Can grandpa understand the Bell's Theorem?

[sfs01]

I still don't understand what the experiment entails. All I have direct experience with is polarization. If light comes through polarized lenses, I can either align it left and right, or up and down... But there is no third way I can polarize it. Also, WHEN it's polarized, you don't get "up" or "down" you just get (up/down) I can't polarize it in the direction of the motion. But in the Bell's experiment there are three variables, each with two available settings.

But I don't even know what particles you are talking about, nor the variables involved, nor the description of how to measure those variables. Obviously these experiments have been carried out over and over again, but nobody describes the experiment... Not in any detail anyway. They just talk about what we expect and what the "surprising" results are in an abstract fashion.

I can't tell whether the results are surprising when the experiment isn't even described!

Apart from orienting your polariser up/down or left/right you can also rotate it to any angle in between the two and it will measure polarisation in that direction. So the 3 variables are the polarisation angle of the incoming light and the 2 angles of the polarisers used to measure that light later on. In a purely classical world, the intensity of light going through the polariser will depend on the angle between the underlying polarisation of the light and the angle of the polariser, e.g. if the angle is 45 degrees, you would expect a certain percentage of the photons to go through and you would expect this to be independent between the two different polarisers at opposite ends of the set-up (the measurement should only depend on the properties of the photon itself and not on what the results of measurements on other photons elsewhere are).

The 'surprising' QM effect is that if you do this experiment with entangled photons and using the same polariser setting of 45 degrees as above on both sides, although each photon individually would still have the same chance of going through either polariser as before, the 2 entangled photons will either both go through their respective polariser or neither of them will - ignoring experimental noise etc.

This result alone could still be explained by adding hidden variables associated with the photons to the classical model. But by going through the other combinations with different angles between the two polarisers, you can find that the combination of all the QM predictions when taken together are inconsistent with Bell's inequalities - which are a more general statistical/information theoretical statement on what kind of correlations between the outcomes on the two sides are possible based on the assumption that each side of the experiment has no prior information about the outcome of the other side.

 

[JesseM]

Sorry but I cannot answer this question because 1) and 2) are two “technical” for me

Really? Could you say which sentences you find too technical? And if you can't understand 1) and 2) which are really pretty simple for anyone who is familiar with the basics of special relativity and electromagnetism, then it's completely absurd that you claim to understand Bell and see flaws in his thinking (you clearly don't understand the notion of "local realism" if you don't understand my 1 and 2), this would be like someone who doesn't understand algebra claiming to find flaws in the proof of a calculus theorem.

and the comments of Arthur Fine are too gossip-like for me to trust him. When I asked you to provide quotes from Einstein's I expect a direct Einstein’s writing and not an interpretation.

Um, did you actually read my post or did you just skim it? The whole first section was from a book that quoted several paragraphs from Einstein's own letter to Schrödinger about the two-box thought-experiment, then I gave my analysis of what he meant which related his comments to my own 1) and 2), only after all that did I quote Fine's comments.

By the way, I don’t believe that a knowledge about the characteristic (spin, polarization, etc.) of one particle yields the complement characteristic of the the correlated particle

According to QM, knowledge of some characteristic of one particle, like whether it's spin-up or spin-down on a particular axis, can allow you to predict with 100% certainty what result we'll get if we do the same measurement on a second entangled particle. Are you disagreeing with QM, or are you just disagreeing that there must have been local properties of the second particle that predetermined what result it would give immediately before measurement, or are you arguing something else? Anyway if you disagree that there were such local properties of the second particle that predetermined its result for that measurement, you are disagreeing with both Einstein and Bell. Of course you are free to disagree with them, but if so there doesn't seem to be any meaningful sense in which your own beliefs are "local realist" ones. And remember, Bell's proof was only meant to show a conflict between local realism and QM, not to say you couldn't have some non local realist model for what's really going on in QM (Bohmian mechanics would be an example of such a non-local realist model).

“Sealed box” and other illustrative examples like Alice, Bob etc. are adding one more layer of interpretation and misinterpretation and in my opinion their use may cause more problems than help.

The sealed box was Einstein's own analogy, again read the beginning of the post that quotes from Einstein's own letter. If you aren't willing to deal with analogies but also are unwilling to try to understand explanations that are the slightest bit "technical" like my 1) and 2), then I don't see any way to try to explain the concept of "local realism" to you, you need to either change your attitude towards these types of explanations or just give up all attempts to understand either local realism or Bell's argument.

Indeed it seems that Bell was sympathetic to Einstein ideas. However the passion with which Bell proclaimed impossibility of local realism and existence on non-locality (as inevitable) tell me that his conclusion was predetermined by strong influence of Copenhagen Interpretation.

You apparently don't understand the most basic aspects of Bell's argument or the meaning of local realism, so you look completely foolish making these pompous pronouncements about where his conclusions came from. And just for your information, Bell wasn't in the least bit sympathetic to Copenhagen, he much preferred nonlocal hidden-variable theories which try to give an objective picture of what's really going on with quantum systems when they're not being measured, like Bohmian mechanics which I mentioned above.

 

[JDoolin]

Aspect's experiments are described in detail in his papers.

An experiment can for example involve a device which generates pairs of photons via a cascade mechanism (which causes the photons to be correlated), some tubes to keep out other sources of light and a pair of observing devices at the ends of the tubes. The observing devices contain devices for splitting polarized light into two separate beams (which could for example simply be a birefringent crystal) with light detection devices (photomultipliers and detectors) to detect photons arriving via each beam. The observing devices can be rotated to different angles around the axis of the tube.

