I Nature Physics on quantum foundations

[Morbert]

As an aside, there is an interesting ambiguity in N&C's account of measurement, since a sequence of projective measurement results can be expressed as a sequence of projectors $E_{m_1}(t_1)E_{m_2}(t_2)\dots E_{m_N}(t_N)$ which can be considered an operator $M_m$ i.e. a single POVM result, since $\sum_m M_m = I$. So by describing the measurement process with a series of projectors or just one operator, we can decide if collapse happens many times or just once.

 

[Demystifier]

The only problem is that the definition of POVM explicitly doesn't include a collapse postulate.

Nor does PVM (projector valued measure).

 

[Demystifier]

As an aside, there is an interesting ambiguity in N&C's account of measurement, since a sequence of projective measurement results can be expressed as a sequence of projectors $E_{m_1}(t_1)E_{m_2}(t_2)\dots E_{m_N}(t_N)$ which can be considered an operator $M_m$ i.e. a single POVM result, since $\sum_m M_m = I$. So by describing the measurement process with a series of projectors or just one operator, we can decide if collapse happens many times or just once.

If you interpret it as one collapse, then you cannot say at what time this collapse happens. Hence, if you postulate that collapse must happen at a definite time, then the one-collapse interpretation is ruled out, so the ambiguity is removed.

Conversely, if you don't make such a postulate, then the collapse may be a process taking a long time, in which case I see no conceptual problem with thinking of a series of collapses as one collapse. It has a classical analog. If you travel by car from New York to Los Angeles, you will have many intermediate stops. You can think of it either as one travel or a series of many travels, there is no any problem with that "ambiguity".

 

[A. Neumaier]

POVM measurement of a measured system can be described as a projective measurement of a larger system, which includes not only the measured system but also a part of its "ancilla" environment. From this bigger point of view, all measurement are projective.

...except that this ancilla is a purely mathematical construct in a nonphysical Hilbert space, hence something without any physical content.

 

[Demystifier]

...except that this ancilla is a purely mathematical construct in a nonphysical Hilbert space, hence something without any physical content.

... except that there are many explicit counterexamples to your claim.

 

[A. Neumaier]

... except that there are many explicit counterexamples to your claim.

Please point to one of them.

 

[Demystifier]

Please point to one of them.

Take e.g. the book A. Peres, Quantum Theory: Concepts and Methods, Sec. 9-6.
In the very first sentence he writes "It will now be shown that there always exists a physical mechanism (that is, a realizable experimental procedure) generating any desired POVM represented by given matrices Aμ" (his italics). The preparation procedure is then explained at page 288.

 

[A. Neumaier]

Take e.g. the book A. Peres, Quantum Theory: Concepts and Methods, Sec. 9-6.
In the very first sentence he writes "It will now be shown that there always exists a physical mechanism (that is, a realizable experimental procedure) generating any desired POVM represented by given matrices Aμ" (his italics). The preparation procedure is then explained at page 288.

This id a counterclaim but not a counterexmple.
Never take words for a valid argument if the details don't support them!

On p.288 Peres assumes already the existence of the ancilla! But the Hilbert space in which the ancilla is constructed - through Neumark's (or Naimark's) theorem as mentioned on p.285 - is non-physical, constructed in a purely mathematical way. The projection matrices defined on p.288 are only formal, not physical, since they are realized not in the physical Hilbert space in which the experimental POVM to be replicated, but only in a mathematical Hilbert space!

Thus the claim about the physicality of the mechanism is unsupported by the details Peres provides.

 

[Demystifier]

Never take words for a valid argument if the details don't support them!

Do you have a reference (not your own) that supports the claim that ancilla is unphysical?

 

[A. Neumaier]

Do you have a reference (not your own) that supports the claim that ancilla is unphysical?

The ancilla is clearly a purely mathematical construct. Thus my claim follows from common sense and needs no reference.

On the other hand, claiming that a purely mathematical construction of a Hilbert space and objects in it is physical is a nontrivial assertion that would require a justification with a reference.

 

[Demystifier]

Thus my claim follows from common sense and needs no reference.

On the other hand, claiming that a purely mathematical construction of a Hilbert space and objects in it is physical is a nontrivial assertion that would require a justification with a reference.

