Deriving Measurement Operators for Realistic Detectors

[Killtech]

Lately I was studying the Bell and CHSH inequalities on Wikipedia (it has proven to be a good source to get an quick idea about everything). The articles are detailed and even provide the core of the proof in a mathematical derivation that is easy to understand. But it leaves me still with a few questions.

1. How exactly does the given proof exclude every possible local realistic theory? For me it seems that it only excludes possibility of the correlation sum to exceed 2 if the physical theory only models the entangled particles and the detectors (all with hidden information). But this seems to neglect the influence of space/vacuum between the detectors which the particles travel through.
   
   For example one could assume a theory where the vacuum is filled with various classical fields that are influenced by the detector setup. This influence may be local i.e. Information may travel with the speed of light (so just like for classical force fields). So the vacuum could polarize in some way according to the detector setup long before the experiment even begins (thus fill the space with the information about the measured axis).
   
   Now I can assume the created entangled particle pair is a highly instable state that collapses immediately (picks a definite spin orientation) after the particles are apart in such a theory. Let’s say the collapse is highly sensitive to initial conditions such that the polarized vacuum randomly picks which spin orientation is chosen and let’s say this happens always in the direction of one of the detectors. This way it seems I can get a theory that violates Bell mathematically maximally to the value of 4.
   
   However is it considered non local since the detectors effectively influence the particles seemingly non-locally via the vacuum? The definition of locality referenced by wiki does not look like any nontrivial vacuum was allowed.
   
   To exclude such a construction I would assume the CHSH setup would require to set the orientation of the detectors after the particles are created and just before they arrive. Or was that done already but it’s just not mentioned in Wikipedia? I’ve read something about an inter-detector communication loophole but it does not sound as if it covers it – although the solutions to close this loophole look similar I do not see how it considers a communication (filling the space) before the experiment begins.
   
   .
2. The second question is about the QM calculation or better said the operator used for the spin measurement. The spin operators act on the wave function in every part of the space but how is that a realistic representation of the detector’s measurement operators? After all each detector can only detect the spin of particles that hit it – so it also is a location detector of some sort, too.
   
   Thus I’d expect a more adequate representation of the detectors to be an operator which eigenstates are a combination of the possible eigenstates of both operators. Thus I thought restricting the spin operators to the space they are actually measured makes sense. While it does not make much difference for a single particle measurement it totally seems to breaks any kind of entanglement (mostly because the wave function only collapses locally so the part of the wave function in the other detector does not experience any change by measuring the spin of the first particle).
   
   So it seems the non-local nature of the spin measurement seems to be a core aspect of breaking Bell’s inequality.
   
   This said my general question is: given a blueprint of a detector, how do I derive the corresponding measurement operator correctly? I never had much to do with experimental physics so I never studied how this is done rigorously. Since I couldn’t find anything specific in wiki, I thought this might be a good place to ask.

 

[Meselwulf]

Lately I was studying the Bell and CHSH inequalities on Wikipedia (it has proven to be a good source to get an quick idea about everything).
[/LIST]

I disagree, I think wiki's explanation is long-winded and there are much more elementary ways of teaching the inequality, I would be happy to demonstrate if one asked?

 

[Killtech]

I disagree, I think wiki's explanation is long-winded and there are much more elementary ways of teaching the inequality, I would be happy to demonstrate if one asked?

it was the first one i found, hehe. well, now i do understand the core of the proof but not its entire framework - say the definitions of the terms used and so on. that's basically where my questions originate from. as a mathematician this all looks a bit vague to me apart from the calculations. but i got used to this when dealing with physics, hehe.

so if you can help me out on these parts i'd appreciate it!

however a small warning: i understand things best when it can be explained by formulas alone and one can skip the text (text can always be interpreted wrong. formulas can't).

 

[DrChinese]

To exclude such a construction I would assume the CHSH setup would require to set the orientation of the detectors after the particles are created and just before they arrive. Or was that done already but it’s just not mentioned in Wikipedia? I’ve read something about an inter-detector communication loophole but it does not sound as if it covers it – although the solutions to close this loophole look similar I do not see how it considers a communication (filling the space) before the experiment begins.

