# Can POVM measurements be explained by projective measurements?

• I
• A. Neumaier
In summary, premises 1 + 2 + 3 give us a quantum mechanically describable ancilla that must exist and premises 4 + 3 + 4 say the measurement scenario involving this ancilla must also be describable with a projective decomposition.
A. Neumaier
TL;DR Summary
This is a new thread that takes up the question in the title, a question discussed (off topic) in another thread.
For the background of the discussion see my Insight artice ''Quantum Physics via Quantum Tomography'' and the posts #405 and later of the thread ''Nature Physics on quantum foundations''.
Morbert said:
Consider a microscopic system ##s## being measured, and the pointer ##M## doing the measuring

Premise 1)
The pointer must be describable with quantum mechanics. I.e. There must be a Hilbert space ##\mathcal{H}_s\otimes\mathcal{H}_M## in principle.

Premise 2)
Given some POVM ##P_k##, there must be an associated measure ##E_k## for the pointer positions

Premise 3)
If the pointer really does measure the microscopic system, then it must be the case that rates are given by

$$p_k = \mathrm{tr}_s\rho_{s}P_k = \mathrm{tr}_{s,M}\rho_{s,M} E_k = \mathrm{tr}_{s,M}P^\dagger_k\rho_{s,M}P_k E_k$$ I.e. The rates must be repoducible by both measures ##P_k## and ##E_k##

Premise 4) Since the pointer positions are mutually exclusive, it must be the case that ##E_kE_{k'} = \delta_{k,k'}##

I think premises 1 + 2 would give us an quantum mechanically describable ancilla that must exist and premises 3 + 4 would say the measurement scenario involving this ancilla must also be describable with a projective decomposition. Which of these would you take issue with?
Trying to translate your statements into precise formulas, and using ##\Pi## in place of ##E## (which to me signifies an energy level, not an operator):

The state of system+detector is ##\widehat\rho=\rho_S\otimes\rho_D##. Premise 2 amounts to

(B1) The ##P_k## form a POVM, i.e., they are Hermitian positive semidefinite operators summing to the identity. This leads to response probabilities
$$p_k=\mathrm{tr}_SP_k \rho_S=\mathrm{tr}(P_k\otimes 1)\widehat\rho.$$
Premise 1 and 4 amount to

(B2) The ##\Pi_k## form a complete family of orthogonal projectors to the joint eigenspaces of a vector of commuting operators corresponding to simultaneous measurement results ##a_k##, interpreted according to Born's rule for projective measurements. This leads to the response probabilities
$$p_k=\mathrm{tr}_D\Pi_k \rho_D=\mathrm{tr}(1\otimes \Pi_k)\widehat\rho.$$
Premise 3 says that the probabilities ##p_k## in (B1) and (B2) are the same.

Now please complete the argument that (B1) and (B2) explain the POVM in terms of the projective measurements.

A. Neumaier said:
In practice, pointers on a continuous scale are readable only approximately. Hence the actual measurements average over some neighborhood, which spoils exact orthogonality
For the moment, assume that the eigenvalues are discrete, so that this is not yet an issue.

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Demystifier
Deleted my initial response because I found a big mistake. Will respond tomorrow.

I believe Sec. 3.3 of my "Bohmian mechanics for instrumentalists" is helpful here, but I think @A. Neumaier finds it too sketchy and wants a more detailed and rigorous analysis.

Morbert
Earlier I tried to impose a consistent histories understanding of measurement to a POVM. Under consistent histories, an apparatus with a pointer observable/'perceptible' ##O:=\sum_k o_k\Pi_k## measures a quantity ##A:=\sum a_k P_k## if a pointer 'outcome' ##o_k## logically implies a 'result' ##a_k##, and vice versa, for every possible outcome and result. This implication can be expressed* with conditional probabilities $$p(a_k|o_k) = p(o_k|a_k) = 1$$or equivalently $$p(a_k) = p(o_k) = p(a_k \land o_k)$$These quantities are computed like so $$\begin{eqnarray*}p(a_k) &=& \mathrm{tr}_SP_k\rho_S\\p(o_k) &=& \mathrm{tr}_D\Pi_k\rho_D\\p(a_k\land o_k) &=& \mathrm{tr}(1\otimes\Pi_k)(P^{1/2}_k\otimes 1)\hat{\rho}(P^{1/2}_k\otimes 1)^\dagger\end{eqnarray*}$$But the quantity ##p(a_k\land o_k)## is not equal to the other two when ##\{P_k\}## are not projectors, as is the case with the quantity ##A## in the Quantum Tomography paper. Premise 3 above is wrong. CH can't immediately relate a POVM to a measurement of a physical quantity. You have to do something like this which might not be immediately relevant to here.
A. Neumaier said:
Now please complete the argument that (B1) and (B2) explain the POVM in terms of the projective measurements.
I think I now agree that a POVM cannot be replaced with a projective measurement on a larger system without losing some of the advantages of the POVM approach. At most, if you present me with a scenario where a detector responds to sources, modeled by a POVM, I can present to you an equivalent scenario where a researcher responds to the detector, and model this "researcher response" with a PVM.