How about this one?
https://pubmed.ncbi.nlm.nih.gov/23004600/

 

[A. Neumaier]

How about this one?
https://pubmed.ncbi.nlm.nih.gov/23004600/

The experimental setting described here uses a physical ancilla (created by means of entangled photon pairs) to measure the terms in an unknown POVM representing the detectur under test (DUT) by collecting measurements on the twin beam with a photon-number-resolving detector (PNR).

If you just measure a photon in the DUT without having prepared it in the very special way, nothing at all happens in the PNR with the ancilla, proving that the procective measurements using the ancilla have in general nothing to do with the POVM measurements in the DUT.

So this is quite different from the earlier task: explaining the performance of a standard POVM measurement by treating this standard measurement itself in terms of projective measurements in a bigger, artificial Hilbert space.

 

[Demystifier]

The ancilla is clearly a purely mathematical construct. Thus my claim follows from common sense and needs no reference.

By that kind of thinking one could say the same for many things in theoretical physics. For example, Noether charge is a purely mathematical construct, so one could claim that Noether charges are unphysical and that it's common sense which does not need any reference. And when confronted with a measurement of conserved energy, one could argue that it's not the same because the experiments didn't measure energy of the whole universe as the Noether theorem requires.

In any case, it is a general view in the community that ancillas (that reduce POVM measurements to projective measurements) are physical. You have a different opinion, and maybe you are right, but as far as I am aware, nobody in the community shares your opinion.

 

[A. Neumaier]

By that kind of thinking one could say the same for many things in theoretical physics. For example, Noether charge is a purely mathematical construct, so one could claim that Noether charges are unphysical

No, since each Noether charge of physical interest actually corresponds to a key observable. Ancillas correspond to nothing physical - only the POVMs do.

it is a general view in the community

You'd need to provide lots of references to justify a claim of this generality!

that ancillas (that reduce POVM measurements to projective measurements) are physical.

No. The general view (which I hold as well, as it is a purely mathematical statement) is that ancillas reduce POVM measurements to projective measurements. This proves that POVMs give a mathematically consistent extension of standard quantum mechanics, which is enough for most users of POVMs.

But not that they are physical; this question is nowhere (that I'd know of) discussed in detail. The single reference you gave assumed the ancilla without any discussion of its physicality.

 

[Demystifier]

No, since each Noether charge of physical interest actually corresponds to a key observable. Ancillas correspond to nothing physical - only the POVMs do.

A related question. In decoherence theory, where a mixed density matrix is obtained from tracing out the unobserved environment degrees of freedom, would you say that the unobserved environment is unphysical?

 

[A. Neumaier]

A related question. In decoherence theory, where a mixed density matrix is obtained from tracing out the unobserved environment degrees of freedom, would you say that the unobserved environment is unphysical?

It depends on the features of the environment model used.

Since every bounded physical system is embedded in its environment, the environment as such is obviously physical, unless one considers the universe as a whole. Of course, models of the environment may still be unphysical. To regard a model as physical one must argue convincingly that the model captures the essence of a true environment.

The problem with the ancilla construction is that the Hilbert space is not defined by physical considerations but by a mathematical construction that doesn't resemble the Hilbert space of a system with a physically natural environment.

 

[Fra]

I think he suggested that neuroscience can be relevant to quantum interpretations. It's not even a weird position, given that many interpretations claim that subjective observations play a fundamental role in quantum interpretations.

I think it's a reasonable association but it's one that is just as easily misinterpreted as the notion of "observer" is.

I personally think the link is neither that QM should based the foundations of QM on neuroscientific models of the human neural network, nor that neuroscientist should look at QM or QFT. I see the more sound link at the more abstract paradigm level.

QM: Given a preparatation, we can compute the expectations of statistics of the future. This "expectation" is not currently given any physical basis, it is just "information" managed "classicaly" in the macroscopic laboratory. The size, mass or information capacity of this lab, never ever enters the "equations".

Neuroscience: Given a history of intreactions and evolution, the brain has a current state of expectations about the future, on which it's actions are based. This is thought to be explained by evolution, ie. it's simply a survivial trait. Here the information is managed by a real physics system, the physical state of the brain. The brain is finite, and has limitations on processing capacity both in terms of memory bandwith and speed.

How to optimize knowledge construction in the brain​

"The act of retrieval is generally thought to alter a memory again, updating it with previously and currently learned or retrieved information. Memories are then suggested to become reconsolidated into existing schemas, presumably altering their features again. This way, schemas are thought to be continuously adjusted to optimize our understanding of the world around us and to allow prediction of future occurrences"
- -https://www.nature.com/articles/s41539-020-0064-y

There is also in here two views on "probability", the descriptive statistical one and the guiding one, ie describing properties of the dice, or the statistics from actual dice throws. In decision making and gamling I think the distinction is important, and also in the foundations of QM.

