Are PVMs more fundamental than POVMs from the view of decoherence?

[atyy]

In quantum mechanics, POVMs can be considered more fundamental than PVMs because PVMs can be considered a special sort of POVM. However, because of the Naimark extension, one can formally consider POVMs to always be derived from PVMs. Accordingly, one could argue that choosing PVMs or POVMs to be fundamental is a matter of taste. From a Copenhagenish standpoint, POVMs are more fundamental, since there is no wave function of the universe and the extension to a larger Hilbert space is not always physical.

However, one interesting explanation for the emergence of a preferred basis is decoherence. Formally, this depends on a unique basis for the Schmidt decomposition when systems+apparatus+environment is considered (reviewed by Schlosshauer in http://arxiv.org/abs/quant-ph/0312059). The Schmidt decomposition is an orthogonal decomposition, so it corresponds to a basis for a PVM. Does this mean that PVMs are more fundamental than POVMs from the standpoint of decoherence?

 

[bhobba]

I don't think so.

Decoherence is basically the phenomena of when you observe one part of an entangled system a superposition becomes a mixed state. I don't think it matters if its a Von-Neumann measurement or a generalised measurement.

Thanks
Bill

 

[atyy]

I don't think so.

Decoherence is basically the phenomena of when you observe one part of an entangled system a superposition becomes a mixed state. I don't think it matters if its a Von-Neumann measurement or a generalised measurement.

But if one considers the basis in which the superposition is diagonal, doesn't that correspond to the basis for a von Neumann measurement?

 

[bhobba]

But if one considers the basis in which the superposition is diagonal, doesn't that correspond to the basis for a von Neumann measurement?

Sure - but that doesn't mean if you do a general measurement it will show interference effects.

You are now getting into things I don't know the detail of, so I really can't help any more than that.

Thanks
Bill

 

[naima]

In this paper the authors show that the choice of a preferre basis is the choice of an "apparatus".
The set of povm setups is wider than the set of pvm setups.

 

[Demystifier]

POVM is more general than PVM, but PVM is more fundamental than POVM.

Perhaps someone has a general problem of understanding how something more general can be less fundamental? For that purpose consider the example of atoms and molecules. Atoms are more fundamental, but molecules are more general (atoms can be viewed as one-atom molecules). Similarly, PVM's can be viewed as "atoms" of POVM's.

If the atom/molecule analogy is not very enlightening, here is an analogy which is much closer to the point. Consider a 3-dimensional vector spece. On this space, consider
i) 3 arbitrary orthonormal vectors.
ii) n (larger or equal to 3) arbitrary vectors which do not lie in the same plane.
An arbitrary vector can be written as a superposition of vectors in i), but also as a superposition as vectors in ii). The former superposition can be done in a unique way, while the latter superposition (for n>3) is not unique. Obviously, ii) is more general than i), and yet i) is more fundamental than ii). Indeed, there is a close relation of i) with PVM and ii) with POVM.

 

[bhobba]

Similarly, PVM's can be viewed as "atoms" of POVM's.

One way to see this is that a POVM results from interacting a probe with a system then doing a Von Neumann measurement on the probe:
http://www.quantum.umb.edu/Jacobs/QMT/QMT_Chapter1.pdf

See section 1.2.3

Thanks
Bill

 

[atyy]

POVM is more general than PVM, but PVM is more fundamental than POVM.

Perhaps someone has a general problem of understanding how something more general can be less fundamental? For that purpose consider the example of atoms and molecules. Atoms are more fundamental, but molecules are more general (atoms can be viewed as one-atom molecules). Similarly, PVM's can be viewed as "atoms" of POVM's.

Actually, why do you say PVMs are more fundamental? One of the questions I'm interested in for this thread is whether there is any "physical" argument that would make PVMs and projective state reduction more fundamental than POVMs and completely positive trace non-increasing maps. From a Copenhagen viewpoint, and I think also a Bohmian viewpoint, POVMs and CP maps seem to be more fundamental. From a Copenhagen viewpoint, I think one argues that the probability distributions we can observe should respect the vector space structure, and the state reduction has to be completely positive so that if we extend the system, the state reduction of the extended system also takes density matrices to density matrices.

There is a good traditional physical argument for the von Neumann projective measurement in that it is repeatable, which is clearly a good condition for the pointer. But I don't know if this can be extended to continuous variables. It can be argued that we only need large but finite dimensional Hilbert spaces (especially since the standard model is probably not UV complete), in which case the von Neumann measurement is certainly fundamental. But here I would like to consider what happens if we also allow infinite dimensional Hilbert spaces and continuous variables to be fundamental in quantum mechanics.

If the atom/molecule analogy is not very enlightening, here is an analogy which is much closer to the point. Consider a 3-dimensional vector spece. On this space, consider
i) 3 arbitrary orthonormal vectors.
ii) n (larger or equal to 3) arbitrary vectors which do not lie in the same plane.
An arbitrary vector can be written as a superposition of vectors in i), but also as a superposition as vectors in ii). The former superposition can be done in a unique way, while the latter superposition (for n>3) is not unique. Obviously, ii) is more general than i), and yet i) is more fundamental than ii). Indeed, there is a close relation of i) with PVM and ii) with POVM.

Yes, the uniqueness of the orthogonal decomposition (when considering system+apparatus+environment) is why I thought maybe decoherence provides a physical viewpoint in which one could argue that PVMs and projective state reduction are more fundamental. Is that right, or can decoherence lead to a POVM (that is not physically derived from a PVM)? Another thought is that even if it is right that decoherence favours PVMs as fundamental, it is not very physical, since there is no exact decoherence?