*  - Adding the reference for this: "The Interpretation of Quantum Mechanics" by Roland Omnes

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Morbert said:
I think I now agree that a POVM cannot be replaced with a projective measurement on a larger system without losing some of the advantages of the POVM approach. At most, if you present me with a scenario where a detector responds to sources, modeled by a POVM, I can present to you an equivalent scenario where a researcher responds to the detector, and model this "researcher response" with a PVM.
Why is the equivalent scenario (in the second sentence) not enough? What advantage of the POVM approach is lost by that?

An additional note. CH starts from the assumption/axiom that probability of a history is given by a formula involving a product of projectors at different times. In CH, all probabilities are derived from that, right? If so, isn't it an indication that, in principle, all probabilities of measurement outcomes in CH can be expressed in terms of projectors, without non-projector POVM's?

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Morbert said:
if you present me with a scenario where a detector responds to sources, modeled by a POVM, I can present to you an equivalent scenario where a researcher responds to the detector, and model this "researcher response" with a PVM.
Yes, I agree that this simulation can always be done. But the quest is whether one can explain all actual POVM measurements in terms of actual (rather than equivalently simulated) projective measurements.

Demystifier said:
Why is the equivalent scenario (in the second sentence) not enough? What advantage of the POVM approach is lost by that?

An additional note. CH starts from the assumption/axiom that probability of a history is given by a formula involving a product of projectors at different times. In CH, all probabilities are derived from that, right? If so, isn't it an indication that, in principle, all probabilities of measurement outcomes in CH can be expressed in terms of projectors, without non-projector POVM's?
In CH, the probabilities derived from products of projectors are quite fundamental, and measurement scenarios are derived from the logic they allow. I was hoping it would be possible to represent a measurement of the 'quantity' ##A:=\sum_ka_kP_k## using histories, but I don't think it is possible. We are limited to an analysis like the one here, where Griffiths presents two options for handling the most general POVMs: associate each outcome ##k## with a different, imperfect measurement of a different set of possible properties (see equation 38 + 40), or use histories that reference just the detector, without any measured quantity (see equation 41).

I.e. While I think CH can provide an interesting analysis of POVMs, the measurement loses its 'measurement' character, as the quantity being measured has no representation (as far as I can tell) in any set of histories. To answer your first question, it's this generalisation of a measured quantity that is the lost advantage. It seems like a quite succinct way to characterise detectors.

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Demystifier
Morbert said:
or use histories that reference just the detector, without any measured quantity (see equation 41).

It seems like a quite succinct way to characterise detectors.
That's exactly how I interpret POVM measurements in my "Bohmian mechanics for instrumentalists", without using CH interpretation. But my point is, if you interpret "measurement" as a determination of the state of detector, then all measurements are projective measurements. POVM measurement is a description from the point of view of "microscopic measured system", but when you look at things from the macroscopic apparatus point of view, then POVM's are not needed and PVM's are enough.

Demystifier said:
That's exactly how I interpret POVM measurements in my "Bohmian mechanics for instrumentalists",
Your proposal in Section 3.3 has the defect that the Hilbert space in which the ancilla constructed from a POVM by Naimark's theorem lives is not part of the Hilbert space of the universe in which your other arguments take place.

"repeating the analysis described in Eqs. (12)-(16)'' that is is based on perceptibles, which must be modeled in the Hilbert space of the universe, proves nothing about formal measurement by an artificial ancilla.

A. Neumaier said:
Your proposal in Section 3.3 has the defect that the Hilbert space in which the ancilla constructed from a POVM by Naimark's theorem lives is not part of the Hilbert space of the universe in which your other arguments take place.

"repeating the analysis described in Eqs. (12)-(16)'' that is is based on perceptibles, which must be modeled in the Hilbert space of the universe, proves nothing about formal measurement by an artificial ancilla.
Yes, but I consider this to be a feature, not a defect, of my proposal.