Photons arriving at all four channels (two at each end) are electronically logged. The results are processed afterwards to select relevant events, which are those where a single photon arrived at each end within an interval consistent with the cascade mechanism. Those events are then used to calculate the correlations. Other calculations are done to estimate for example the rate of unmatched single photons and the rate at which those unmatched photons just happened to arrive within the right interval. When all the statistics are sorted out, the result matches QM predictions very well, and in some cases the experiments have been refined to the extent that the raw results violate Bell inequalities without even having to allow for experimental inefficiencies.

Further refinements include inserting a switching device (effectively involving an electronically controlled mirror) that can either let the original photons go to the original observation device or divert it to another observation device configured at a different angle. This device can be switched faster that the light travel time between the ends of the device, ensuring that the results at both ends cannot be determined via communication between the ends.

~So this cascade mechanism... Is it similar to the "Stimulated Emission of Radiation" that goes on inside a laser? Do the two photons come out in exactly opposite directions?

~There would be some doubt as to whether two photons were really coming from the same event, or were just coincidentally happened at the same time. I think you could overcome this doubt via a statistical argument.

~If I follow one beam, if I understand correctly, it comes upon a birefringent crystal. I would expect it would have three possible outcomes; one is to reflect off the surface, two and three are to polarize according to the crystal structure and pass on through. If it reflects off the surface, of course, you don't get a reading on both photons, so it's not counted. (Edit: This possibility seems missing in the scratch-lottery-ticket analogy. When you scratch, you could get a lemon or a cherry, but shouldn't there also be the possibility that the lottery ticket just disintegrates in your hand and is thrown out of the experiment?)

~At the end, if the photon goes through the crystal, it ends up at one of two photomultipliers. Based on which photomultiplier it goes into you can tell which of the available polarizations the photon has.

~the idea that the result of this test could be either "up" or "down" is misleading, since the two possible results of polarization are not 180 degrees from each other, (nor are they opposite) but 90 degrees from one another.

~Your choice of "what" to measure is determined by what angle you place the birefringent crystal.

So if I have this much right, (if not, let me know) then what are your set-ups with the bi-refringent crystals? I know there are angles involved, but are you just using angles like 0, 120, and 240 degrees around the axis (parallel to the light ray), or are you also rotating along an axis perpendicular to the light ray, or are you using different faces of the birefringent crystal?

 

[Jonathan Scott]

~So this cascade mechanism... Is it similar to the "Stimulated Emission of Radiation" that goes on inside a laser? Do the two photons come out in exactly opposite directions?

The cascade mechanism involves pumping energy into an atom and then letting it decay via a pair of photons, so there is some resemblance. I don't expect the photons would come out in exactly opposite directions in general, but enough of them would do so to make the experiment work.

~There would be some doubt as to whether two photons were really coming from the same event, or were just coincidentally happened at the same time. I think you could overcome this doubt via a statistical argument.

Yes, this is checked via the statistics for unmatched photons.

~If I follow one beam, if I understand correctly, it comes upon a birefringent crystal. I would expect it would have three possible outcomes; one is to reflect off the surface, two and three are to polarize according to the crystal structure and pass on through. If it reflects off the surface, of course, you don't get a reading on both photons, so it's not counted. (Edit: This possibility seems missing in the scratch-lottery-ticket analogy. When you scratch, you could get a lemon or a cherry, but shouldn't there also be the possibility that the lottery ticket just disintegrates in your hand and is thrown out of the experiment?)

This is true, and it is very difficult to eliminate this loophole with photons, as non-detection is always an option. Other equivalent experiments have been done involving pairs of atoms in which non-detection is not an option, but in those experiments there are other loopholes. If QM was devious enough to exploit different loopholes in different cases, which of course seems very implausible, then I don't think experiments have been done which would eliminate all possible loopholes in the same experiment.

~At the end, if the photon goes through the crystal, it ends up at one of two photomultipliers. Based on which photomultiplier it goes into you can tell which of the available polarizations the photon has.

~the idea that the result of this test could be either "up" or "down" is misleading, since the two possible results of polarization are not 180 degrees from each other, (nor are they opposite) but 90 degrees from one another.

The "up"/"down" terminology is from the equivalent fermion experiment, observing spins. For photons, an example result might be H/V for Horizontal/Vertical polarization, but I think that in practice most experiments use left/right cyclic polarization.

~Your choice of "what" to measure is determined by what angle you place the birefringent crystal.

So if I have this much right, (if not, let me know) then what are your set-ups with the bi-refringent crystals? I know there are angles involved, but are you just using angles like 0, 120, and 240 degrees around the axis (parallel to the light ray), or are you also rotating along an axis perpendicular to the light ray, or are you using different faces of the birefringent crystal?

I think again that we have a mixture of the two versions of the experiment here. The whole observation device (including beam splitter - I don't recall whether it really is a birefringent crystal or what) is physically rotated to different angles around the axis. For photon polarization, the interesting angles are 22.5 degrees either way of a reference direction at either end. In practice, one also tries the whole experiment with both ends rotated to different angles to see whether the set-up is rotationally uniform, as it should be.

The interesting sets of results with photons are with both ends aligned to get perfect correlation, one end turned by 22.5 degrees (which should cause approximately 15% of results to be different), the other end turned the other way by 22.5 degrees (again changing 15% of results) and finally with both ends turned, so the angle between them is 45 degrees and 50% of the results should be different, for which Bell's Theorem tells us there is no possible local realistic explanation, as 15%+15% cannot exceed 30%.