Without diverging more, it seems to be clear if you think about this, and the list of open question in physics such as the origin of interactions/symmetries. Does it make sense to try to understand this as an "initial value problem" - I think not. Then what other paradigms is there to consider? Here at the abstract paradigm och model type level, I think there are insights to be gained from associating QM foundations with trying to understandn the brain. But it's not at the level of signal substances or cell biology, but at the more abstract modelling paradigm evel. This is my opinon at least.

/Fredrik

 

[Demystifier]

The problem with the ancilla construction is that the Hilbert space is not defined by physical considerations but by a mathematical construction that doesn't resemble the Hilbert space of a system with a physically natural environment.

But that's a matter of some specific constructions existing in literature, not a matter of principle. Perhaps most explicit constructions that can be found in literature are physically unrealistic in this sense. But it doesn't mean that POVM measurements cannot in principle be described by a realistic ancilla, it only means that a realistic ancilla is too complex to analyze in detail, which is why in practice we construct simplified models of ancilla which are not fully realistic.

 

[A. Neumaier]

it doesn't mean that POVM measurements cannot in principle be described by a realistic ancilla

It means that unless the contrary is made plausible.Most things in physics cannot be ruled out in principle - so your argument would amount to having to leave all questions open...

Realistic POVMs arise from many theoretical models by tracing out various sorts of realistic or at least physically plausible simplified environments. None of these has a built-in ancilly.

The only known construction for an ancilla is via Naimark's theorem, which is wholly artificial. The ancilla encodes the result structure, not physical properties of the environment.

 

[Demystifier]

Realistic POVMs arise from many theoretical models by tracing out various sorts of realistic or at least physically plausible simplified environments. None of these has a built-in ancilly.

Now I'm confused. In my view, a traced out environment is a kind of ancilla. It seems that you define ancilla differently, so how would you define the difference between environment and ancilla?

 

[A. Neumaier]

Now I'm confused. In my view, a traced out environment is a kind of ancilla.

This means that your view is a kind of misunderstanding of the literature.

It seems that you define ancilla differently, so how would you define the difference between environment and ancilla?

Look at the definition in Peres' book!

For a system in a general environment one just has a tensor product and gets the reduced density operator on one of the factors (the system) by tracing out the other factor (the environment). This is a process completely independent of measurement. It involves neither POVMs nor projections.

An ancilla is a very special artificial environment constructed mathematically for any given POVM, in a way that allows one to simulate this particular POVM by projective measurements.

 

[Demystifier]

Look at the definition in Peres' book!

An ancilla is a very special artificial environment constructed mathematically for any given POVM, in a way that allows one to simulate this particular POVM by projective measurements.

OK, so we agree that ancilla is a kind of environment. The difference is that I think that it can be physical, while you think that it's purely mathematical. But Peres also considers it to be physical.

 

[vanhees71]

Well, the problem I still have with the POVMs is that it is a very abstract mathematical concept too, i.e., I don't see, how to define a POVM for a given measurement apparatus. That must obviously somehow be modeled given the specifics of the apparatus.

As a very simple example one could use a single-photon double-slit experiment and use a variant of von Weizsäcker's discussion of the "Heisenberg microscope". You observe the photons behind the slit putting a lens. If you put the observational screen/photo plate in the image plane, you'll have a sharp image of the slits, i.e., you can observe through which slit each photon came, and you don't see interference/diffraction patterns. If you put the screen in the focal plane you measure the direction of the photon momenta sharply and thus have no which-way information, getting an interference pattern with full contrast. Both of these special settings seem to realize (FAPP) POVs, i.e., von-Neumann filter measurements, where one observable ("which way"/"position of the photon source" or "which momentum") is precisely measured. You can of course also place the screen in any other plane. Then you neither measure "position" nor "momentum" accurately. It's clear, how to calculate the probability distribution for photon detection. One just has to use the energy-density expectation value of the em. field given the setup (double slit + lens + position of the screen). But since here I do a kind of "weak measurement" of "position" and/or "momentum", shouldn't it be possible to formulate this in terms of a "POVM" measurement and, if this is possible, to discuss it as a POV measurement introducing an ancilla and whether this ancilla has a clear physical interpretation?