Demystifier said:
Yes, but I consider this to be a feature, not a defect, of my proposal.
I agree that it is an artificial feature of your proposal, without any relation to physics.

A. Neumaier said:
Your proposal in Section 3.3 has the defect that the Hilbert space in which the ancilla constructed from a POVM by Naimark's theorem lives is not part of the Hilbert space of the universe in which your other arguments take place.

"repeating the analysis described in Eqs. (12)-(16)'' that is is based on perceptibles, which must be modeled in the Hilbert space of the universe, proves nothing about formal measurement by an artificial ancilla.
I'm not sure I understand this. The measured system is microscopic and part of the universe. The ancilla is the macroscopic apparatus in the lab that is also part of the universe. Are you saying the ancilla you get from Naimark's theorem cannot correspond to the actually existing macroscopic apparatus?

Demystifier
A. Neumaier said:
I agree that it is an artificial feature of your proposal, without any relation to physics.
But it's based on von Neumann theory of measurement. Are you saying that von Neumann theory of measurement is artificial, without any relation to physics?

Demystifier said:
But it's based on von Neumann theory of measurement. Are you saying that von Neumann theory of measurement is artificial, without any relation to physics?
My impression is that von Neumann theory of measurement is indeed very artificial and dangerously misleading. There "are too" many pure states in von Neumann theory of measurement!

Morbert said:
I'm not sure I understand this. The measured system is microscopic and part of the universe. The ancilla is the macroscopic apparatus in the lab that is also part of the universe. Are you saying the ancilla you get from Naimark's theorem cannot correspond to the actually existing macroscopic apparatus?
The ancilla constructed in Naimark's theorem makes no reference to macroscopic objects, so making a connection would have to be justified separately, which hasn't been done.

This means that Nikolic's discussion in Section 3.3 is unconnected to his discussion in Section 3.2, in spite of the opposite claim.

Demystifier said:
But it's based on von Neumann theory of measurement. Are you saying that von Neumann theory of measurement is artificial, without any relation to physics?
His theory is about measurements in the physical Hilbert space - the Hilbert space of the universe traced over various environments, since he never considers the whole universe - , not an artificially constructed one.

Can't we just stipulate the isometry ##J:\mathcal{H}_\textrm{microscopic system}\rightarrow\mathcal{H}_\textrm{macroscopic apparatus}## such that $$P_k = J^\dagger\Pi_\textrm{pointer property k}J$$Is this what is considered artificial?

A. Neumaier said:
His theory is about measurements in the physical Hilbert space - the Hilbert space of the universe traced over various environments, since he never considers the whole universe - , not an artificially constructed one.
I still don't understand why do you think that my apparatus states are unphysical.

Morbert said:
Can't we just stipulate the isometry ##J:\mathcal{H}_\textrm{microscopic system}\rightarrow\mathcal{H}_\textrm{macroscopic apparatus}## such that $$P_k = J^\dagger\Pi_\textrm{pointer property k}J$$Is this what is considered artificial?
Which isometry? You'd need to find one that corresponds to the behavior of the actual measurement devices as predicted by quantum dynamics!
Demystifier said:
I still don't understand why do you think that my apparatus states are unphysical.
The unphysicality problem is not in Section 3.2 where you have physical apparatus states. But in Section 3.3, your apparatus states are unphysical since they are not related to something in the Hilbert space of the physical universe but reside in the purely mathematically constructed abstract Hilbert space given through Naimark's theorem.

On the other hand, there is also a problem in Section 3.2, where the 'Hence' before (4) - in the archiv version which I have - does not follow logically but is just a hope. Indeed, (4) is problematic and probably wrong when the two pointer results are very close.

A. Neumaier said:
The unphysicality problem is not in Section 3.2 where you have physical apparatus states. But in Section 3.3, your apparatus states are unphysical since they are not related to something in the Hilbert space of the physical universe but reside in the purely mathematically constructed abstract Hilbert space given through Naimark's theorem.
You misunderstood Sec. 3.3, there the apparatus states are as physical as those in Sec. 3.2. The explicit construction in Sec. 3.3 does not use the Naimark's theorem.
A. Neumaier said:
On the other hand, there is also a problem in Section 3.2, where the 'Hence' before (4) - in the archiv version which I have - does not follow logically but is just a hope. Indeed, (4) is problematic and probably wrong when the two pointer results are very close.
No, (4) is true whenever the outcomes are distinguishable. If the pointer results are so close that (4) is wrong, then the outcomes are not distinguishable.