 

[miosim]

You do persist in missing the point, don't you?
Bell's theorem itself is very simple and very robust. It proves that either QM is wrong or the assumption of local realism (which is normally taken as part of Special Relativity) is wrong.

Or you are wrong, because nobody as I know, including Einstein didn't claim that QM is wrong, but incomplete only.

You probably missed my main objection to Bell's theorem that it is based on incorrect initial conditions that is the Bell's misinterpretation of Einstein's "local realism" . It is obvious to me that the Bell's model of the "local realism" is wrong, because it violates the very basic principle of Einstein's argument related to interpretation of QM but not altering its result. And one don't have wait for the "proof" provided by Bell to understand the obvious difference between QM and Bell's "local realism". As soon Bell (or anyone else) found that his "very reasonable" (as he called it) model of "local realism" contradicts with prediction of QM Bell should stop and go back to the drawing board to find out what is wrong with his model.

 

[JesseM]

It is obvious to me that the Bell's model of the "local realism" is wrong, because it violates the very basic principle of Einstein's argument related to interpretation of QM but not altering its result.

And I've already pointed out that this is a completely silly argument since Einstein had no way of knowing that there was a conflict between his concept of local realism (which was identical to the concept Bell used) and QM, since Bell hadn't proved that when Einstein was alive. I made this point in two previous posts and you never responded, are you just ignoring it?

As an analogy, you might as well say that since Ptolemy didn't intend for his astronomical system to conflict with any astronomical observations, and yet we now have observations that clearly show the Earth revolves around the Sun, this "proves" that Ptolemy couldn't have really believed the Sun revolves around the Earth! Hopefully you can see that this is argument is nonsense because Ptolemy didn't know that after his time we would find a conflict between astronomical observations and Earth-centered astronomical systems.

 

[miosim]

According to QM, knowledge of some characteristic of one particle, like whether it's spin-up or spin-down on a particular axis, can allow you to predict with 100% certainty what result we'll get if we do the same measurement on a second entangled particle. Are you disagreeing with QM...

I am disagree with "... 100% certainty" regardless it is predicted by QM or EPR model. Please prove me wrong and provide the experiment(s) where both (individual) correlated photons were proved to have 100% correlation. The difference I am expecting is within the principle of uncertainly (similar to particle's position/momentum uncertainly). We may discuss the details of this result later.

 

[JesseM]

I am disagree with "... 100% certainty" regardless it is predicted by QM or EPR model.

My statement was about what is predicted by QM, not an empirical claim. Read it again:

According to QM, knowledge of some characteristic of one particle, like whether it's spin-up or spin-down on a particular axis, can allow you to predict with 100% certainty what result we'll get if we do the same measurement on a second entangled particle.

Of course QM might turn out to be wrong, but this isn't relevant to Bell's theorem (and to Einstein's EPR argument), since Bell's theorem is just dealing with the issue of whether the theory of QM is compatible with an underlying local realist model. Also, as I pointed out in my [post=3275052]last post to billschnieder[/post], Bell did derive a more general inequality known as the CHSH inequality which would still be expected to hold in a local realist theory which does not assume perfect correlations with the same detector setting.

The difference I am expected is within the principle of uncertainly (similar to particle's position/momentum uncertainly). We may discuss the details of this result later.

You should really learn the basics of the areas of physics you're talking about instead of confidently spouting nonsense (your posts seem like a perfect example of the Dunning-Kruger effect), the uncertainty principle only applies to non-commuting operators like position and momentum, in QM there is no uncertainty relation if you measure the same variable twice in quick succession, or if you measure the same variable for two particles which are entangled in that variable.

 

[miosim]

It is obvious to me that the Bell's model of the "local realism" is wrong, because it violates the very basic principle of Einstein's argument related to interpretation of QM but not altering its result.

And I've already pointed out that this is a completely silly argument since Einstein had no way of knowing that there was a conflict between his concept of local realism (which was identical to the concept Bell used) and QM, since Bell hadn't proved that when Einstein was alive. I made this point in two previous posts and you never responded, are you just ignoring it?

It would be a valid point if Bell reproduces Einstein’s concept without any deviations. Instead Bell’s model lacks the major requirement for Einstein’s argument; do not contradict with the predicted result of QM. Bell doesn’t need Einstein to be around to adhere with this basic requirement.
If Bell couldn’t adhere with this requirements, than Bell’s theorem should be exclusively about (his) impossibility to construct such model and not about influence over a distance.

Regarding a possibility to have a model that satisfies both Einstein’s realism and the prediction of QM, I wonder if we can modify the Bell’s model for the Aspect’s experiment as follows:
The EPR correlated photons have 100% predictable polarization before interacting with polarizer, but polarizer rotates this polarization per Malus’ law (as cos^2). Would this model produce the result in agreement with Aspect’s experiment?

You should really learn the basics of the areas of physics you're talking about instead of confidently spouting nonsense (your posts seem like a perfect example of the Dunning-Kruger effect), the uncertainty principle only applies to non-commuting operators like position and momentum, in QM there is no uncertainty relation if you measure the same variable twice in quick succession, or if you measure the same variable for two particles which are entangled in that variable.

You are right that “I … confidently spouting nonsense”. Thank you for providing links.

