Need help with some aspects of Bell’s theorem

[miosim]

Looking for the help with a conceptual framework of the Bell’s theorem and would appreciate any assistance.

The Bell theorem has been verified in numerous experiments (lets assume that those experiments are free form loopholes).

As I understand the essence of those proves are based on comparing correlations predicted by QM with the correlations derived from the hidden variables theories and after ruling them out, the QM (non-locality) is the only option.

If the above makes a sense, should these hidden variables theories have some prerequisites to be used in the Bell theorem? Should they be in compliance with QM equations and yield the same experimental results as predicted by QM. If they don’t complies with QM, they are not real hidden variables (QM) theories but surrogatess only.

If these theories indeed comply with QM prerequisites , why their correlations aren’t calculated based on the QM prediction, but the ‘classical’ approach is used instead?

 

[PeterDonis]

the essence of those proves are based on comparing correlations predicted by QM with the correlations derived from the hidden variables theories and after ruling them out, the QM (non-locality) is the only option.

Yes.

Should they be in compliance with QM equations and yield the same experimental results as predicted by QM.

No; that's the whole point of Bell's theorem. Bell's theorem says that it is impossible to have a local hidden variable theory that makes the same predictions for experimental results as QM. That's why the experimental proofs rule out hidden variable theories: the experiments show that the predictions of QM are correct, so any theory that makes different predictions--and Bell's theorem says any local hidden variable theory must make different predictions--is ruled out by experiment.

In your terminology, Bell's theorem says that it is impossible to have a local hidden variable theory that "complies with QM prerequisites".

 

[DrChinese]

Looking for the help with a conceptual framework of the Bell’s theorem and would appreciate any assistance.

The Bell theorem has been verified in numerous experiments (lets assume that those experiments are free form loopholes).

As I understand the essence of those proves are based on comparing correlations predicted by QM with the correlations derived from the hidden variables theories and after ruling them out, the QM (non-locality) is the only option.

If the above makes a sense, should these hidden variables theories have some prerequisites to be used in the Bell theorem? Should they be in compliance with QM equations and yield the same experimental results as predicted by QM. If they don’t complies with QM, they are not real hidden variables (QM) theories but surrogatess only.

If these theories indeed comply with QM prerequisites , why their correlations aren’t calculated based on the QM prediction, but the ‘classical’ approach is used instead?

miosim, I will give you a short reply but will otherwise bow out to allow others to answer you. I don't believe I have been effective in getting the message across on this subject in earlier threads. Hopefully, the following will help a little.
----------------------------------------------------------

With Bell, you MUST distinguish between LOCAL hidden variable theories and NON-LOCAL ones. You didn't do that above, and it makes it nearly impossible to answer your questions. Here is the summary:

1. LOCAL HV theories are incompatible with QM and Bell tests.
2. Bell tests do NOT prove NON-LOCALITY. They prove what is call QUANTUM NON-LOCALITY.
3. The reason for this different term (with Quantum added) is that there are LOCAL NON-REALISTIC theories that are QUANTUM NON-LOCAL. That may seem like a contradiction, but it is not.
4. There are many interpretations of QM that display Quantum Non-locality. They include: Copenhagen, Bohmian, Many Worlds, Time Symmetric, etc. All of these predict as per orthodox QM. So all of these are either NON-LOCAL or NON-REALISTIC (or both).
5. The observed correlations in Bell tests are not as predicted by classical statistics, so that approach is discarded completely when calculating the predictions. As you can see from the graph you presented previously, a classical approach to the statistics does not even come close to the actual results. The other approach is what is sometimes called semi-classical. That approach is closer to actual, but is still ruled out.

Good luck,

-DrC

 

[miosim]

Bell's theorem says that it is impossible to have a local hidden variable theory that makes the same predictions for experimental results as QM

This is exactly what I can't grasp, doesn't matter how references I read.

I thought that Bell (and other) select some generic candidate for hidden variable theory and applying some sort of classical approach (foreign for the QM system) calculate correlation.
As I understand the REAL hidden variable theory should have the same predictions as orthodox QM but only having different interpretation of the observable results. Therefore it may not be fair to calculate correlation not deriving from QM.

