}\rho_{ij}$ for suitable complex numbers $P_{kji}$. If the Hilbett space has finite dimension $n$, these coefficients can be found operationally by approximately measuring the rates for at least $n^2$ linearly independent states $\rho$ and solving the resulting linear least squares problem for the coefficients. This is called quantum detection tomography.

Introducing the matrices $P_k$ with $(j,i)$ entries $P_{kji}$, this can be written as
$$p_k=Tr~\rho P_k,$$
thus providing a derivation of the POVM extension of Born's probability formula from very simple first principles. The properties of the matrices can be deduced from the fact that the $p_k$ are rates of a stationary process. Hence they are nonnegative and sum to a constant. Since $p_k$ is real for all states $\rho$, the $P_k$ must be Hermitian. Picking arbitrary pure states $\rho=\psi\psi^*$ shows that $P_k$ is positive semidefinite. Summing the probabilities shows that the sum of the $P_k$ is a multiple of the identity. Requiring this multiple to be 1 is conventional and amounts to a choice of units for the rate in such a way that if the state of the surce is normalized to trace 1, the $p_k$ bdcome probabilities rather than rates. Thus the $P_k$ form a POVM and we have derived everything.

If there are a large number of detector elements, the detection event are usually encoded numerically. The value assigned to the $k$th detection event is pure convention, and can be any number $a_k$, or even a vector when the elements are arranged in a multidimensional array. It is whatever has been written on the scale the pointer points to, or whatever has been programmed to be written by an automatic digital recording device.

The state dependent formula for the expectation of the observable measured that follows from POVM together with the value assignment is $\langle A\rangle=Tr~\rho A$ with the operator (or operator vector) $A=\sum a_kP_k$. We may say, the detector measures an observable represented by the operator (vector) $A$
Note that the same operator $A$ in the expectation can be decomposed in many ways into a linear combinaion of many POVM terms; thus there may be many different POVMs measuring observables corresponding to the same operator $A$.

By picking the values carefully one can choose them to approximate a particular operator $X$ of interest, for example the position operator. This corrsponds to the classical situation of labeling the scale of a meter to optimally match a desired observable.

If the detector can be tuned by adjusting parameters $\theta$ affecting its responses, the $P_k=P_k(\theta)$ depend on these these parameters, giving $A=\sum a_kP_k(\theta)$. Now both the labels $a_k$ and the parameters $\theta$ can be tuned to improve the accuracy with which the desired $X$ is approximated. This is the process called calibration. Constructing detector devices that allow a high quality measurement corresponding to theoreticlly important operators is the challenge of high precision experimental physics.

The derivation just given is simple, intuitive, and complete. It tells everything needed to check and if necessary calibrate arbitrary detectors for their claimed measurement properties.

The derivation is far simpler, far more intuitive, and far more complete than what is needed to introduce students new to quantum physics to Born's rule, with its initially very weird formula for probabilities in a pure state.

Born's rule in expectation form is the very idealized case (realized experimentally only approximately, in very special situations) where the $P_k$ are orthogonal projectors, .e., $P_k^2=P_k=P_k^*$ and $P_jP_k=P_kP_j$ for all $j,k$. In this special case case, and only in this case, the components of $A$ commute and have a joint discrete spectrum, given by the $a_k$. This special case is distinguished in that by relabeling the values $a_k$ to $f(a_k)$, the same detector also measures any function $f(A)$ of $A$.

To get Born's rule in its traditional textbook form, one has to specialize further the state to be a normalized pure state, $\rho=\psi\psi^*$ with $\psi^*\psi=1$, and one finds $p_k=\psi^*P_k\psi$.

Well, this is how I introduce my teachers students to QM too, of course without the POVM formalism. I start with polarization experiments with (of course idealized) polaroids, letting one linear-polarization component of a classical em. field through and blocking the perpendicularly polarized linear-polarization component completely. Then I argue that when dimming the corresponding laser more and more at some point the stochastic nature of the em. field becomes observable (of course also cautioning that these are detection events for single photons though the state is not a single-photon state). Then you get the probabilistic interpretation and the entire QT-formalism in terms of Hilbert-space vectors, self-adjoint (hermitean is of course fine in this case, because we deal with a finite-dimensional unitary space here) including Born's rule in the usual way.