According to QM, knowledge of some characteristic of one particle, like whether it's spin-up or spin-down on a particular axis, can allow you to predict with 100% certainty what result we'll get if we do the same measurement on a second entangled particle. Are you disagreeing with QM, or are you just disagreeing that there must have been local properties of the second particle that predetermined what result it would give immediately before measurement, or are you arguing something else? Anyway if you disagree that there were such local properties of the second particle that predetermined its result for that measurement, you are disagreeing with both Einstein and Bell. Of course you are free to disagree with them, but if so there doesn't seem to be any meaningful sense in which your own beliefs are "local realist" ones. And remember, Bell's proof was only meant to show a conflict between local realism and QM, not to say you couldn't have some non local realist model for what's really going on in QM (Bohmian mechanics would be an example of such a non-local realist model).

Sorry, I miss-read your question first time. To have a complete answer I would need to explore my beliefs that are indeed different from QM and EPR. However my beliefs aren’t relevant to my arguments against Bell theorem and I don’t want to derail this thread. Therefore I shouldn’t mention my disagreement with a “100% certainty.” My fault.

P.S.
Regarding Bell's proof that “..was only meant to show a conflict between local realism and QM…” I am increasingly uncomfortable with the label of “local realism”. You pointed few times to your definition of “local realism” 1) and 2), but do you have Bell’s and Einstein’s definitions (in theirs own words)?

 

[JesseM]

It would be a valid point if Bell reproduces Einstein’s concept without any deviations. Instead Bell’s model lacks the major requirement for Einstein’s argument; do not contradict with the predicted result of QM. Bell doesn’t need Einstein to be around to adhere with this basic requirement.

But Einstein didn't know his concept of a local and objective model conflicted with QM in the first place! He thought it could be possible to come up with such a model that does not contradict QM, but he was wrong! Do you really not get this?

Einstein: I'm hoping for a theory which has features X and Y, and which reproduces the predictions of QM.

Bell: Here is a proof that any theory with features X and Y must automatically conflict with QM.

What miosim says: But Einstein required a theory which "reproduces the predictions of QM", if Bell's proof shows a conflict between X and Y and QM, that just proves that X and Y were not actually what Einstein meant!

What Einstein would have said: Oh, I didn't realize there was a basic conflict between X and Y and QM. Very interesting, unless QM's predictions are disproven I guess this means I must abandon the hope for a theory with features X and Y.

If you still don't get why Einstein's realism could be identical to Bell's realism, could you address my Ptolemy analogy? Do you agree that Ptolemy designed his astronomical models with the assumption that they should fit with astronomical observations? Do you therefore think the fact that we now know an Earth centered system doesn't fit with astronomical observations is proof that any summary of Ptolemy's system which states that the Sun revolves around the Earth must not really be what Ptolemy had in mind?

Regarding a possibility to have a model that satisfies both Einstein’s realism and the prediction of QM, I wonder if we can modify the Bell’s model for the Aspect’s experiment as follows:
The EPR correlated photons have 100% predictable polarization before interacting with polarizer, but polarizer rotates this polarization per Malus’ law (as cos^2).

Malus' law has nothing to do with "rotating the polarization", it involves reducing the intensity of a polarized beam by cos^2 of the angle between the beam and the polarizer's angle. If you want to fantasize that the polarizers rotate the polarization in a deterministic way that's fine, but then you need to specify how that altered polarization determines which of two results we get (going through the polarizer or deflected by it). If the final angle determines the result in a deterministic way, then if both photons end up with the same final angle you can have 100% correlation, but in this case the original angle predetermines what the final angle will be and what the final angle will be, so you still have each photon having local properties that predetermine what response they will give to any measurement, so Bell's theorem still applies.

Sorry, I miss-read your question first time. To have a complete answer I would need to explore my beliefs that are indeed different from QM and EPR. However my beliefs aren’t relevant to my arguments against Bell theorem and I don’t want to derail this thread. Therefore I shouldn’t mention my disagreement with a “100% certainty.” My fault.

OK, so think about Einstein's box analogy again. Obviously the simplest way to explain the 100% correlation between the results of opening each box is to say that prior to being opened, each box had "hidden" local properties that predetermined their results--one has a ball hidden inside it, one is empty. Einstein said it would be "absurd" to think that there was no definite truth about what was in each box before they were opened. And since he was using this analogy to specifically explain the ideas that he had wanted the EPR paper to explain, do you disagree that he thought the same way about perfect correlations in QM? That he thought, for example, that if two entangled particles are 100% guaranteed to have the same magnitude of momentum, that must mean that even before being measured they both had "hidden" local properties that predetermined they would give that result if their momentum was measured?

P.S.
Regarding Bell's proof that “..was only meant to show a conflict between local realism and QM…” I am increasingly uncomfortable with the label of “local realism”. You pointed few times to your definition of “local realism” 1) and 2), but do you have Bell’s and Einstein’s definitions (in theirs own words)?

Neither of them used the exact term "local realism", it's a later term intended to summarize the type of theories that Einstein and Bell were discussing. Bell did use the similar term "local causality", and if you look at the links and discussion I gave of the La nouvelle cuisine paper in [post=3248153]this post[/post] you can see how Bell assumes "local beables" which are the same concept as my "local facts" in 1) (also see Bell's paper The Theory of Local Beables), and you can also see how he assumes the value of local beables can only be influenced by events in their past light cone, identical to my 2) (you said my discussion was too "technical" for you, do you understand what a "light cone" is? If not you could start here). As for Einstein, it seems he never gave any systematic exposition, but again read the direct quotes about his two-box explanation for perfect correlations in QM in [post=3270631]this post[/post], and Bell also quotes some other relevant comments of Einstein's on p. 7-8 of this paper (starting with the paragraph that begins "If one asks what, irrespective of quantum mechanics, is characteristic of the world of ideas of physics...")