 

[PeterDonis]

I thought that Bell (and other) select some generic candidate for hidden variable theory and applying some sort of classical approach (foreign for the QM system) calculate correlation.

First, as DrChinese noted, you need to carefully distinguish between local hidden variable theories and nonlocal hidden variable theories. Bell's Theorem only applies to local ones.

What Bell did was to show that any local hidden variable theory must have certain properties that restrict the correlations it can predict; and that the correlations predicted by QM are outside those restrictions. He didn't pick any particular local hidden variable theory, not even a "generic candidate"; he just used general properties that any local hidden variable theory must have.

As I understand the REAL hidden variable theory should have the same predictions as orthodox QM

It's perfectly possible for a nonlocal hidden variable theory to make the same predictions as QM. An example of such a theory is the De Broglie-Bohm pilot wave theory. But it's not possible for a local hidden variable theory to make the same predictions as QM. So one way of stating the conclusion of Bell's Theorem is that, if you want the "real" hidden variable theory to make the same predictions as QM, it has to be nonlocal.

 

[miosim]

Thank you for both of you, it helps to focus .

Dr. DrChinese

I am back reading tutorials on your site ...

http://drchinese.com/David/Bell_Theorem_Easy_Math.htm

... but stumble on the following sentence:

“…Suppose we consider a single particle (photon) of light. We ask a simple question: does it have a definite polarization at the following three angles: 0 degrees (A), 120 degrees (B), and 240 degrees (C)? According to the EPR paper, its polarization at these 3 angles correspond to actual elements of reality…”

My question is: How a single photon may have “definite polarization at the THREE angles”?

 

[miosim]

My question is: How a single photon may have “definite polarization at the THREE angles”?

I gess you mean that according to EPR a single photon have definite polarization regardless of the angle of polarizer.

 

[DrChinese]

I gess you mean that according to EPR a single photon have definite polarization regardless of the angle of polarizer.

Exactly! In other words: your car (if you have one :) ) has a length, a height, and a width at all times. Does a photon? Obviously photons do not have a length, a height, or a width. But do they have a polarization at 0, 120, 240 degrees at one point in time (simultaneously) ?

The EPR ipremise was that they did. (And you could call that a "semi-classical" idea because it would match the perfect correlations they expected for entangled systems. Please note that "semi-classical" has somewhat different meanings to different people in QM concepts.) In other words, EPR did not deny entanglement as phenomenon. They envisioned a day in which such a theory could be formulated (it never was of course).

So EPR ASSUMED realism (elements of reality) but could not prove it rigorously. Bell formulated a challenge to that assumption.

 

[jtbell]

What Bell did was to show that any local hidden variable theory must have certain properties that restrict the correlations it can predict;

For a given experimental setup, this restriction can usually be described using an inequality relation for the quantity being measured. This is often called a Bell inequality.

and that the correlations predicted by QM are outside those restrictions.

That is, the predictions of QM violate the Bell inequality for the experiment in question; whereas the predictions of any local realistic theory must satisfy that inequality. To date all experiments of this type have violated their Bell inequality.

 

[miosim]

But do they have a polarization at 0, 120, 240 degrees at one point in time (simultaneously)

I think this sentence may confuse reader, because I read it as the 'same photon may have 3 different polarizations (simultaneously) prior interacting with polarizer'.

I have other difficulties to read explanations on your site, but I don't want to "bug" this thread with my silly questions about every sentence.

I understand all comments provided so far, but I feel I need go back to reading and than I may have a better question.

Thank you all

 

[DrChinese]

I think this sentence may confuse reader, because I read it as the 'same photon may have 3 different polarizations (simultaneously) prior interacting with polarizer'.

You don't think it strange that any ordinary object has a length/width/height, but have a hard time imagining polarizations at 3 angles? EPR imagined the extra attributes existed (of course I follow QM which does not support that).

Different sites use different words, and that may help you too.

 

[Nugatory]

I think this sentence may confuse reader, because I read it as the 'same photon may have 3 different polarizations (simultaneously) prior interacting with polarizer'.

