Eigenvalue degeneracy in real physical systems

[A. Neumaier]

I do sympathize with this framework as it (correct me if I am wrong) allows avoiding usage of spectral theorem.

Nothing needs correction, except your interpretation of Naimak's theorem.

That you can simulate a POVM in a - different, nonphysical - Hilbert space doesn't have any practical relevance. POVMs work not because of Naimark's theorem but because of agreement with experiments.

 

[atyy]

However, POVMs can't resolve the mathematical computability issue that ErikZorkin brought up, since they can always be seen as PVMs on a larger Hilbert space, so if they could resolve the issue, then the issue with the PVMs would also be resolved, which is apparently impossible. I think the physical resolution is what I have written in posts #17 and #19 and of course it can also be formulated using POVMs.

I think Naimark's equivalence between POVMs and PVMs depending on small or large Hilbert spaces only applies to the Born rule part of the observables, not the collapse (I don't think the projection postulate exists for continuous variables).

However, I do agree with you that POVMs are not the solution, since the question asked by the OP can be stated for discrete variables. It needs some generalization for continuous variables, but the discrete version is not misleading.

 

[ErikZorkin]

I think Naimark's equivalence between POVMs and PVMs depending on small or large Hilbert spaces only applies to the Born rule part of the observables, not the collapse (I don't think the projection postulate exists for continuous variables).

However, I do agree with you that POVMs are not the solution, since the question asked by the OP can be stated for discrete variables. It needs some generalization for continuous variables, but the discrete version is not misleading.

Actually, I was thinking of a simple discrete example in the first place, not even continuous. Take Stern-Gerlach experiment, for instance. There, you can easily demonstrate the degeneracy problem. If the beam splitting is solely undetectable, how can you "project" your state correctly? I thought POVMs could at least describe rigorously what an approximate measurement is (in terms of measuring range, not exact value). Which is, at least for me, a reminiscent of post #17. Because POVMs describe what the final state is explicitly without application of spectral decomposition, as far as I understand.

 

[atyy]

Actually, I was thinking of a simple discrete example in the first place, not even continuous. Take Stern-Gerlach experiment, for instance. There, you can easily demonstrate the degeneracy problem. If the beam splitting is solely undetectable, how can you "project" your state correctly? I thought POVMs could at least describe rigorously what an approximate measurement is (in terms of measuring range, not exact value). Which is, at least for me, a reminiscent of post #17. Because POVMs describe what the final state is explicitly without application of spectral decomposition, as far as I understand.

For discrete observables, POVMs and the old-fashioned projection rule are equivalent, depending on how big of a Hilbert space one chooses to work with. I think rubi gave you the answer back around post #17? The generalization of the von Neumann rule that includes degenerate spaces is called the Luders rule http://arxiv.org/abs/1111.1088v2.

 

[ErikZorkin]

For discrete observables, POVMs and the old-fashioned projection rule are equivalent, depending on how big of a Hilbert space one chooses to work with. I think rubi gave you the answer back around post #17? The generalization of the von Neumann rule that includes degenerate spaces is called the Luders rule http://arxiv.org/abs/1111.1088v2.

Thanks for the hint!

Well, rubi gave a good answer, but then he himself pointed out some difficulties with it. See post #52

 

[atyy]

Thanks for the hint!

Well, rubi gave a good answer, but then he himself pointed out some difficulties with it. See post #52

They are not real difficulties, as long as the variable is discrete. Within Copenhagen, what counts as a "measurement" is subjective. So we can always take the Luders rule and add any unitary operation to it, and count the (Luders rule + unitary operation) as the "measurement".

However, it should be said that if one considers the spirit of Copenhagen to be a "smaller Hilbert space" view, in the sense that a sensible interpretation of the wave function of the universe is not available, then POVMs are more fundamental than projection measurements. http://mattleifer.info/wordpress/wp-content/uploads/2008/11/commandments.pdf

 

[ErikZorkin]

They are not real difficulties, as long as the variable is discrete

Well, discrete means all distinct? Because otherwise, we are in trouble. If we simply approximate the spectrum and projections, we might drop off something important.

By the way, what about other interpretations? Does, say, Bohmian pilot wave interpretation also suffer from spectral decomposition?

 

[atyy]

Well, discrete means all distinct? Because otherwise, we are in trouble. If we simply approximate the spectrum and projections, we might drop off something important.