Demystifier said:
You misunderstood Sec. 3.3, there the apparatus states are as physical as those in Sec. 3.2. The explicit construction in Sec. 3.3 does not use the Naimark's theorem.
What then is the role of his theorem, why do you cite it? How do you arrive at (17) given a POVM measurement setup?

Demystifier said:
No, (4) is true whenever the outcomes are distinguishable. If the pointer results are so close that (4) is wrong, then the outcomes are not distinguishable.
But if you measure the position of pixels on a screen there will be ambiguities at the borders of the pixels, which are correctly resolved only if there is a nontrivial overlap. This is accounted for by coherent state POVMs but not by your setting.

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A. Neumaier said:
What then is the role of his theorem, why do you cite it?
It's a general remark, not essential for the rest of the analysis.
A. Neumaier said:
How do you arrive at (17) given a POVM measurement?
I explained it to you three years ago here:
(EDIT: Does the link work correctly for you? When I click it, it moves me to an entirely unrelated thread.)
A. Neumaier said:
But if you measure the position of pixels on a screen there will be ambiguities at the border of the pixels, which are correctly resolved only if there is a nontrivial overlap. This accounted for by coherent state POVMs but not by your setting.
I don't understand your point. The pixels on the screen can in practice be reasonably well (not perfectly well) distinguished precisely because their overlap is small, i.e. the because the area of the border is much smaller than the area of the pixel.

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Demystifier said:
It's a general remark, not essential for the rest of the analysis.
Then its misleading since it plays no role anywhere at all!
Demystifier said:
I explained it to you three years ago here:
(EDIT: Does the link work correctly for you? When I click it, it moves me to an entirely unrelated thread.)
The link does not lead to a post from you. Maybe copy the intended post to here!
Demystifier said:
I don't understand your point. The pixels on the screen can in practice be reasonably well (not perfectly well) distinguished precisely because their overlap is small, i.e. the because the area of the border is much smaller than the area of the pixel.
But unless there is a significant overlap of their wave functions (not the visbile pixels themselves), particles arriving at positions directly at the border between them will not be measured, unlike what one would expects in practice - 50% chance to be registered on each side.

A. Neumaier said:
The link does not lead to a post from you. Maybe copy the intended post to here!
See the link in post #24.
A. Neumaier said:
But unless there is a significant overlap of their wave functions (not the visbile pixels themselves), particles arriving at positions directly at the border between them will not be measured, unlike what one would expects in practice - 50% chance to be registered on each side.
Yes, but in practice that probability can be very small, so it's not important in the FAPP sense.

A. Neumaier said:
Then its misleading since it plays no role anywhere at all!
Perhaps it's not misleading if one thinks of it in the following way. One should distinguish the statement of the Naimark's theorem from a proof of the Naimark's theorem. The theorem only states that an appropriate PVM exists, it states nothing about the explicit construction of an appropriate PVM. The proof, on the other hand, may involve a concrete construction which is physically artificial, but nobody says that there are no other more physical constructions. Hence referring to the statement of the theorem, but not to its proof, does not assume any particular construction and hence contains nothing physically artificial.

A side note: It reminds me of the Godel theorem, the proof of which involves a very artificial self-referring sentence of the form "This sentence can't be proved", but the statement of the theorem contains nothing artificial. Indeed, there are non-artificial non-self-referring demonstrations of the Godel theorem, e.g. the Goodstein's theorem is a "normal" mathematical statement which is true but cannot be proved from first order Peano axioms.

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Demystifier said:
Hence referring to the statement of the theorem, but not to its proof, does not assume any particular construction
Without construction it is just an empty statement. To apply the statement in your context you need to give a construction in terms of projective measurements that can be performed (at least in principle) in the physical universe.

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A. Neumaier said:
Without construction it is just an empty statement. To apply the statement in your context you need to give a construction in terms of projective meaurements that can be performed (at least in principle) in the physical universe.
And I gave such a construction.

If the space ##\mathcal{H}_\textrm{measurement apparatus}## is large enough to specify microscopic configuration of the measurement apparatus, and the 'pointer projectors' ##\{\Pi_k\}## project to 1D subspaces, then these pointers would be impossible to read by any mortal physicist, and measurement would be impossible. In reality, the measurement apparatus would be represented with some simplified model of its human-readable collective degrees of freedom, or ##\{\Pi_k\}## would correspond to a highly coarse-grained decomposition of the identity, projecting to enormous subspaces.

Demystifier
Demystifier said:
And I gave such a construction.
where?

A. Neumaier said:
where?
Eq. (18) and the text before it. More details in the link in post #24.

Demystifier said:
Eq. (18) and the text before it. More details in the link in post #24.
This is not a construction but a sketchy outline of a program that to make it work one would have to justify by a serious derivation.