 

[Demystifier]

Actually, why do you say PVMs are more fundamental? One of the questions I'm interested in for this thread is whether there is any "physical" argument that would make PVMs and projective state reduction more fundamental than POVMs and completely positive trace non-increasing maps. From a Copenhagen viewpoint, and I think also a Bohmian viewpoint, POVMs and CP maps seem to be more fundamental. From a Copenhagen viewpoint, I think one argues that the probability distributions we can observe should respect the vector space structure, and the state reduction has to be completely positive so that if we extend the system, the state reduction of the extended system also takes density matrices to density matrices.

There is a good traditional physical argument for the von Neumann projective measurement in that it is repeatable, which is clearly a good condition for the pointer. But I don't know if this can be extended to continuous variables. It can be argued that we only need large but finite dimensional Hilbert spaces (especially since the standard model is probably not UV complete), in which case the von Neumann measurement is certainly fundamental. But here I would like to consider what happens if we also allow infinite dimensional Hilbert spaces and continuous variables to be fundamental in quantum mechanics.

Yes, the uniqueness of the orthogonal decomposition (when considering system+apparatus+environment) is why I thought maybe decoherence provides a physical viewpoint in which one could argue that PVMs and projective state reduction are more fundamental. Is that right, or can decoherence lead to a POVM (that is not physically derived from a PVM)? Another thought is that even if it is right that decoherence favours PVMs as fundamental, it is not very physical, since there is no exact decoherence?

Let me put it this way.

If you think that quantum physics is fundamentally about (i) quantum formalism and (ii) classical macroscopic apparata (which is a rather Copenhagen view), then you are right, POVM is more fundamental.

But if you think that ALL nature is fundamentally quantum, then POVM cannot be fundamental. Let me explain. If all nature is fundamentally quantum, then you want to EXPLAIN why macroscopic world LOOKS classical, despite the fact that it is really not classical. (In the Copenhagen approach you do not try to explain it, you just take it for granted.) To explain it, you need a quantum theory of measurement which takes into account not only the quantum state of the microscopic measured system (which is what a Copenhagen approach does), but also the quantum state of the macroscopic measuring apparatus (which is typicaly ignored in a Copenhagen approach). When you do that, then the role of decoherence becomes essential. You are right that there is no exact decoherence and hence no exact PVM in such an approach. Yet, the PVM description of such systems is an almost perfect approximation valid FAPP. Moreover, the validity of such an approximation is essential for the decoherence approach to work. If you had POVM's which could not be well approximated by PVM's, then the decoherence approach would not work and you could not explain how the quantum measurement works and why macroscopic world looks classical despite the fact that it is really quantum.

Concerning continuous variables, in practice you can never measure a continuous spectrum. But you can measure (and do measure) a pseudo-continuous spectrum, that is, a discrete spectrum with a very small difference between values that can be experimentally distingushed. Here it is important to stress that in a non-Copenhagen approach it does not mean that discreteness is fundamental, because fundamentally (in a non-Copenhagen approach) quantum mechanics is NOT only about measurements.

In fact, in a non-Copenhagen approach neither PVM nor POVM are truly fundamental. Both are tools for describing measurements, and measurement is not fundamental in a non-Copenhagen approach. But PVM is more IMPORTANT than POVM, because it plays an important (even if only approximate) role in explaining the classical appearance of the quantum world at the macroscopic level.

I am not sure that this removes all your doubts, but I hope that it still helps. If you have further questions, I can try to answer them too.

 

[naima]

Consider measurements which destroy the particles.
How can we know if they are PVM or POVM?
Is it a physical or a mathematical problem?

 

[Demystifier]

Consider measurements which destroy the particles.
How can we know if they are PVM or POVM?
Is it a physical or a mathematical problem?

It is a POVM in the space of one-particle states, but a PVM in a larger QFT space of arbitrary number of particles. I would call it physical.

 

[naima]

Look at these setups

They seem to be "povm apparatuses". Is there a way to see these measurements as PVMs ? What would be the probe?

 

[atyy]

@Demystifier, thanks for the post above!

So would it be ok to say that decoherence is basically a "physical" Naimark extension?

When you talk about PVMs being more important in a non-Copenhagen decoherence-based approach, do you mean decoherence within Many-Worlds?

Or do you also include Bohmian Mechanics (I assume BM needs decoherence in the pre-measurement)?

I haven't read http://arxiv.org/abs/quant-ph/0308038 yet, which is about how POVMs arise from Bohmian Mechanics, but it seems relevant.

 

[naima]

One way to see this is that a POVM results from interacting a probe with a system then doing a Von Neumann measurement on the probe:
http://www.quantum.umb.edu/Jacobs/QMT/QMT_Chapter1.pdf

See section 1.2.3

Thanks
Bill

You give this link to show that PVM are "atoms of POVM". This link is interesting but it is about how to imitate POVM measurements with a probe. Not with direct POVM setups.
I gave a link which tells how to construct POVM devices for polarized photons. Where do you find underlying PVM in them?

 

[Demystifier]

Look at these setups

They seem to be "povm apparatuses". Is there a way to see these measurements as PVMs ? What would be the probe?

They certainly can be seen as PVM's if you introduce the wave function of the apparatus.

 

[Demystifier]

So would it be ok to say that decoherence is basically a "physical" Naimark extension?

Yes.

When you talk about PVMs being more important in a non-Copenhagen decoherence-based approach, do you mean decoherence within Many-Worlds?

Or do you also include Bohmian Mechanics (I assume BM needs decoherence in the pre-measurement)?

I include both many worlds and Bohmian mechanics. And yes, BM needs decoherence in (pre)measurement.

I haven't read http://arxiv.org/abs/quant-ph/0308038 yet, which is about how POVMs arise from Bohmian Mechanics, but it seems relevant.

I think the paper is relevant.