Yes, the detector settings were changed mid-flight in some Bell experiments. This is done in such a way that the new setting would need to be communicated to the other part of the apparatus faster than c. The gold standard for a Bell test, which I am sure is referenced in Wiki:

Violation of Bell's inequality under strict Einstein locality conditions

Gregor Weihs, Thomas Jennewein, Christoph Simon, Harald Weinfurter, Anton Zeilinger (University of Innsbruck, Austria, submitted on 26 Oct 1998)

We observe strong violation of Bell's inequality in an Einstein, Podolsky and Rosen type experiment with independent observers. Our experiment definitely implements the ideas behind the well known work by Aspect et al. We for the first time fully enforce the condition of locality, a central assumption in the derivation of Bell's theorem. The necessary space-like separation of the observations is achieved by sufficient physical distance between the measurement stations, by ultra-fast and random setting of the analyzers, and by completely independent data registration.

http://arxiv.org/abs/quant-ph/9810080

 

[Killtech]

thanks DrChinese,
that's the thing i was looking for!

though it still leaves the second question open

 

[DrChinese]

thanks DrChinese,
that's the thing i was looking for!

though it still leaves the second question open

You asked about the realism requirement, saying the following: "The spin operators act on the wave function in every part of the space but how is that a realistic representation of the detector’s measurement operators? After all each detector can only detect the spin of particles that hit it – so it also is a location detector of some sort, too. Thus I’d expect a more adequate representation of the detectors to be an operator which eigenstates are a combination of the possible eigenstates of both operators. Thus I thought restricting the spin operators to the space they are actually measured makes sense. While it does not make much difference for a single particle measurement it totally seems to breaks any kind of entanglement (mostly because the wave function only collapses locally so the part of the wave function in the other detector does not experience any change by measuring the spin of the first particle)."

I would like to comment on this. First, QM lacks the realism requirement of course. QM is not realistic because of the HUP.

Second, the spin operators ARE separate from position operators. It is possible to collapse one without collapsing the other.

Third, your statement "the wave function only collapses locally so the part of the wave function in the other detector does not experience any change by measuring the spin of the first particle" does not have any actual experimental support. This is essentially the issue that Bell was trying to address. If you assume that to be the case, then you run afoul of the Bell contradiction. Which is why it is rejected.

Lastly, the realism requirement is a necessary part of Bell. Of course, realism can have somewhat different meanings. For example, an interpretation of QM in which locality is preserved but the future can affect the past is considered to be compatible with Bell. That is because such interpretation is not realistic.

 

[Killtech]

I would like to comment on this. First, QM lacks the realism requirement of course. QM is not realistic because of the HUP.

HUP = Heisenberg uncertainty? why does it exclude realism? perhaps it does so for an true particle based understanding but as the equations of motions of QM clearly describes these 'particles' as pure waves i have always understood the wave function as just another field - except that it behaves weird when measured and each 'particle' requires an own field (though this may be due to an inefficient formulation of the theory). i thought this understanding was a sort of realism.

Second, the spin operators ARE separate from position operators. It is possible to collapse one without collapsing the other.

hmm, true. my bad. thus my attempt to restrict the operator to a space region was the wrong way to go then. but that makes me want know how to derive the correct operator for a given measurement setup even more.

Third, your statement "the wave function only collapses locally so the part of the wave function in the other detector does not experience any change by measuring the spin of the first particle" does not have any actual experimental support. This is essentially the issue that Bell was trying to address. If you assume that to be the case, then you run afoul of the Bell contradiction. Which is why it is rejected.[/QUOTE]
my statement was a result of using a space restricted spin operator and i know that it yields wrong results. my intention was to understand if a more intuitive operator (better representing the detector) would still yield the right results. however i have chosen the wrong approach for the operator anyway.

Lastly, the realism requirement is a necessary part of Bell. Of course, realism can have somewhat different meanings. For example, an interpretation of QM in which locality is preserved but the future can affect the past is considered to be compatible with Bell. That is because such interpretation is not realistic.

sure it is to be expected that if you drop either one of the prerequisites you will be able to circumvent Bell. otherwise it would not be requirement.
...but Bell must have used some specific well defined understanding of realism and it would be nice to see it.

 

[DrChinese]

1. HUP = Heisenberg uncertainty? why does it exclude realism?

2. ...but Bell must have used some specific well defined understanding of realism and it would be nice to see it.

1.The Heisenberg Uncertainty Principle (as you guessed) does not EXCLUDE realism, it just does not offer it. EPR thought that meant that QM was incomplete. So if QM IS complete, then realism does not survive.

2. Sadly, this is not exactly the case. I have never seen Bell talk too much about it, certainly not in the original paper. You will find the definition debated, so I think it is best to refer back to the 1935 EPR paper that Bell was responding to.