You should read that as "the photon may be in a state such that if I measure the polarization at a given angle I will get a particular result". A three-dimensional object works that way - I can say that if I measure its height I'll get some number, if I measure its width I'll get some number, if I measure its length I'll get some number. We call these three numbers the height, width, and length respectively, and we have no problem saying that the object has definite height, width, and length whether we make the measurement or not.

Quantum mechanical measurements don't work that way. The formalism of QM says that if we don't measure something then it doesn't have a definite value (not "it has a value but we don't know what it is because we didn't measure it"). Bell's theorem says that any assignment of definite values to these unmeasured quantities will, under some circumstances, lead to results that differ from the quantum mechanical prediction.

 

[morrobay]

First, as DrChinese noted, you need to carefully distinguish between local hidden variable theories and nonlocal hidden variable theories. Bell's Theorem only applies to local ones.

What Bell did was to show that any local hidden variable theory must have certain properties that restrict the correlations it can predict; and that the correlations predicted by QM are outside those restrictions. He didn't pick any particular local hidden variable theory, not even a "generic candidate"; he just used general properties that any local hidden variable theory must have.
It's perfectly possible for a nonlocal hidden variable theory to make the same predictions as QM. An example of such a theory is the De Broglie-Bohm pilot wave theory. But it's not possible for a local hidden variable theory to make the same predictions as QM. So one way of stating the conclusion of Bell's Theorem is that, if you want the "real" hidden variable theory to make the same predictions as QM, it has to be nonlocal.

EPR assumed realism but the conclusion for the inequality violation is usually non - locality. Why not non realism alone ?
What would be a real (complete) local non realism hidden variable model that would make same predictions as QM ?
Besides the many worlds interpretation.

 

[PeterDonis]

EPR assumed realism but the conclusion for the inequality violation is usually non - locality.

That's because Bell used the assumption of locality to derive his theorem; he didn't use any assumption of "realism". AFAIK the MWI, since it's just an "interpretation" of QM and makes all the same predictions, violates the locality assumption as well, at least as Bell formulated it.

 

[miosim]

Bell's theorem says that any assignment of definite values to these unmeasured quantities will, under some circumstances, lead to results that differ from the quantum mechanical prediction.

I am trying to understand why it happens. What are the KEY physical characteristics of any deterministic LV theories that case a lower correlation compare to correlations derived from QM?

I wonder if the difference is as follow: in the deterministic model the spin/polarization are not affected by position of detectors, but according to QM the position of detectors influences (positively?) the result of measurement thus causing a higher correlation.
(see below Bell's article "On the E...P...R... paradox (1964) describing the deterministic LV model: "The vital assumption for this model is that the result of measurement B doesn't depend on setting of detectors."

https://www.physicsforums.com/attachments/74296

It's perfectly possible for a nonlocal hidden variable theory to make the same predictions as QM. An example of such a theory is the De Broglie-Bohm pilot wave theory.

It seams that Broglie–Bohm theory (see quote from Wiki below) has the same prediction as QM because according to this theory the measurement device is also influences the particle's property (spin, polarization, etc.) and thus increases (compare to deterministic model) correlation. Does it make sense?

http://en.wikipedia.org/wiki/De_Broglie%E2%80%93Bohm_theory
Thus, for the de Broglie–Bohm theory, the particle's spin is not an intrinsic property of the particle—instead spin is, so to speak, in the wave function of the particle in relation to the particular device being used to measure the spin.

 

[Nugatory]

I am trying to understand why it happens. What are the KEY physical characteristics of any deterministic LV theories that case a lower correlation compare to correlations derived from QM?

I wonder if the difference is as follow: in the deterministic model the spin/polarization are not affected by position of detectors, but according to QM the position of detectors influences (positively?) the result of measurement thus causing a higher correlation.
(see below Bell's article "On the E...P...R... paradox (1964) describing the deterministic LV model: "The vital assumption for this model is that the result of measurement B doesn't depend on setting of detectors."

Yes, that's pretty much it. Bell's proof proceeds from the assumption that the probability distribution of the measurement at a detector can be written as a function of the initial state of the entangled pair and the setting of that detector, but not the setting of the other detector. That's pretty much the operational definition of a local theory, and altogether consistent with EPR's sense of locality.