Of course, in an advanced lecture for experts the POVM formalism is important and can for sure be nicely taught along the lines of your posting and Insights article. That the classical Maxwell theory is so close to QFT is clear, because you can get very far with linear-response theory, and there the equations for classical fields and their operator analogues don't differ very much. It's even hard to find true examples for the necessity of field quantization. The most simple one is spontaneous emission, which afaik cannot be described in the semiclassical approximation (i.e., with the "matter", i.e., in quantum optics mostly electrons, treated quantum-theoretically and the em. field as classical "background field"). There you are back at the historical development, i.e., the thermodynamical origin of QT, but I'd quote Einstein's derivation of the Planck Law using kinetic arguments, where he clearly discovered the necessity for spontaneous emission.

It's also clear that a complete classical theory of macroscopic electrodynamics needs statistical physics, e.g., that the intensity of light is the temporal average over rapidly oscillating energy-momentum tensor components of the em. field (i.e., quadratic functionals of the field).

 

[Demystifier]

One conceptual problem with POVM measurements is that it is not so clear what does it correspond to for classical measurements.

 

[A. Neumaier]

One conceptual problem with POVM measurements is that it is not so clear what does it correspond to for classical measurements.

How is this special to POVMs? It is also not clear what Born's rule corresponds to for classical measurements.

 

[Demystifier]

How is this special to POVMs? It is also not clear what Born's rule corresponds to for classical measurements.

In the usual projective measurement, you measure an observable (position, momentum, energy, ...) that has a classical counterpart. But what is measured in a generalized POVM measurement?

 

[A. Neumaier]

In the usual projective measurement, you measure an observable (position, momentum, energy, ...) that has a classical counterpart. But what is measured in a generalized POVM measurement?

Exactly measured is the operator $A$ constructed in post #1 from the POVM and the assigned values $a_k$. To measure a prescribed operator $X$ with a classical counterpart (e.g., position, momentum, energy, ...) you must create a detector whose $A$ reproduces $X$ sufficiently well.

Apart from the simplicity and the straightforward motivation, the advantage of the POVM setting in #1 is that it is absolutely clear what measurement amounts to and how accurate it is in operational terms.

In contrast, Born's rule is completely silent about the notion of measurement, specifying only the statistics of the results of a mysterious measurement process. It tells one nothing about how to tune a concrete detector to produce accurate results.

 

[A. Neumaier]

Well, this is how I introduce my teachers students to QM too, of course without the POVM formalism.

But probably also without deriving the density operator and the Schrödinger equation from the classical optics. This is new in my approach; I have never seen it anywhere. Together with the POVM approach in post ##1, this provides a fully intelligible motivation for all basic features of quantum mechanics.

In the usual treatments, these basic features are addressed by just postulating the required items.

when dimming the corresponding laser more and more at some point the stochastic nature of the em. field becomes observable

Actually, following the semiclassical description of the photoeffect in the book by Mandel and Wolf, the correct explanation is that the stochastic nature of the detector response (to a sufficiently dim classical or quantum electromagnetic field) becomes observable.

 

[Demystifier]

Exactly measured is the operator $A$ constructed in post #1 from the POVM and the assigned values $a_k$. To measure a prescribed operator $X$ (e.g., position, momentum, energy, ...) you must create a detector whose $A$ reproduces $X$ sufficiently well.

Apart from the simplicity and the straightforward motivation, the advantage of the POVM setting in #1 is that it is absolutely clear what measurement amounts to and how accurate it is in operational terms, whereas Born's rule is completely silent about the notion of measurement, specifying only the statistics of the results of a mysterious measurement process. It tells one nothing about how to tune a concrete detector to produce accurate results.

I understand all this, but I suspect that a student of physics familiar with classical concepts who learns QM for the first time will not find this very illuminating.

Anyway, an introductory QM textbook that teaches POVM's (without calling them so) is
https://www.amazon.com/dp/052187534X/?tag=pfamazon01-20
Sec. 9.5 Measurements on open systems.

 

[A. Neumaier]

I suspect that a student of physics familiar with classical concepts who learns QM for the first time will not find this very illuminating.

But surely not less illuminating than the standard introduction of Born's rule!

 

[Demystifier]

But surely not less illuminating than the sandard introduction of Born's rule!

I agree, but at least one has a feeling that one understands what is measured. QM is shocking at many levels, but perhaps one does not need to get all the shocks at the same time.

 

[A. Neumaier]

I agree, but at least one has a feeling that one understands what is measured. QM is shocking at many levels, but perhaps one does not need to get all the shocks at the same time.

Well, what is measured is not spelled out at all by Born's rule, only what you get when you measure it. So how can one get a feeling that one understands what is measured?