 

[JDoolin]

The cascade mechanism involves pumping energy into an atom and then letting it decay via a pair of photons, so there is some resemblance. I don't expect the photons would come out in exactly opposite directions in general, but enough of them would do so to make the experiment work.



Yes, this is checked via the statistics for unmatched photons.



This is true, and it is very difficult to eliminate this loophole with photons, as non-detection is always an option. Other equivalent experiments have been done involving pairs of atoms in which non-detection is not an option, but in those experiments there are other loopholes. If QM was devious enough to exploit different loopholes in different cases, which of course seems very implausible, then I don't think experiments have been done which would eliminate all possible loopholes in the same experiment.


The "up"/"down" terminology is from the equivalent fermion experiment, observing spins. For photons, an example result might be H/V for Horizontal/Vertical polarization, but I think that in practice most experiments use left/right cyclic polarization.



I think again that we have a mixture of the two versions of the experiment here. The whole observation device (including beam splitter - I don't recall whether it really is a birefringent crystal or what) is physically rotated to different angles around the axis. For photon polarization, the interesting angles are 22.5 degrees either way of a reference direction at either end. In practice, one also tries the whole experiment with both ends rotated to different angles to see whether the set-up is rotationally uniform, as it should be.

The interesting sets of results with photons are with both ends aligned to get perfect correlation, one end turned by 22.5 degrees (which should cause approximately 15% of results to be different), the other end turned the other way by 22.5 degrees (again changing 15% of results) and finally with both ends turned, so the angle between them is 45 degrees and 50% of the results should be different, for which Bell's Theorem tells us there is no possible local realistic explanation, as 15%+15% cannot exceed 30%.

~I take this as a correction of post 2 in this thread, where you use 45 and 90 degrees instead of 22.5 and 45 degrees. These new figures are consistent with Malus Law, where cos^2(22.5)=.85, and cos^2(45) = .5


~You could get the same math as above with two polarizers. If you place two polarizers at 22.5 degrees, 85% of the light that gets through the first polarizer will get through the second polarizer. If you place them at 45 degrees, 50% of the light that gets through the first polarizer gets through the second polarizer. In each case, you shouldn't ignore the fact that the first polarizer blocks a significant portion of the light. It also forces the light that goes through into polarization in the same direction. It's only with the second polarizer that you get to apply Malus Law.

It seems to me that in this version of photon entanglement, the results are exactly as should be expected from polarization with a "hidden variable..." No, not even hidden; a variable that you just don't happen to know. The variable is the angle of polarization which is a continuous variable, with some value between 0 and 360 degrees, and a variable which can be changed (if it is not blocked) by running it through a polarizer

But of course, we're going to run into trouble explaining this result if we insist on treating the polarization as a hidden three dimensional binary variable with some value of
{000,001,010,011,100,101,110,111}.

~I don't believe nondetection is as much of an issue as I was thinking before. (Edit: a polarizer either blocks or does not block the light. With birefringent crystal, it appears to pass just about everything, just at different angles.)

 

[JesseM]

In figure 4 it states that a hidden-variable theory will be a straight line. Why is it assumed that any hidden-variable theory would result in a straight line?

FIG. 4: Predicted polarization correlations for a quantum mechanical entangled state (solid curve) and a hidden-variable theory (dashed line).

I don't think the assumption of local hidden variables actually uniquely implies a straight line, rather I think that straight line is just meant to be the closest a local hidden variables theory can get to the curve predicted by quantum mechanics...I don't really understand the details of where they get the straight line, but see [post=3240160]post #20[/post] for some comments by DrChinese that seem to suggest this (perhaps he can comment here and clarify this issue?)

 

[Jonathan Scott]

~I take this as a correction of post 2 in this thread, where you use 45 and 90 degrees instead of 22.5 and 45 degrees. These new figures are consistent with Malus Law, where cos^2(22.5)=.85, and cos^2(45) = .5

In that post, I was describing the case of two spin-1/2 particles, which works in exactly the same way except that the angles are doubled. In that case the beam-splitter is a Stern-Gerlach device.

(People often use electrons as an example, but I've heard that this can't be made to work in practice. Atoms can however be used.)

~You could get the same math as above with two polarizers. If you place two polarizers at 22.5 degrees, 85% of the light that gets through the first polarizer will get through the second polarizer. If you place them at 45 degrees, 50% of the light that gets through the first polarizer gets through the second polarizer. In each case, you shouldn't ignore the fact that the first polarizer blocks a significant portion of the light. It also forces the light that goes through into polarization in the same direction. It's only with the second polarizer that you get to apply Malus Law.

That would work for single photons passing in succession through two polarizers. It would also apply to pairs of photons emitted from a common source if the initial state happened to be polarized in the direction of one of the two observation devices (as in that case, that one would give 100% correlation with the source and the other would be determined by Malus' law). However, if you change BOTH observations to some other angle relative to the initial angle, then unless there is magic feedback from the observation devices to the source there is no way for the direction of polarization to match one of the devices in all four cases (both same, turn one, turn other, turn both).

People occasionally spot that it is possible to produce a local realistic model which will reproduce Malus' law and match QM if instead of turning both, you simply turn one device twice as much. However, that is simply equivalent to the QM special case where the emitted particles are prepared with polarization aligned with the initial observation devices, and does not cover the general case addressed by Bell's Theorem.

 

[edguy99]

I don't think the assumption of local hidden variables actually uniquely implies a straight line, rather I think that straight line is just meant to be the closest a local hidden variables theory can get to the curve predicted by quantum mechanics...I don't really understand the details of where they get the straight line, but see [post=3240160]post #20[/post] for some comments by DrChinese that seem to suggest this (perhaps he can comment here and clarify this issue?)