It seams that Broglie–Bohm theory (see quote from Wiki below) has the same prediction as QM because according to this theory the measurement device is also influences the particle's property (spin, polarization, etc.) and thus increases (compare to deterministic model) correlation. Does it make sense?

The dBB theory is indeed non-local in this sense, and thus is capable of reproducing the QM predictions and experimental results. It's a matter of personal taste and an endless interpretational rathole whether it is strictly correct that the measuring device "influences the particle's property"; the most that you can say without slipping in some additional assumptions is that the settings of the measuring devices influence the correlation of the results of the measurements.

 

[miosim]

Bell's proof proceeds from the assumption that the probability distribution of the measurement at a detector can be written as a function of the initial state of the entangled pair and the setting of that detector, but not the setting of the other detector.

Do you mean that per QM (and not per deterministic model) the measurement at a closest detector can be written as a function of the initial state of the entangled pair and the setting of that detector, while the second (more distal) detector interacts with the particle that lost entanglements?

 

[Nugatory]

Do you mean that per QM (and not per deterministic model) the measurement at a closest detector can be written as a function of the initial state of the entangled pair and the setting of that detector, while the second (more distany) detector interacts with the particle that lost entanglements?

No I do not mean that. It's easy to set up thought experiments in which the distances to the detectors and hence the relative ordering of the two observations is different for different observers, so there is no meaningful way of saying that one particle interacts before the other. The quantum mechanical prediction is that the after you've done both measurements, in either order, they will be correlated; bell's theorem says this correlation will have statistical properties that cannot be reproduced by any theory in which the probability distribution of results at one detector is not a function of, among other things, the setting of the other detector.

 

[miosim]

It's easy to set up thought experiments in which the distances to the detectors and hence the relative ordering of the two observations is different for different observers, so there is no meaningful way of saying that one particle interacts before the other.

Say we set up a thought experimentscase of entangled photons, in which the first detectors is just few meters away from the experimenter, while the second detectors is on Mars. Would the experimenter receive the results of measurement from the first detector in a few nanoseconds instead of few minutes (time that light travel to Mars)?
This question is not about Bell correlation but about QM in general. Is the photon that will reach the detector on Mars will be still entangled?

 

[DrChinese]

That's because Bell used the assumption of locality to derive his theorem; he didn't use any assumption of "realism".

Don't mean to be contradictory, but...

Bell did not identify the spot where he introduced realism... but he did introduce it and I can show you where. Look at his original paper. His (2) is the locality condition, and then after his (14) he makes the leap to a THIRD element of reality (in addition to a and b): "It follows that c is another unit vector..." This is referenced in EPR when they say that they assume that elements of reality that individually must exist (height/length/width are our examples of simultaneous realism in this thread) must also exist simultaneously. Bell is making the mathematical implementation of that statement by mixing a, b and c in the next few equations. (QM of course would reject all this, either (2) or (14+) or both.)

 

[DrChinese]

Say we set up a thought experimentscase of entangled photons, in which the first detectors is just few meters away from the experimenter, while the second detectors is on Mars. Would the experimenter receive the results of measurement from the first detector in a few nanoseconds instead of few minutes (time that light travel to Mars)?
This question is not about Bell correlation but about QM in general. Is the photon that will reach the detector on Mars will be still entangled?

Nugatory answered that and you quoted his answer: there is no relevance to the ordering or the distance involved. You can make any deductions you like from that, because QM makes no further statement about it at all.

 

[miosim]

Nugatory answered that and you quoted his answer: there is no relevance to the ordering or the distance involved. You can make any deductions you like from that, because QM makes no further statement about it at all.

OK, fair enough.

 

[Nugatory]

Say we set up a thought experiment case of entangled photons, in which the first detectors is just few meters away from the experimenter, while the second detectors is on Mars. Would the experimenter receive the results of measurement from the first detector in a few nanoseconds instead of few minutes (time that light travel to Mars)?

yes, but an observer aboard a spaceship flying through the solar system at sufficient speed will find that the detector on Mars triggers before the one on earth. Both observers are equally correct, and there is no way to claim that one order of detections is more real or right than the other. You'll have to learn some special relativity to understand what's going on here - google for "relative of simultaneity" and "space-like separation".