Instead you make strong assumptions about arbitrary measurements that go far beyond what has been demonstrated in the literature (where always a nondemolition assumption was involved to get decoherence). I had criticised your approach in detail in that thread. I reread the whole thread and still think my criticism is fully valid and that you give nothing but empty phrases and expressions of hope asserted as truth.

Moreover, POVMs appear nowhere - your argument does not depend in any way on the properties of POVMs, which should you make deeply suspicious of your arguments having anything to do with them.

A. Neumaier said:
Instead you make strong assumptions about arbitrary measurements that go far beyond what has been demonstrated in the literature (where always a nondemolition assumption was involved to get decoherence).
I'm not sure I understand your point. Are you saying that in the literature only the non-demolition is assumed, while I assume something additional? What is this additional assumption that I make?

Demystifier said:
I'm not sure I understand your point. Are you saying that in the literature only the non-demolition is assumed, while I assume something additional? What is this additional assumption that I make?
No. You make fewer assumptions but pretend that essentially the same results follow, without telling why this should be so.

A. Neumaier said:
No. You make fewer assumptions but pretend that essentially the same results follow, without telling why this should be so.
I make fewer explicit assumptions, but the assumptions are implicitly there. That's because I think like physicist, not mathematician, so I try to explain how nature works, not to present a mathematical proof. I don't make explicit assumptions which seem obvious to me from a physical point of view, because I see such details as distraction from really important ideas. But I perfectly understand that you, as a mathematician, don't like this type of reasoning.

<h2>1. What is a POVM measurement?</h2><p>A POVM (positive operator-valued measure) measurement is a type of quantum measurement that allows for a more general description of the outcomes compared to projective measurements. It involves a set of positive operators that sum to the identity operator, and each operator corresponds to a possible measurement outcome.</p><h2>2. How is a POVM measurement different from a projective measurement?</h2><p>A projective measurement is a type of quantum measurement that involves projecting a quantum state onto one of the eigenstates of the measured observable. In contrast, a POVM measurement allows for a wider range of possible outcomes and does not require the measured observable to have discrete eigenvalues.</p><h2>3. Can POVM measurements be explained by projective measurements?</h2><p>Yes, POVM measurements can be explained by projective measurements in the sense that any POVM measurement can be decomposed into a set of projective measurements. However, this decomposition may not be unique and may involve multiple projective measurements for each POVM element.</p><h2>4. What are the advantages of using POVM measurements?</h2><p>POVM measurements have several advantages over projective measurements. They allow for a more general description of measurement outcomes, can be used to measure non-commuting observables, and can be implemented more easily in experiments. Additionally, POVM measurements can provide more information about the quantum state being measured.</p><h2>5. Are POVM measurements commonly used in quantum experiments?</h2><p>Yes, POVM measurements are commonly used in quantum experiments, particularly in areas such as quantum information and quantum computing. They have also been used in various applications such as quantum cryptography and quantum metrology. However, projective measurements are still more widely used in many experimental setups due to their simplicity and ease of implementation.</p>

## 1. What is a POVM measurement?

A POVM (positive operator-valued measure) measurement is a type of quantum measurement that allows for a more general description of the outcomes compared to projective measurements. It involves a set of positive operators that sum to the identity operator, and each operator corresponds to a possible measurement outcome.

## 2. How is a POVM measurement different from a projective measurement?

A projective measurement is a type of quantum measurement that involves projecting a quantum state onto one of the eigenstates of the measured observable. In contrast, a POVM measurement allows for a wider range of possible outcomes and does not require the measured observable to have discrete eigenvalues.

## 3. Can POVM measurements be explained by projective measurements?

Yes, POVM measurements can be explained by projective measurements in the sense that any POVM measurement can be decomposed into a set of projective measurements. However, this decomposition may not be unique and may involve multiple projective measurements for each POVM element.

## 4. What are the advantages of using POVM measurements?

POVM measurements have several advantages over projective measurements. They allow for a more general description of measurement outcomes, can be used to measure non-commuting observables, and can be implemented more easily in experiments. Additionally, POVM measurements can provide more information about the quantum state being measured.

## 5. Are POVM measurements commonly used in quantum experiments?

Yes, POVM measurements are commonly used in quantum experiments, particularly in areas such as quantum information and quantum computing. They have also been used in various applications such as quantum cryptography and quantum metrology. However, projective measurements are still more widely used in many experimental setups due to their simplicity and ease of implementation.

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