 

[Demystifier]

Then you neither measure "position" nor "momentum" accurately.

But you have some definite measurement outcomes. In general, the POVM is defined by the set of possible measurement outcomes. Perhaps in this case the measurement outcomes can be associated with certain coherent states $|z\rangle=|x,p\rangle$ with blurred position and momentum, so the POVM is the set of operators $\{|z\rangle\langle z|\}$.

Ancilla can be qualitatively described as follows. The full system consisting of both the measured system and the measuring apparatus is in a state of the form
$$|\Psi\rangle = \sum_z c_z |z\rangle |A_z\rangle \equiv \sum_z c_z |\Psi_z\rangle$$
where $|A_z\rangle$ are the "ancilla" states of the apparatus, which are macroscopically distinguishable and hence orthogonal (at least approximately). Hence the operators $\{|\Psi_z\rangle\langle \Psi_z|\}$ define a PVM, i.e. $|\Psi_z\rangle\langle \Psi_z|$ are projectors and the measurement can be viewed as a projective measurement.

Here the apparatus states $|A_z\rangle$ are physical, but in practice we don't know their explicit form. To make them explicit we can model them somehow, but the model will be a simplification which is not fully realistic. I think this is why @A. Neumaier thinks that ancilla is "unphysical".

 

[A. Neumaier]

OK, so we agree that ancilla is a kind of environment. The difference is that I think that it can be physical, while you think that it's purely mathematical. But Peres also considers it to be physical.

... without giving the slightest justification for it. Just calling something physical doesn't solve the problem! You need to show how it is at least approximately realized in the environment.

 

[Demystifier]

... without giving the slightest justification for it. Just calling something physical doesn't solve the problem! You need to show how it is at least approximately realized in the environment.

See post #374.

 

[A. Neumaier]

the problem I still have with the POVMs is that it is a very abstract mathematical concept too, i.e., I don't see, how to define a POVM for a given measurement apparatus. That must obviously somehow be modeled given the specifics of the apparatus.

This may be nontrivial, but even more nontrivial (and indeed impossible) would be the analogous quest for seeing how to define a projection valued measure (POM) needed without the use of POVMs. Whenever you can do the latter you can do the former, since every PVM is a POVM. But you cannot do the latter if the statistics of results corresponds to a POVM that is not a PVM).

You can of course also place the screen in any other plane. Then you neither measure "position" nor "momentum" accurately.

How do you analyze this case using the standard approach? Surely you get measurement results for both position and momentum, with some uncertainty. But they are not described by Born's rule!

In my quantum tomography paper I gave constructions of POVMs for quite a number of experimental situations, including joint position-momentum measurements, though not for your particular experiment but for what is done at CERN.

 

[A. Neumaier]

See post #374.

Just postulating the ancilla and saying that it is physical,

Here the apparatus states |Az⟩ are physical, but in practice we don't know their explicit form.

is not enough. One can answer arbitrary questions in the same way.

To make them explicit we can model them somehow, but the model will be a simplification which is not fully realistic. I think this is why @A. Neumaier thinks that ancilla is "unphysical".

At least you need to give such a simplified model, so that it can be discussed for its merits of being physical or not.

 

[Demystifier]

At least you need to give such a simplified model, so that it can be discussed for its merits of being physical or not.

An explicit model can be seen in https://ieeexplore.ieee.org/document/6769581 .
It is based on von Neumann model of wave function of the measuring apparatus.

 

[vanhees71]

But you have some definite measurement outcomes. In general, the POVM is defined by the set of possible measurement outcomes. Perhaps in this case the measurement outcomes can be associated with certain coherent states $|z\rangle=|x,p\rangle$ with blurred position and momentum, so the POVM is the set of operators $\{|z\rangle\langle z|\}$.

I'd rather think the POVM is not defined by something like a Wigner distribution, which is of course not a phase-space-distribution function, because it's not positive semidefinite. I'd think it's rather a Husimi function, i.e., some coarse-grained Wigner distribution.

This may be nontrivial, but even more nontrivial (and indeed impossible) would be the analogous quest for seeing how to define a projection valued measure (POM) needed without the use of POVMs. Whenever you can do the latter you can do the former, since every PVM is a POVM. But you cannot do the latter if the statistics of results corresponds to a POVM that is no PVM).

