Nature Physics on quantum foundations

[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).

 

[vanhees71]

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##.

If this is your POVM for a joint measurement of ##x## and ##p##, so how is this concretely realized in an experiment? Maybe it's easier to give the POVM (math) first, and then give a measurement device?

 

[Morbert]

As an aside: The discussion between @Demystifier and @A. Neumaier might have some relevance to consistent histories.

For every possible preparation of a microscopic system ##s##, the detector response theorem let's us compute the mean rates ##p_k## of measurement outcomes ##k## from a quantum measure ##P_k## like so $$p_k = \mathrm{tr}_{s}\rho_sP_k$$ Luis and Sanchez-Soto give a similar expression, but they consider a Hilbert space that includes both the microscopic system and ancilla ##a## relevant to the measurement process $$p_k = \mathrm{tr}_{s,a}\rho_s \rho_aU^\dagger|k\rangle\langle k|U$$My understanding is that @A. Neumaier maintains that there is not necessarily a physical ancilla ##a## that would let us recover projectors ##|k\rangle\langle k|## from the measure ##P_k## for every physical measurement scenario. i.e. $$P_k = \mathrm{tr}_a\rho_aU^\dagger|k\rangle\langle k|U$$This is actually something that consistent histories relies upon, as physical properties are associated with projective decompositions of the identity (PDI), not POVMs. If there are measurements for which there is no PDI, then that wold mean trouble for consistent histories, as we would have physical properties (the macroscopic measurement outcomes) that have no representation in consistent histories quantum theory.

 

[A. Neumaier]

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

No, these are meters that produce the measured results, not ancillas in the sense Peres (and everyone else) is using the term.

 

[A. Neumaier]

If this is your POVM for a joint measurement of ##x## and ##p##, so how is this concretely realized in an experiment? Maybe it's easier to give the POVM (math) first, and then give a measurement device?

A concrete experimental realization is given in the paper Demystifier referred to in post #379; see also #399.

 

[A. Neumaier]

My understanding is that @A. Neumaier maintains that there is not necessarily a physical ancilla a that would let us recover projectors

Yes. It is a one-way street.

If you happen to have an ancilla in an environment and use it to define a measurement process, you get a POVM.

But given an arbitrary experimental arrangement for quantum detection one always has a quantum measure (by my detector response theorem) and hence a POVM , but usually no ancilla in the quantum description of the whole experiment. One can construct an associated ancilla by Naimark's theorem, but this ancilla does not live in the Hilbert space of the environment one started with but in an artificially constructed Hilbert space.

 

[Demystifier]

No, these are meters that produce the measured results, not ancillas in the sense Peres (and everyone else) is using the term.

Well, those meters are "ancillas" in the sense I used that term in post #374, for which you asked me to give a specific model. Maybe others don't use the word "ancilla" in that sense, but in that sense the ancilla is manifestly physical. If it's really so non-standard to think of "ancilla" in that sense, maybe I should write a paper about it, titled something like "Measurement apparatus as ancilla".

 

[vanhees71]

A concrete experimental realization is given in the paper Demystifier referred to in post #379; see also #399.

In #379 it's just standard QT. No POVM is constructed.

 

[Demystifier]

In #379 it's just standard QT. No POVM is constructed.

No POVM is explicitly constructed in that paper, but POVM is implicitly there as I explained in #374.
I also discuss this stuff briefly in my "Bohmian mechanics for instrumentalists", Sec. 3.3.

 

[vanhees71]

From our discussion here, I've gotten once more the impression that the POVM approach is entirely theoretical. Concrete understanding of measurement protocols is rather modeled by standard theory of open quantum systems and thus entirely within the standard interpretation of the quantum state using Born's rule.

 

[Morbert]

Yes. It is a one-way street.

If you happen to have an ancilla in an environment and use it to define a measurement process, you get a POVM.

But given an arbitrary experimental arrangement for quantum detection one always has a quantum measure (by my detector response theorem) and hence a POVM , but usually no ancilla in the quantum description of the whole experiment. One can construct an associated ancilla by Naimark's theorem, but this ancilla does not live in the Hilbert space of the environment one started with but in an artificially constructed Hilbert space.