The projection postulate only holds for discrete variables. If a position measurement is made, the state after that cannot be a position eigenstate, because the position eigenstate is not a valid state (not square integrable).

By the way, what about other interpretations? Does, say, Bohmian pilot wave interpretation also suffer from spectral decomposition?

I didn't quite understand the spectral decomposition problem. I was referring to the collapse rule needing an additional assumption to become defined (the instrument and measurement operators, as can be seen for the generalized collapse rule for POVMs).

 

[ErikZorkin]

I didn't quite understand the spectral decomposition problem.

Spectral decomposition is uncomputable. You can't even DEFINE eigenvalues/vectors/spaces/projections. Only in approximate format or if it is known beforehand that eigenvalues are distinct.

 

[atyy]

Spectral decomposition is uncomputable. You can't even DEFINE eigenvalues/vectors/spaces/projections. Only in approximate format or if it is known beforehand that eigenvalues are distinct.

Really, could you give a reference?

 

[ErikZorkin]

Really, could you give a reference?

Sure.

 

[atyy]

Sure.

But one just needs to know how many distinct eigenvalues there are. Presumably this should be known from experiment.

 

[ErikZorkin]

Presumably this should be known from experiment.

No.

Take an example of Stern-Gerlach experiment and suppose that beam splitting is completely undetectable (that it's beyond Planck scale, for instance -- the process of collapse still happens but we have no idea about the final state). See the image:

 

[atyy]

No.

Take an example of Stern-Gerlach experiment and suppose that beam splitting is completely undetectable (that it's beyond Planck scale, for instance -- the process of collapse still happens but we have no idea about the final state). See the image:

Then just omit the spin variable.

 

[ErikZorkin]

Then just omit the spin variable.

Eh, what?

And by the way, how can you deduce the number of eigenvalues from the experiment? It's a mathematical property of the operator.

 

[atyy]

Eh, what?

If there is no spin, then you won't get any splitting in the Stern-Gerlach experiment. So there will be no splitting in your theory, and no observed splitting, and all will be well.

And by the way, how can you deduce the number of eigenvalues from the experiment? It's a mathematical property of the operator.

OK, I don't know. In all practical cases, we don't seem to have a problem. For example, whether one needs a degeneracy or not depends on how good the experimentalist is. For example, in non-relativistic QM, the simplest Hamiltonians have lots of degeneracy. But then experiments get better, and one sees splittings, so one adds terms or uses better computations to get theory and experiment to match, eg. Zeeman, Lamb shift etc.

 

[ErikZorkin]

If there is no spin, then you won't get any splitting in the Stern-Gerlach experiment.

There is spin and there is splitting! They are just not detectable.

OK, I don't know. In all practical cases, we don't seem to have a problem.

Well, only because physicists don't care about math much. It is a fact that spectrum is uncomputable. Rubi was right, the only thing you can measure is that an eigenvalue lies in so or so range. You can't try to measure something, that you postulated mathematically, but what it uncomputable. It's nonsense. I am simply looking for a consistent explanation that is used in physics. Classical spectral decomposition and projection postulate are simply wrong when it comes to real experiments.

 

[ErikZorkin]

so one adds terms or uses better computations

What's that? "Adjusting" operators to better match with the reality (i.e. remove degeneracy)?

 

[bhobba]

What's that? "Adjusting" operators to better match with the reality (i.e. remove degeneracy)?

The value assigned to an observational outcome is entirely arbitrary.

The link I gave before explains clearly what's going on with POVM's etc etc:
https://www.physicsforums.com/threads/the-born-rule-in-many-worlds.763139/page-7

The value assigned to the elements of the POVM are irrelevant hence degeneracy is irrelevant and easily avoided - or not - it doesn't matter. The POVM is the important thing.

Thanks
Bill

 

[bhobba]

Well, only because physicists don't care about math much.

Why anyone would believe that has me beat. Some physicists have won Fields medals.

Thanks
Bill

 

[ErikZorkin]

Well, in the Stern-Gerlach experiment, the outcomes are quite certain, right? It's either spin up or down.

 

[bhobba]

Well, in the Stern-Gerlach experiment, the outcomes are quite certain, right? It's either spin up or down.

Spin up and spin down do not appear in the observable - you must assign it a number. That number is entirely arbitrary - it could be 1 and 0, 1 and 2, 1 and -1 or 1 an 1 - in which case you have degeneracy.