But here you simply measure an observable "accurately". You don't need to specify how you do that, and then you have the simple probabilities from Born's rule. $P(a)=\sum_{b} \langle a,b|\hat{\rho}|a,b \rangle$, where $|a,b \rangle$ is a complete set of eigenvectors of $\hat{A}$ (representing the measured observable $A$ with eigenvalues $a$ and some other observables $\hat{B}=(\hat{B}_1,\ldots,\hat{B}_n)$, which together with $\hat{A}$ form a complete independent set of compatible observables.

For the POVM you need to be more specific concerning an apparatus, where you don't measure any observable accurately. That's why I brought up this simple Heisenberg-von-Weizsäcker-microscope setup, where you measure approximately the "position" of the photon and approximately its momentum, with the accuracy depending on which position of the screen behind the lens you choose. It should be possible to define the corresponding POVM for this. The detection-probability distribution is in his case of course given by the expectation value of the electromagnetic wave's energy density (and this is of course calculated based on Born's rule). That somehow should be translatable in the POVM formalism, but I'm not familiar enough with the formalism to do this quickly ;-)).

How do you analyze this case using the standard approach? Surely you get measurement results for both position and momentum, with some uncertainty. But they are not described by Born's rule!

I calculate the expectation value of the em.-field-energy density across the screen using Born's rule.

In my quantum tomography paper I gave constructions of POVMs for quite a number of experimental situations, including joint position-momentum measurements, though not for your particular experiment but for what is done at CERN.

I'll make another try to understand this. I guess, you mean this paper

https://arxiv.org/abs/2110.05294

Where is a concrete (!) example where explicitly a POVM is constructed given a specific setup? I don't think that it's simpler to construct for a CERN experiment than for this very simple Heisenberg-von-Weizsäcker experiment. There you know what should come out ;-)).

 

[Demystifier]

I'd rather think the POVM is not defined by something like a Wigner distribution, which is of course not a phase-space-distribution function, because it's not positive semidefinite. I'd think it's rather a Husimi function, i.e., some coarse-grained Wigner distribution.

Yes, the coherent states $|z\rangle$ that I discussed define the Husimi distribution. For a pure state $|\psi\rangle$ it is
$$\rho_{\rm Husimi}(z)=\langle\psi|z\rangle \langle z|\psi\rangle$$

 

[A. Neumaier]

I'll make another try to understand this. I guess, you mean this paper

https://arxiv.org/abs/2110.05294

Yes. I don't think the principles are difficult to understand. In fact, they are simpler than to explain self-adjoint operators and the spectral theorem, needed to state Born's rule in its most conventional form.

Where is a concrete (!) example where explicitly a POVM is constructed given a specific setup?

The terminology in my paper is slightly different, to simplify understanding. An overview is in Section 1.3 on pp.9-13. Page and section numbers are for version v4, which I uploaded today. (The older version 3 from January 2022 has a different arrangement.)

The equivalence of a POVM with a quantum measure (a family of positive semidefinite Hermitian operators $P_k$ summing to 1) together with a scale (the mapping of $k$ to a measured value) is explained in Section 6.1 on p.41.

Section 4.2 on p.31 describes the quantum measure and scale for minimal uncertainty measurements (similar to what Demystifier alluded to). Here the quantum measure is given explicitly in terms of coherent states and a partition of unity.

Section 4.4 on p.32 describes the quantum measure and scale for a time projection chamber measurement of position, momentum, and energy. The trivial part is the specification of the detection elements. The resulting quantum measure is constructed in the proof of the general detection response theorem on p.19. Less trivial is the specification of the scale.

I don't think that it's simpler to construct for a CERN experiment than for this very simple Heisenberg-von-Weizsäcker experiment. There you know what should come out ;-)).

The point is that I spent already time to figure out how to model the CERN measurement, while I would need to think again about how to do it for Heisenberg's Gedankenexperiment, for which I currently don't have the time.

 

[A. Neumaier]

You can of course also place the screen in any other plane. Then you neither measure "position" nor "momentum" accurately.

How do you analyze this case using the standard approach? Surely you get measurement results for both position and momentum, with some uncertainty. But they are not described by Born's rule!

I calculate the expectation value of the em.-field-energy density across the screen using Born's rule.

But how does this give position and momentum?

 

[vanhees71]

It doesn't give position and momentum but a prediction of the observed intensity on the screen, i.e., for a single photon the detection probability distribution for detecting the photon at a given place on the screen.

 

[A. Neumaier]

I calculate the expectation value of the em.-field-energy density across the screen using Born's rule.