Regarding http://arxiv.org/abs/quant-ph/0205171 and also the DrChinese post here again showing that LR predicts a straight line. It is true that a spinning ball that is not allowed to have any other properties (ie a second direction of spin as in precession) in which you measure every particle will be a straight line, but lots of other particles will have a curved line if not all the particles reach the detectors. From the discussion in the linked article, it appears they are only checking the particles (ie photons) that reach the detectors.

An example of such a particle is posted https://www.physicsforums.com/showthread.php?t=489944".

 

[Stephen Tashi]

I recall seeing explanations of Bell's inequality that have nothing to do with probability or physical experiments. It was explained as a simple consequence of classifying a set of objects with respect to several properties. It was illustrated it by an inventory office supplies or something that mundane. Can anyone provide a link to that type of explanation?

 

[JesseM]

I recall seeing explanations of Bell's inequality that have nothing to do with probability or physical experiments. It was explained as a simple consequence of classifying a set of objects with respect to several properties. It was illustrated it by an inventory office supplies or something that mundane. Can anyone provide a link to that type of explanation?

Sure, see here for example. But this type of explanation doesn't remove the need to think about probability or the meaning of local realism, because you need local realism to derive the conclusion that each particle must have an identical set of properties that predetermine what result they will give for each possible detector setting (in order to explain why they always give the same result when measured with the same setting), and then you need the assumption that the choice of detector settings isn't statistically correlated with the properties of the particles in order to go from this:

Number(A, not B) + Number(B, not C) greater than or equal to Number(A, not C)

To this:

(Number of trials where particle 1 measured for property A and particle 2 measured for property B, and particle 1 found to have A and 2 found to have not-B)

+

(Number of trials where particle 1 measured for property B and particle 2 measured for property C, and particle 1 found to have B and 2 found to have not-C)

greater than or equal to

(Number of trials where particle 1 measured for property A and particle 2 measured for property C, and particle 1 found to have A and 2 found to have not-C)

 

[Stephen Tashi]

I found a link to what I had in mind. http://theworld.com/~reinhold/bellsinequalities.html

 

[JesseM]

Regarding http://arxiv.org/abs/quant-ph/0205171 and also the DrChinese post here again showing that LR predicts a straight line.

But in that post DrChinese doesn't say LR automatically predicts a straight line, he says that a straight line is the closest a LR theory can come to the quantum prediction:

The LR(Theta) line, in blue, is a straight line ranging from 1 at 0 degrees to 0 at 90 degrees. This matches the values that an LR would need to come closest to the predictions of QM, shown in Red. Other LR theories might posit different functions, but if they are out there then they will lead to even greater differences as compared to QM. Keep in mind that the QM predicted values match experiment closely.

As for the paper you linked to, they just say the straight line is a prediction for "a hidden-variable theory", not "all hidden-variable theories". They also say on p. 6 that "our HVT is very simple", implying that one could come up with more complex HV theories that give different predictions, although they note Bell's result that no local HVT could match the predictions of QM.

It is true that a spinning ball that is not allowed to have any other properties (ie a second direction of spin as in precession) in which you measure every particle will be a straight line, but lots of other particles will have a curved line if not all the particles reach the detectors. From the discussion in the linked article, it appears they are only checking the particles (ie photons) that reach the detectors.

An example of such a particle is posted https://www.physicsforums.com/showthread.php?t=489944".

It seems like in that thread you are talking about a model where certain particles have properties that make them "defective" for particular measurement settings, if so I commented briefly on such models at the end of [post=3270631]this post[/post]:

I should also note that Fine thinks there is some possibility of getting around Bell's theorem by use of a "prism model" in which some particles are intrinsically "defective" for certain types of measurements, so if we try to measure a given property (like spin in a particular direction) some fraction of the particles just won't show up in our measurements and thus won't be included in our dataset, which means the choice of what to measure can no longer be considered independent of the properties that the particle had immediately before measurement in our dataset (if this is unclear, billschnieder explained this type of model in terms of my own lotto card analogy in posts [post=2767632]113[/post] and [post=2767828]115[/post] on an older thread). Bell does assume in most of his proofs that there is no correlation between particle properties before measurement and the choice of detector setting, but it seems to me that these prism models would be themselves contradict the predictions of QM, so they aren't really relevant to a theoretical proof showing that local realism is incompatible with QM. But in terms of the possibility that something like this could be true experimentally, I think this loophole is just one version of what's called the"detection efficency loophole", and there are modified versions of Bell inequalities which take into account that not all particle pairs are successfully measured, see here. There have been Bell tests with ions that managed to close the detector efficiency loophole, see [post=2851208]this post[/post], although they didn't simultaneously close the locality loophole (though experiments with photons have closed that one, none have yet closed both simultaneously. It seems pretty unlikely that we could have a non-contrived-looking local realist theory where both types of loopholes were being exploited at once, though.)

 

[DrChinese]

Regarding http://arxiv.org/abs/quant-ph/0205171 and also the DrChinese post here again showing that LR predicts a straight line. It is true that a spinning ball that is not allowed to have any other properties (ie a second direction of spin as in precession) in which you measure every particle will be a straight line, but lots of other particles will have a curved line if not all the particles reach the detectors. From the discussion in the linked article, it appears they are only checking the particles (ie photons) that reach the detectors...

Yes, there are many graphs possible for LR theories. But if you follow the EPR constraint that you always get the same answer at the same angle settings, there is basically just the straight line one as a possibility. (There are more, but they are even further from the Bell test results.)