EPR clearly defined “elements of reality”, which cumulatively constitute “realism”. Such elements are defined as anything you can predict with 100% certainty without disturbing the system. Entangled particle attributes fit this definition. The second part of their definition is more subtle and not introduced until the last 2 paragraphs of the EPR paper. They then say that it is reasonable to include as elements of reality anything which can be so predicted, even if they cannot all be predicted simultaneously.

It is this requirement which becomes the cornerstone of Bell, as he needs 3 simultaneous elements of reality for his magic to work. In Bell’s original paper, they are a, b and c; introduced after his (14).

 

[Killtech]

1.The Heisenberg Uncertainty Principle (as you guessed) does not EXCLUDE realism, it just does not offer it. EPR thought that meant that QM was incomplete. So if QM IS complete, then realism does not survive.

wow, that's a lot of conclusions in a short statement and it's a bit hard to for me to follow. i am quite confused about what definition 'complete' may be in this context and and even more so how that excludes realism. as i understand the bell experiment now dropping locality is enough to preserve realism unless one uses a strange definition that is already excluded by something else.

as for the 'completeness' of any QM any meaning seems very far off considering that the theory isn't even able to provide an well defined time evolution of the wave function with the strongly contradicting von-neumann wave collapse and QM equations of motion and no defined mechanism to decide which one to chose when (one needs to define exactly within the terms of the theory what interaction on the wave function counts as a measurement).

and yes, i think HUP actually provides some reason for realism: if one tries to build any model in probability theory using this principle one finds it is impossible to model it with random variables only. it seems to be necessary to introduce a deterministic function/field/distribution (i.e. the wave function) from which the probability densities of both random variables (location and impulse) can be derived. interestingly the HUP is primarily a statement about this underlying deterministic object.

2. Sadly, this is not exactly the case. I have never seen Bell talk too much about it, certainly not in the original paper. You will find the definition debated, so I think it is best to refer back to the 1935 EPR paper that Bell was responding to.

EPR clearly defined “elements of reality”, which cumulatively constitute “realism”. Such elements are defined as anything you can predict with 100% certainty without disturbing the system. Entangled particle attributes fit this definition. The second part of their definition is more subtle and not introduced until the last 2 paragraphs of the EPR paper. They then say that it is reasonable to include as elements of reality anything which can be so predicted, even if they cannot all be predicted simultaneously.

It is this requirement which becomes the cornerstone of Bell, as he needs 3 simultaneous elements of reality for his magic to work. In Bell’s original paper, they are a, b and c; introduced after his (14).

hmm, sounds a bit like the definition EPR had in mind is something like: if you were god and wanted to create a world a realistic theory would be good enough to be used as the underlying physics for this world. plus i read you it implies determinism, too which prohibits god from using a random number generator for the physics, hehe. but i guess i'll need to have a longer look at Bell's proof to understand what properties of realism are actually needed. guess it might also help to study what is not counted as realism and getting that into a usable formula.

 

[DrChinese]

wow, that's a lot of conclusions in a short statement and it's a bit hard to for me to follow. i am quite confused about what definition 'complete' may be in this context and and even more so how that excludes realism.

The concepts are a bit difficult to follow, and the terminology can easily get in the way.

Realism is not about defining reality, it is making a statement about things that may or may not exist simultaneously. In the HUP, position is undefined when momentum is well defined. So realism is not present.

If you believe, as EPR did, that a more complete specification of the system might be possible, you are a realist. EPR thought QM to be incomplete, and that was an extremely reasonable position. Of course, not based on facts. However, most realists - like Einstein - also believed that superluminal influences were not possible. I.e. locality should be respected.

So Bell combined these 2 ideas to demonstrate that there was a contradiction if QM IS correct. Either QM is flat out wrong in its predictions (so far not the case) or at least one of the Bell assumptions is incorrect (locality/separability and/or realism). You pick.

 

[Killtech]

The concepts are a bit difficult to follow, and the terminology can easily get in the way.

Realism is not about defining reality, it is making a statement about things that may or may not exist simultaneously. In the HUP, position is undefined when momentum is well defined. So realism is not present.

hmm i think i see the issue here: the question is not whether the theory itself is realistic but more about what objects/constructs within it are.
a pure particle concept is evidently not realistic due to the HUP - yet may still be thought of as local.
but at the same time an electron understood an object without a fixed form represented by its changing charge density and current (derived from WF) is well defined at every location of space time and completely determines the time evolution thus is a realistic (and even deterministic) object (in terms of the equations of motion) - but therefore the measurement interaction/collapse cannot be described in an einstein local manner according to Bell.
so is realism rather a question what terms you chose to understand the theory in without any meaningful mathematical effect?

is it that so far all our measurements are of the form of the prior that one prefers a non-realism approach to QM? however doesn't the theory itself state that the field like degrees of freedom of the wave functions of QM objects produce distinguishable outcomes implying that the underlying wave function is necessary and irreducible for a adequate physical description and thus must be measurable itself in principle?