Thanks
Bill

 

[ErikZorkin]

As far as I remember, not all degenerates may be removed.

 

[bhobba]

As far as I remember, not all degenerates may be removed.

Hmmmm. Theoretically it should be possible - but likely not in a natural way.

Thanks
Bill

 

[ErikZorkin]

Theoretically it should be possible

How, for example?

 

[bhobba]

How, for example?

Read the link I gave.

An observation is a mapping to a POVM. The value assigned is entirely arbitrary.

An observation/measurement with possible outcomes i = 1, 2, 3 ... is described by a POVM Ei such that the probability of outcome i is determined by Ei, and only by Ei, in particular it does not depend on what POVM it is part of.

Thanks
Bill

 

[atyy]

What's that? "Adjusting" operators to better match with the reality (i.e. remove degeneracy)?

Yes. https://en.wikipedia.org/wiki/Fine_structure

 

[ErikZorkin]

Yes. https://en.wikipedia.org/wiki/Fine_structure

Nice example! But, again, it's not always possible to split the eigenvalues, as far as I understand.

An observation is a mapping to a POVM.

I sympathize with POVM approach as it (seemingly) avoids direct use of spectral decomposition (or?). Apparently, there has been a bit of discussion as to how it really addresses the question in the first place. For instance, POVM don't seem suitable for discrete variables.

 

[bhobba]

I sympathize with POVM approach as it (seemingly) avoids direct use of spectral decomposition (or?). Apparently, there has been a bit of discussion as to how it really addresses the question in the first place. For instance, POVM don't seem suitable for discrete variables.

I think you mean continuous variables.

Yes there is an issue - but it really needs a thread of its own.

That said continuous values don't ever actually occur.

Thanks
Bill

 

[vanhees71]

This is a phantastic example for the strange inability of mathematicians and physicists to communicate with each other. I must admit, I lost track what's the problem with "degeneracy" here. There is no such problem from a physicist's point of view, because it's all well defined in terms of observables (discribed by self-adjoint operators of Hilbert space) and states (described by a self-adjoint trace-class operator with trace 1, the statistical operator):

Any complete set of independent compatible observables $A_j$ ($j \in \{1,2,\ldots,n\}$) defines via the common (generalized) eigenvectors of the corresponding self-adjoint operators $\hat{A}_j$ (that are pairwise commuting by definition), denoted by $|a_1,\ldots,a_n \rangle$. Here the the components of the tupel $(a_1,\ldots,a_n) \in \mathbb{R}^n$ can run over discrete and/or continuous domains.

If the state of the system is represented by the statistical operator $\hat{\rho}$ the probability/probability density to simultaneously find the values $(a_1,\ldots,a_n)$ measuring the observables $A_j$ is given by (Born's rule):
$$P_{\rho}(a_1,\ldots,a_n)=\langle a_1,\ldots,a_n|\hat{\rho}|a_1,\ldots,a_n \rangle.$$
Everything else follows from the standard rules of probability theory. If I measure only von observable, e.g., $A_1$, I just have to sum/integrate over all other $a_2,\ldots,a_n$ over the corresponding spectrum, i.e.,
$$P_{\rho}(a_1)=\sum_{a_2,\ldots,a_n} P_{\rho}(a_1,\ldots,a_n).$$
This refers to measuring precisely the observable $A_1$. Of course in this case the eigenspaces of $\hat{A}_1$ alone are degenerate, because you need the other independent compatible observables to completely define the orthonormalized basis vectors (up to phase factors for each basis vector of course, but these cancel always out in physical results that are all dictated by the above given probability measure according to Born's rule). QT as a mathematical theory is consistent, as "weird" as some physicists and mathematicians seem to think it might be ;-))!

If you measure something not precisely you can envoke more complicated descriptions in terms of POVMs. Strictly speaking that you must do in any case if you measure an observable in the continuous part of the spectrum, because then you necessarily must use an apparatus with finite resolution, i.e., you can make, e.g., a position measurement of a particle, only up to a certain resolution determined by a the technology available for the measurement apparatus (which can be as fine as you like but not absolutely exact ever).

Now you can of course make this into a precise and well-defined mathematical formalism, and this is nice too, but one should not forget this not that difficult to understand physics and "metrological" constraints, which are very clear to everybody who as ever done experiments (and be it as a theoretician only in the standard lab practice mandatory to attend for every physics student wherever physics is taught).