The example posted by edguy99 fails to meet the basic standards. (It is a bit hard to follow because something called "loss of momentum" is thrown in. While hypothetical effects are nice as an "escape" to Bell, they always fail when you follow the example through.) You still, for example, need to deliver results that match the QM predictions and this model won't do that. As a proof of that, all you need to do is consider the angle settings 0, 120 and 240 degrees. The 0/45/90 degree examples are not meaningful because simple models can approach these predictions.

 

[JDoolin]

In that post, I was describing the case of two spin-1/2 particles, which works in exactly the same way except that the angles are doubled. In that case the beam-splitter is a Stern-Gerlach device.

(People often use electrons as an example, but I've heard that this can't be made to work in practice. Atoms can however be used.)



That would work for single photons passing in succession through two polarizers. It would also apply to pairs of photons emitted from a common source if the initial state happened to be polarized in the direction of one of the two observation devices (as in that case, that one would give 100% correlation with the source and the other would be determined by Malus' law). However, if you change BOTH observations to some other angle relative to the initial angle, then unless there is magic feedback from the observation devices to the source there is no way for the direction of polarization to match one of the devices in all four cases (both same, turn one, turn other, turn both).

People occasionally spot that it is possible to produce a local realistic model which will reproduce Malus' law and match QM if instead of turning both, you simply turn one device twice as much. However, that is simply equivalent to the QM special case where the emitted particles are prepared with polarization aligned with the initial observation devices, and does not cover the general case addressed by Bell's Theorem.

On further thought, I realized that if the experiment worked to my expectation, then even when the two crystals are perfectly aligned, you would not get perfect agreement. I think this is what you are referring to that I boldfaced above. If the source creates photon pairs of a random polarization with "uniform distribution" then most of the time the photon would not be aligned with either polarizer. For instance if the polarization was 0, or 90 degrees, there would be a 100% agreement, but if the polarization was 45 degrees off, there would only be a 50% agreement. I made up a spreadsheet to take the avereage agreement of all angles from 0 to 357 every 3 degree increment, and found that at best, you can expect a 75% agreement rate. (There's probably a more elegant method of doing this with calculus).

By this method, I also got these values:
Perfectly aligned crystals: 75% agreement
22.5 degrees off: 68% agreement
45 degrees off: 50% agreement


But if I understand correctly, the actual experiment yields:
Perfectly aligned crystals: 100% agreement
22.5 degrees off: 85% agreement
45 degrees off: 50% agreement

...and as you said, that would be "equivalent to the QM special case where the emitted particles are prepared with polarization aligned with [one of] the initial observation devices."

Almost equivalent, but not quite... if the polarizations were aligned, the two would always agree the same way. Both would always be vertical, for instance. If the polarizations are not aligned, then you'd have both always agreeing, but horizontal half the time and vertical the other half the time.

 

[edguy99]

...It seems like in that thread you are talking about a model where certain particles have properties that make them "defective" for particular measurement settings, if so I commented briefly on such models at the end of [post=3270631]this post[/post]:

I would not call them defective, but they do have the property that they cannot be measured in certain directions. Specifically, if you are talking about a bloch sphere as shown http://en.wikipedia.org/wiki/Rabi_problem" . The article talks about the "pseudo-spin vector" (ie, the axis of the spinning vector that is rotating around the axis of measurement). Notice, to quote the article, they are "throwing out terms with high angular velocity". Specifically, they are talking about trying to measure a spinning particle at 90 degrees to its spin and the inability of a bloch sphere to react to that measurement.

Videos of spinning bloch spheres subjected to a magnetic measuring field can be seen http://www.animatedphysics.com/videos/larmorfrequency.htm" . Notice the amount of precession going on in the lower right hand video where you have a higher angle of precession and imagine this same particle tilted a full 90 degrees. The green pseudo-spin vector would have to be perpendicular to the spin and you can imagine lots of particles that could not handle this.

 

[Jonathan Scott]

The 0/45/90 degree examples are not meaningful because simple models can approach these predictions.

I find this statement slightly confusing, perhaps requiring more context. For photon polarization, the angles 0/45/90 degrees mean that if one end is a pure state the other end is too, so a classical explanation is trivial. However, for observing the spin of a fermion (where observation angles are doubled), 0/45/90 are a useful set of relative angles for illustrating Bell's theorem, in the same way that 0/22.5/45 are for photons, provided that the last angle is achieved by turning both ends by the middle angle.

 

[JesseM]

I would not call them defective, but they do have the property that they cannot be measured in certain directions.

That's all that Fine meant by "defective" (I was just mirroring his terminology)--if you try to measure them at a given angle you won't get a + or - result that can be included in your data set. Anyway, like I said this isn't relevant to the purely theoretical question of whether Bell's theorem is correct, since the theorem deals with the incompatibility between local realism and the theoretical predictions of QM which don't include any such notion of particles that are impossible to measure at certain angles. And for the experimental question, this possibility is part of the "detector efficiency loophole" which can be closed by using a version of a Bell inequality that takes into account limits on detector efficiency.

 

[DrChinese]

...I don't really understand the details of where they get the straight line...

To tell the truth, I don't think I've actually seen an formal analysis of this (not that it is needed for anything). Intuitively, I just think of that as being a solution in which the slope is constant and nonzero. That constant slope being needed to handle the case like this (where actually the -22.5 degrees and 22.5 degrees values could be anything):

f(-22.5,0) = f(0,22.5)
f(-22.5,0) + f(0,22.5) = f(-22.5, 22.5)

So I guess that means I haven't added anything useful.