The truth is that one assumes in an introductory course that the student already knows what is measured, or at least trusts the physics community that it knows. The student would be hard pressed to explain which ingredients ensure that what is actually measured is what is claimed to be measured.

The POVM version in post #1 has no such problems. It gives actual understanding, not only the appearance of it.

 

[Demystifier]

It gives actual understanding, not only the appearance of it.

Then perhaps you would also agree with the pedagogy of the book
https://www.amazon.com/dp/9814579394/?tag=pfamazon01-20
which starts with QFT and ends with classical mechanics.

 

[A. Neumaier]

Then perhaps you would also agree with the pedagogy of the book
https://www.amazon.com/dp/9814579394/?tag=pfamazon01-20
which starts with QFT and ends with classical mechanics.

Perhaps, but I don't currently have access to it, neither to the book you cited in post #14. Thus I cannot comment unless you summarize the essential points that differ from traditional expositions.

 

[vanhees71]

But probably also without deriving the density operator and the Schrödinger equation from the classical optics. This is new in my approach; I have never seen it anywhere. Together with the POVM approach in post ##1, this provides a fully intelligible motivation for all basic features of quantum mechanics.

In the usual treatments, these basic features are addressed by just postulating the required items.

Actually, following the semiclassical description of the photoeffect in the book by Mandel and Wolf, the correct explanation is that the stochastic nature of the detector response (to a sufficiently dim classical or quantum electromagnetic field) becomes observable.

Of course, I derive the Schrödinger equation in the traditional way too. Also the density operator comes much later (if at all). As I said, you need to start with the simple heuristic things first to understand the refined modern views of POVMs.

That the photoeffect does not prove the necessity of field quantization is quite old. I guess it was derived very early in the history of QM using the semiclassical approximation. Ironically the field quantization met much scepticism first: as far as I know the first appearance in the literature is in the famous "Dreimännerarbeit" by Born, Jordan, and Heisenberg, with the QFT part mostly due to Jordan. The usual argument was that the quantization of the em. field was "too much", and indeed the early experimental "proofs" of QM (photoeffect, hydrogen spectrum, Compton effect) don't need field quantization in the lowest order of approximation. There the semiclassical treatment is sufficient. The QFT treatment of the em. field had to be reinvented shortly later by Dirac (1927 or 1928).

 

[A. Neumaier]

Of course, I derive the Schrödinger equation in the traditional way too.

What is this traditional way? The tradition is to postulate the Schrödinger equation, not to derive it.

That the photoeffect does not prove the necessity of field quantization is quite old.

Given that you know this I don't understand how you can

argue that when dimming the corresponding laser more and more at some point the stochastic nature of the em. field becomes observable

because that ''the photoeffect does not prove the necessity of field quantization'' proves that the stochastic nature of the em. field is irrelevant for the occurrence of the photoeffect (and is needed only for quantitative predictions in case nonclassical light is used - not a topic for an introductory course).

Instead, one can argue only that

the stochastic nature of the detector response (to a sufficiently dim classical or quantum electromagnetic field) becomes observable.

Indeed, this is what follows from the semiclassical analysis of Wentzel 1926 (as found in Mandel and Wolf).

 

[vanhees71]

Of course you are right. I argue that you can do the polarization experiment with true single-photon states nowadays, but that you cannot really understand before the QT formalism is established.

Of course one cannot "derive" any fundamental equation of physics in a mathematical sense. The standard "derivation" argues via the Einstein-de Broglie relations $E=\hbar \omega$, $\vec{p}=\hbar \vec{k}$. It's of course also heuristic and in no way strict.

 

[A. Neumaier]

By picking the values carefully one can choose them to approximate a particular operator X of interest, for example the position operator. This corrsponds to the classical situation of labeling the scale of a meter to optimally match a desired observable.

See also the parallel discussion in another thread.

 

[ftr]

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.

 

[A. Neumaier]

Would you tie POVM concept to TI in your "textbook"?

No. This is completely independent of the TI, though it was inspired by it.

For your other questions you should open a new thread as the TI is off-topic here.

 

[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]

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]

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]

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.,

• https://arxiv.org/pdf/atom-ph/9609001.pdf
• https://arxiv.org/pdf/quant-ph/0410207.pdf
• https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6463478/
• https://iopscience.iop.org/article/10.1088/1367-2630/14/8/083009/pdf
• http://photonicquantum.info/Research/publications/Puentes PRL 102, 080404 (2009).pdf

 

[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.