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?

 

[A. Neumaier]

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

I don't understand the precise intended meaning of 'associated'?

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

If the ##E_k## are measures, how can their product be a number?

In practice, pointers on a continuous scale are readable only approximately. Hence the actual measurements average over some neighborhood, which most likely spoils exact orthogonality of whatever you precisely mean by ##E_k##

 

[Morbert]

Oops, yes I meant ##E_kE_{k'} = \delta_{k,k'}E_k## or ##\mathrm{tr}\rho E_kE_{k'} = \delta_{k,k'}\mathrm{tr}\rho E_k##. And yes continuous measurement would need a clearer treatment.

 

[Morbert]

I don't understand the precise intended meaning of 'associated'?

I'm trying to relate the POVM used to predict the mean rates, with a "pointer observable" the experimenter directly looks at (like a dial position) to read off results, as I think that is the pointer system is the ancilla that can be added to recover projective measurement. So this association would be an isometry ##J:\mathcal{H}_s\rightarrow\mathcal{H}_M## such that ##J^\dagger E_k J = P_k##.

 

[A. Neumaier]

I'm trying to relate the POVM used to predict the mean rates, with a "pointer observable" the experimenter directly looks at (like a dial position) to read off results,

So the preparation of the projective measurement may involve computer calculations that go into computing the pixels that determine the number on an electronic display which the experimenter looks at?

as I think that is the pointer system is the ancilla that can be added to recover projective measurement. So this association would be an isometry ##J:\mathcal{H}_s\rightarrow\mathcal{H}_M## such that ##J^\dagger E_k J = P_k##.

So ##E_k## is a projection matrix, not a measure?

 

[Morbert]

So the preparation of the projective measurement may involve computer calculations that go into computing the pixels that determine the number on an electronic display which the experimenter looks at?

So ##E_k## is a projection matrix, not a measure?

Yes, sorry I though the former was a special case of the latter

 

[A. Neumaier]

Yes, sorry I though the former was a special case of the latter

well, not quite. There is an isomorphism (though not s canonical one) between measures on the set of values and projection valued measures indexed by the set of values. This means that, informally, there are many more PVMs than measures, though each PVM defines a measure.

But whereas before you had used set brackets to distinguish sets from individuals, you now talked about ##E_k## without brackets, which would be single projections. These have no relation to measures.

 

[Morbert]

But whereas before you had used set brackets to distinguish sets from individuals, you now talked about ##E_k## without brackets, which would be single projections. These have no relation to measures.

After I looked through your paper I tried to adopt the convention used there but I should have been explicit in this change. E.g. This sentence

is defining a quantum measure as the collection of ##P_k##s right?

 

[A. Neumaier]

After I looked through your paper I tried to adopt the convention used there but I should have been explicit in this change. E.g. This sentence

View attachment 316289

is defining a quantum measure as the collection of ##P_k##s right?

Yes, but you forgot to add the term quantum, which makes a difference. Without that qualification, measure has the standard meaning as everywhere in mathematics. (A measure may be considered as a quantum measure where all ##P_k## are multiples of the identity.)

 

[Demystifier]

The editors of high impact journal Nature Physics explain why the field of quantum foundations is important for physics.
https://www.nature.com/articles/s41567-022-01766-x

As a reaction to this, another paper in the same journal is published yesterday. I can't see the whole paper (behind the paywall), but given the authors it must be about superdeterminism.
https://www.nature.com/articles/s41567-022-01831-5

 

[A. Neumaier]

As a reaction to this, another paper in the same journal is published yesterday. I can't see the whole paper (behind the paywall), but given the authors it must be about superdeterminism.
https://www.nature.com/articles/s41567-022-01831-5

the observed violations of Bell’s inequality can be said to show that maintaining local causality requires violating statistical independence. [...] Types of local hidden variables theories that violate statistical independence include those that are superdeterministic, retrocausal [or] supermeasured.