 

[DrChinese]

I find this statement slightly confusing, perhaps requiring more context. For photon polarization, the angles 0/45/90 degrees mean that if one end is a pure state the other end is too, so a classical explanation is trivial. However, for observing the spin of a fermion (where observation angles are doubled), 0/45/90 are a useful set of relative angles for illustrating Bell's theorem, in the same way that 0/22.5/45 are for photons, provided that the last angle is achieved by turning both ends by the middle angle.

That's correct. I always use photon examples. Yours works too.

 

[DrChinese]

I would not call them defective, but they do have the property that they cannot be measured in certain directions.

Don't you see that this is easily testable with entangled pairs?

Or are you saying that BOTH of a pair are "invisible"? In which case nothing is explained vis a vis Bell.

 

[Jonathan Scott]

...and as you said, that would be "equivalent to the QM special case where the emitted particles are prepared with polarization aligned with [one of] the initial observation devices."

Almost equivalent, but not quite... if the polarizations were aligned, the two would always agree the same way. Both would always be vertical, for instance. If the polarizations are not aligned, then you'd have both always agreeing, but horizontal half the time and vertical the other half the time.

Your interpretation is what I meant; I should have made it clearer that by "aligned" I meant oriented in such a way that the initial polarization was in a pure state relative to the observation direction.

 

[JDoolin]

On further thought, I realized that if the experiment worked to my expectation, then even when the two crystals are perfectly aligned, you would not get perfect agreement. I think this is what you are referring to that I boldfaced above. If the source creates photon pairs of a random polarization with "uniform distribution" then most of the time the photon would not be aligned with either polarizer. For instance if the polarization was 0, or 90 degrees, there would be a 100% agreement, but if the polarization was 45 degrees off, there would only be a 50% agreement. I made up a spreadsheet to take the avereage agreement of all angles from 0 to 357 every 3 degree increment, and found that at best, you can expect a 75% agreement rate. (There's probably a more elegant method of doing this with calculus).

By this method, I also got these values:
Perfectly aligned crystals: 75% agreement
22.5 degrees off: 68% agreement
45 degrees off: 50% agreement


But if I understand correctly, the actual experiment yields:
Perfectly aligned crystals: 100% agreement
22.5 degrees off: 85% agreement
45 degrees off: 50% agreement

...and as you said, that would be "equivalent to the QM special case where the emitted particles are prepared with polarization aligned with [one of] the initial observation devices."

Almost equivalent, but not quite... if the polarizations were aligned, the two would always agree the same way. Both would always be vertical, for instance. If the polarizations are not aligned, then you'd have both always agreeing, but horizontal half the time and vertical the other half the time.

Your interpretation is what I meant; I should have made it clearer that by "aligned" I meant oriented in such a way that the initial polarization was in a pure state relative to the observation direction.

Okay... just to make sure, because I don't have the results of any experiment. I want to verify that the results of the experiment yield 100%, 85%, 50%... Not 75%, 68%, 50%.

If you do yield 100%, 85%, 50% then it makes me think "you MUST have a polarized source." But you could check that by turning both crystals together. If the results change, then you probably have a polarized source.

If the results don't change, but you still have (aligned) 100%, (22.5 degrees) 85%, and (45 degrees) 50%, then you've got an unpolarized source, but its somehow being forced into the same polarization at both ends. At this point, I'd say you've already got "spooky action at a distance," and you don't need to go into Bell's Theorem to recognize it.

 

[Jonathan Scott]

Okay... just to make sure, because I don't have the results of any experiment. I want to verify that the results of the experiment yield 100%, 85%, 50%... Not 75%, 68%, 50%.

Yes, the 100%, 85%, 50% values are the probabilities of the two-state results matching according to QM theory and experiment. The results of a run are usually expressed as correlations, equal to probability(same) minus probability(different), giving roughly 1, 0.7, 0.

I think that the obvious classical model which uses Malus' law independently for each photon with a random initial polarization of the pair gives exactly half the correlation of QM, like the QM result diluted by a similar amount of "noise". In that case, the expected match rates would indeed be around 75%, 68%, 50%, giving correlations of 0.5, 0.36, 0.

One of the objections to some early experiments in this area was that statistical adjustments to eliminate "noise" would also hide the distinction between QM and classical predictions, but improved experiments eliminated this problem.

 

[DrChinese]

Yes, the 100%, 85%, 50% values are the probabilities of the two-state results matching according to QM theory and experiment. The results of a run are usually expressed as correlations, equal to probability(same) minus probability(different), giving roughly 1, 0.7, 0.

I think that the obvious classical model which uses Malus' law independently for each photon with a random initial polarization of the pair gives exactly half the correlation of QM, like the QM result diluted by a similar amount of "noise". In that case, the expected match rates would indeed be around 75%, 68%, 50%, giving correlations of 0.5, 0.36, 0.

One of the objections to some early experiments in this area was that statistical adjustments to eliminate "noise" would also hide the distinction between QM and classical predictions, but improved experiments eliminated this problem.

A. The Product State statistics follow the formula (for matches):

.25+ (cos^2(theta)/2)

which yields the other series you mention. In a local realistic model that follows Malus, that is what you would expect to see. Thus the match rate ranges from .25 to .75.

B. Obviously, that is far away from the QM prediction of cos^2(theta), which ranges from 0 to 1. There aren't any local realistic models that follow this prediction, of course. You can also have the local realistic model which DOES range from 0 to 1 on a straight line. Of course, that then does NOT follow Malus.