How does this give position and momentum?

It doesn't give position and momentum but a prediction of the observed intensity on the screen, i.e., for a single photon the detection probability distribution for detecting the photon at a given place on the screen.

This is a standard projective measurement, hence there is no difficulty.

But the quest was to give a description of an imperfect joint position-momentum measurement in terms of Born's rule!

 

[vanhees71]

No, it was my request to construct a POVM for this joint position-momentum measurement in the hope, how the abstract, mathematical concept for predicting probabilities from the quantum formalism in a more general way than the straight-forward standard one, is to be constructed for a real physical simple case. From a mathematical point of view, I think I understand the idea, i.e., why it provides a valid probability measure for some measurement. What I don't understand is, how to construct the POVM for a given physical situation. You are right in demanding that one has to define, how the observed intensity pattern provides a weak common measurement of position and momentum of the incoming photon (which is difficult of course, because there's no position operator for photons to begin with, but here I understand it as the position on the screen where the photon was detected; you can of course also use neutrons or other massive particles, where this formal quibble doesn't occur).

This optical double-slit version of the Heisenberg microscope is, by the way, realized in a more sophisticated way with entangled photon pairs in Dopfner's PhD thesis. So it's simple but not completely academic:

http://people.isy.liu.se/jalar/kurser/QF/assignments/Dopfer1998.pdf

 

[A. Neumaier]

No, it was my request to construct a POVM for this joint position-momentum measurement

Since you describe it as a joint position-momentum measurement, you'd be able to interpret the measurement in terms of Born's rule. But then you said you use Born's rule instead to measure the photon distribution. What is the connection between the two kinds of measurements?

You are right in demanding that one has to define, how the observed intensity pattern provides a weak common measurement of position and momentum of the incoming photon (which is difficult of course, because there's no position operator for photons to begin with, but here I understand it as the position on the screen where the photon was detected; you can of course also use neutrons or other massive particles, where this formal quibble doesn't occur).

You may assume massive particles. But this does not yet explain how you go from

I calculate the expectation value of the em.-field-energy density across the screen using Born's rule.

to a joint distribution of position and momentum:

You can of course also place the screen in any other plane. Then you neither measure "position" nor "momentum" accurately.

It is this step that I 'd need to be able to tell you how to construct the POVM.

 

[vanhees71]

Since you describe it as a joint position-momentum measurement, you'd be able to interpret the measurement in terms of Born's rule. But then you said you use Born's rule instead to measure the photon distribution. What is the connection between the two kinds of measurements?

My problem is precisely that I don't understand the claim that POVM measurements can describe such a joint measurement of incompatible observables as long as the construction of this POVM for a given concrete setup is not clarified. What obviously is measured here is the distribution of the photons on the screen, and this can be predicted with standard QFT (detection theory as found in any theoretical quantum-optics textbook and based on the use of Born's rule for calculating the energy-density of the em. field at the screen), but how from these observables a joint weak measurement of position and momentum is constructed in terms of a POVM is not clear to me, and that's why I ask about it.

You may assume massive particles. But this does not yet explain how you go from

to a joint distribution of position and momentum:

It is this step that I 'd need to be able to tell you how to construct the POVM.

That's precisely the step, I don't understand and what I'm thus asking about!

 

[A. Neumaier]

how from these observables a joint weak measurement of position and momentum is constructed in terms of a POVM is not clear to me, and that's why I ask about it.

But the question is whether it is such a joint measurement at all. What are your reasons for believing it is? If you only have a feeling that it might be the case, this is not enough.

I could analyze the time projection chamber measurements since the literature actually tells us how the numbers measured are produced.

That's precisely the step, I don't understand and what I'm thus asking about!

I can only find POVMs for experiments that actually produce somehow numbers for position and momentum. If these numbers cannot be produced by some explicitly specified protocol, how can it be a joint measurement?

 

[vanhees71]

Then the answer is that this experiment cannot be used to define a POVM for such a measurement?

 

[Morbert]

I'm a bit confused about joint distribution of position + momentum. Is it a distribution that resolves a POVM that looks something like $\{\frac{1}{2}|p\rangle\langle p|, \frac{1}{2}|x\rangle\langle x|\}$ where $$\frac{1}{2}\left[\sum |p\rangle\langle p| + \sum|x\rangle\langle x|\right] = I$$

 

[A. Neumaier]

Then the answer is that this experiment cannot be used to define a POVM for such a measurement?