If you believe, as EPR did, that a more complete specification of the system might be possible, you are a realist. EPR thought QM to be incomplete, and that was an extremely reasonable position. Of course, not based on facts. However, most realists - like Einstein - also believed that superluminal influences were not possible. I.e. locality should be respected.

i'm still not sure what 'complete' refers to so i don't know what my position is. is this the question whether the probabilistic part of QM is actually explainable? is it determinism? what you write at least does not sound like the completeness of the definitions i.e. if the theory is well defined in mathematical terms (i.e. yields unambiguous predictions).

So Bell combined these 2 ideas to demonstrate that there was a contradiction if QM IS correct. Either QM is flat out wrong in its predictions (so far not the case) or at least one of the Bell assumptions is incorrect (locality/separability and/or realism). You pick.

yeah, I'm think i understand this part now.

 

[DrChinese]

1. hmm i think i see the issue here: the question is not whether the theory itself is realistic but more about what objects/constructs within it are.

2. so is realism rather a question what terms you chose to understand the theory in without any meaningful mathematical effect?

3. is it that so far all our measurements are of the form of the prior that one prefers a non-realism approach to QM? however doesn't the theory itself state that the field like degrees of freedom of the wave functions of QM objects produce distinguishable outcomes implying that the underlying wave function is necessary and irreducible for a adequate physical description and thus must be measurable itself in principle?

4. I'm still not sure what 'complete' refers to so i don't know what my position is. is this the question whether the probabilistic part of QM is actually explainable? is it determinism? what you write at least does not sound like the completeness of the definitions i.e. if the theory is well defined in mathematical terms (i.e. yields unambiguous predictions).

1. Yes, pretty much saying that there are elements of reality corresponding to things that can be measured (and predicted), and then whether those things exist independently of the act of measurement.1

2. The mathematical effect comes in when (or IF) you assume that you these elements exist independently of the act of observation. If you say they do, then Bell kicks in. See starting with his (14). "Let c be a unit vector..." where c is not simultaneously measurable, only a and b are. Or actually only 2 out of: {a, b, c, d, e, ...}

3. QM only makes statements about what is measurable. So when you say something is measurable "in principle", you must really be referring to that which could be measured using some actual setup. And not something that you already know cannot be read. For example, in QM you cannot know the position and momentum of a particle simultaneously.

4. The word "complete" gets a treatment all on its own. I think the best way to think of it is: is a "more complete" specification of the system possible? To me, the answer to that is "no" and that has 2 implications: a) you cannot access non-commuting properties simultaneously; b) you cannot explain the reason for a particle collapsing to a particular eigenvalue. On the other hand, a more complete theory would supply one or the other of these. Bohmian mechanics attempts to supply these. On the other hand, it is axiomatic with BM that actual values are not to be provided because the input variables cannot be sufficiently known. So again, you pick your poison.

 

[harrylin]

[..]
however a small warning: i understand things best when it can be explained by formulas alone and one can skip the text (text can always be interpreted wrong. formulas can't).

Then a bigger counter warning may be at its place. Formula's with very precise definitions of the symbols and their claimed field of application can hardly be misinterpreted.
However, Formula's without text can easily be misinterpreted (not a rare occurrence).

Also experimental set-ups as well as the results can be misinterpreted. And what "non-realists" say that "realists" mean with "realism" doesn't always correspond to what "realists" themselves say that they mean. Nothing easy about this!

 

[Killtech]

1. Yes, pretty much saying that there are elements of reality corresponding to things that can be measured (and predicted), and then whether those things exist independently of the act of measurement.1

2. The mathematical effect comes in when (or IF) you assume that you these elements exist independently of the act of observation. If you say they do, then Bell kicks in. See starting with his (14). "Let c be a unit vector..." where c is not simultaneously measurable, only a and b are. Or actually only 2 out of: {a, b, c, d, e, ...}

3. QM only makes statements about what is measurable. So when you say something is measurable "in principle", you must really be referring to that which could be measured using some actual setup. And not something that you already know cannot be read. For example, in QM you cannot know the position and momentum of a particle simultaneously.