If this experiment really measures position and momentum jointly you need to give me somehow the recipe how these numbers are extracted from the experiment. Then I can look for a POVM to describe it.

But if there is no prescription for how one gets position and momentum from the experiment one has no reason to call the latter a joint position-momentum measurement. Thus there is also no reason why a POVM should exist for it.

 

[A. Neumaier]

I'm a bit confused about joint distribution of position + momentum. Is it a distribution that resolves a POVM that looks something like $\{\frac{1}{2}|p\rangle\langle p|, \frac{1}{2}|x\rangle\langle x|\}$ where $$\frac{1}{2}\left[\sum |p\rangle\langle p| + \sum|x\rangle\langle x|\right] = I$$

No. A POVM with only two operators can only describe an experiment with a binary result. The number of terms in the POVM must be equal to (or larger) than the number of possible measurement results.

Thus the operators must be indexed by pairs $(x,p)$ of position and momentum.

 

[Morbert]

No. A POVM with only two operators can only describe an experiment with a binary result. The number of terms in the POVM must be equal to (or larger) than the number of possible measurement results.

So like $\{M_i\}$ where $M_i = \frac{1}{2}(|p_i\rangle\langle p_i| + |x_i\rangle\langle x_i|)$ (i.e. where you are certain of a value returned by an outcome/pointer, but uncertain of whether it belongs to position or momentum)?

 

[A. Neumaier]

So like $\{M_i\}$ where $M_i = \frac{1}{2}(|p_i\rangle\langle p_i| + |x_i\rangle\langle x_i|)$ (i.e. where you are certain of a value returned by an outcome/pointer, but uncertain of whether it belongs to position or momentum)?

Like in any joint measurement of two numbers you are certain of two measured values, as far as one can be certain about a measurement result. But they are related to position and momentum through some incomplete argument only - e.g., by stating that the marginal distributions reproduce those of position and momentum. The POVM is the way to tell you in exact terms what you really measured.

 

[A. Neumaier]

So like $\{M_i\}$ where $M_i = \frac{1}{2}(|p_i\rangle\langle p_i| + |x_i\rangle\langle x_i|)$ (i.e. where you are certain of a value returned by an outcome/pointer, but uncertain of whether it belongs to position or momentum)?

Like $\{M_{px} \mid p\in P, x\in X\}$, where $M_{px}= \frac{1}{2}(|p\rangle\langle p| + |x\rangle\langle x|)$. (Or rather some more physical expression such as an integral over nearby coherent states - I doubt that your expression can be realized.)

The measurement picks one $M_{px}$, from which you can read off both $p$ and $x$.

 

[Morbert]

Would that be possible?

I'm thinking a POVM $\{M_{ij}\}$ where each $M_{ij}$ gives us two probability distributions $$P_{ij}(x) = \mathrm{Tr}\left[M_{ij}\rho M^\dagger_{ij}|x\rangle\langle x |\right]$$$$P_{ij}(p) = \mathrm{Tr}\left[M_{ij}\rho M^\dagger_{ij}|p\rangle\langle p |\right]$$ I don't think there is any POVM where these distributions single out both a $p$ and a $x$ i.e. $P_{ij}(x) = \delta_{x,x_i}$ and $P_{ij}(p) = \delta_{p,p_j}$

 

[A. Neumaier]

I'm thinking a POVM $\{M_{ij}\}$ where each $M_{ij}$ gives us two probability distributions $$P_{ij}(x) = \mathrm{Tr}\left[M_{ij}\rho M^\dagger_{ij}|x\rangle\langle x |\right]$$$$P_{ij}(p) = \mathrm{Tr}\left[M_{ij}\rho M^\dagger_{ij}|p\rangle\langle p |\right].$$

By definition, the probabilities associated with a POVM $\{P_{ij}\}$ are $$p_{ij}=\mathrm{Tr} \rho P_{ij}.$$ If you change the rules, you shouldn't call your construction a POVM!

It seems you are thinking in terms of quantum instruments (also called quantum operations), not POVMs.

 

[A. Neumaier]

An explicit model can be seen in https://ieeexplore.ieee.org/document/6769581 .
It is based on von Neumann model of wave function of the measuring apparatus.

I got the paper. Where is the ancilla in this model?

 

[Demystifier]

I got the paper. Where is the ancilla in this model?

There are two ancillas, they are called meters in the paper, see the sentence after Eq. (1).