2. i think i get it now. for me this aspect was hard to follow because most physicists bind the realism aspect to a particle understanding of QM objects. anyhow, so the statement that the wave function is not real or just a mathematical tool should be understood mathematically as this object is not guaranteed to be a well defined in every situation? this is a problem i have with QM because it implies that you cannot make gedankenexperiemnts imagining how the wave function will behave since you don't know if it is the correct and adequate representation of the real physical system at hand. that said this is what i would canonically understand as an 'uncomplete theory'. so the beef over realism and other interpretations is whether a theory actually exists where all terms used by it are well defined at all times? (note: this does not require determinism but merely proper definitions).

3. well by the measurement postulates itself the wave function cannot be measured, well unless you allow non-linear operators as observables that is. however the equations of motion/correspondence principle are a distinct set of axioms that can be used to tackle the question, too and they allow for greater freedoms. i said 'in priciple' because i don't see anything that would prevent this to be used to construct a system where one can measure a wave function. after all the measurement postulates are not derived from the EOM so they do not need to be consistent with its results. to give a rough idea of what construction i mean is to think of a complex quantum system that works like an ear: composed of many independent resonator type systems that can only be exited by the corresponding element of Fourier decomposition of the to-be-measured wave function such that they pick up the Fourier coefficients of the decomposition as energy which can be measured afterwards for each resonator independently. if the energy specturms are fine enough it should suffice to reconstruct parts of the decomposed wave function from its measured base coefficients (unless you can set up infinitely many resonators to pick them all up).

1. Then a bigger counter warning may be at its place. Formula's with very precise definitions of the symbols and their claimed field of application can hardly be misinterpreted.
However, Formula's without text can easily be misinterpreted (not a rare occurrence).

2. Also experimental set-ups as well as the results can be misinterpreted. And what "non-realists" say that "realists" mean with "realism" doesn't always correspond to what "realists" themselves say that they mean. Nothing easy about this!

1. well if all objects are defined purely by their relationship to other objects within the same framework you'll need little text in between. what i meant is i prefer a bourbaki style presentation and physics are rarely written in such a format such that you have to find all the important underlying details yourself.
the only thing you really need text for is the interpretation, i.e. which objects of the theory correspond to which objects in reality. everything else is math.

2. hehe, true.

 

[harrylin]

[..] well if all objects are defined purely by their relationship to other objects within the same framework you'll need little text in between. what i meant is i prefer a bourbaki style presentation and physics are rarely written in such a format such that you have to find all the important underlying details yourself.
the only thing you really need text for is the interpretation, i.e. which objects of the theory correspond to which objects in reality. everything else is math. [..]

Interesting, I did not know of Bourbaki.

It will be interesting to see if Bell's theorem (which includes the physical interpretation of his inequality) can be "reduced to math" in that sense; I don't think that it can!

 

[DrChinese]

2. i think i get it now. for me this aspect was hard to follow because most physicists bind the realism aspect to a particle understanding of QM objects. anyhow, so the statement that the wave function is not real or just a mathematical tool should be understood mathematically as this object is not guaranteed to be a well defined in every situation? this is a problem i have with QM because it implies that you cannot make gedankenexperiemnts imagining how the wave function will behave since you don't know if it is the correct and adequate representation of the real physical system at hand. that said this is what i would canonically understand as an 'uncomplete theory'. so the beef over realism and other interpretations is whether a theory actually exists where all terms used by it are well defined at all times? (note: this does not require determinism but merely proper definitions).

Well, you will find a lot of different viewpoints about these comments. First, I think there are plenty of physicists who see the wave function as "real" in some sense. I certainly think it is more than a mathematical artifact. After all, you can manipulate it!

"Uncomplete?" Maybe. Of course every version of a label like this begs the question, "what more is there to have?". Certainly we aren't going to get a local realistic extension, so once you are past that you shouldn't be further disappointed.

Finally, contradictions arise quickly if you attempt to assert any form of realism - even non-local ones. This is part of the reason Bohmian Mechanics (BM/dBB class theories) is not better accepted. Although this provides a solution to the Bell problem, it skirts the issue of counterfactual measurements because it is still contextual. And there are a LOT of papers out there that strongly imply that realism is not viable in any form.

So my point is: be careful before you dismiss some of the issues as being semantic ("proper definitions") in nature. I would definitely recommend that you get a good handle on what kind of experiments are being done today so you can see that: the state of the art on these issues is quite advanced.