This question is not about Bell correlation but about QM in general. Is the photon that will reach the detector on Mars will be still entangled?

In QM it makes no sense to talk about the time in between the two measurements, nor about a single member of a pair being entangled. Entanglement is a statement about the results of two measurements, after they've both been made - and nothing more.

 

[miosim]

but an observer aboard a spaceship flying through the solar system at sufficient speed will find that the detector on Mars triggers before the one on earth.

I think I have a general understanding of this concept. However for an observer who is in the same time reference as Earth or Mars and taking in account comparatively small difference in speed between Earth and Mars - would the time dilution will be too small to "wash out" the difference between nanoseconds and minutes ?

 

[miosim]

I actually don't concern much about "space-like separation" question and is satisfied with your answer and the fact that "QM makes no further statement about it at all"

I am glad to see the light in the end of a tunnel called Bell's theorem. I will have few more questions but first I need to do some homework.

Thank you all for help.

 

[Nugatory]

I think I have a general understanding of this concept. However for an observer who is in the same time reference as Earth or Mars and taking in account comparatively small difference in speed between Earth and Mars - would the time dilution will be too small to "wash out" the difference between nanoseconds and minutes ?

There will be some observers for whom the detection on Earth "obviously" happens first, and they'll be the ones who are more or less at rest relative to Earth and Mars - Mars is moving at several kilometers per second relative to earth, but compared with the speed of light that's nothing.

There will be other observers for whom the detection on Mars just as "obviously" happens first, and they'll be the ones who are flying past the solar system at a relative speed that's very close to that of light.

There is no way that you can arrange the two detectors so that all observers everywhere in the universe will agree about which one happened first. That's the significance of the "space-like separation" that I suggested you google for earlier.

 

[miosim]

I am still troubled with how Bell theorems treats the Local Hidden Variable (LHV) theories by reducing them to classical objects, like magnets, socks, boxes, etc.

J. S. Bell, “Bertlmann’s socks and the nature of reality” 1981
“.. We formulated these considerations first for pairs of socks, moving with considerable confidence in those familiar objects. But why not reason similarly for the pairs of particles of the EPR experiment? …”

My answer to this rhetorical question would be: - no, because in general we should not use the same reasoning for classical objects (socks, magnets, etc.) and for quantum entities (as LHV models are).

Lets continuer with this Bell’s paper:

“General argument.
So the following argument will not mention particles, nor indeed fields, nor any other particular picture of what goes on at the microscopic level. Nor will it it involve any use of the words "quantum mechanical system", which can have an unfortunate effect on the discussion. The difficulty is not created by any such picture or any such terminology. It is created by the predictions about the correlations in the visible outputs of certain conceivable experimental set-ups. Consider the general experimental set-up of figure. …”

Following these lines, the LHV model, striped from all quantum and even physical “privileges”, was used to calculate correlation based on the formal logic of socks and boxes. Eventually this “stripped to bones” LHV model was extended to All LHV models and eventually was “successfully” stricken down because it does not match with experiment.

OK, let's be a little more creative with LHV and conceder one that incorporates the “algorithm” that assists photon passing polarizer (or any other obstacle). The example of such type of “algorithms” is principle of minimal action and principle of minimum free energy, etc.

Could the correlation for such LHV model match with correlation for QM model?

 

[Nugatory]

OK, let's be a little more creative with LHV and conceder one that incorporates the “algorithm” that assists photon passing polarizer (or any other obstacle). The example of such type of “algorithms” is principle of minimal action and principle of minimum free energy, etc.

Could the correlation for such LHV model match with correlation for QM model?

No, not as long as it is local. You will notice that Bell's $\lambda$ is a very general construct, and it's quite capable of incorporating complex and "smart" algorithms. The only constraint that Bell places on the $\lambda$ "algorithm" is that the setting at detector A is not allowed to be an input for the result at detector B, and vice versa.

It's a fun exercise (I've done it just for fun - but also google for "quantum Randi challenge") to actually code up a computer simulation in which you can plug in any functions you please to assign any values you please to any variables associated with each particle; and use any functions you please to calculate the probability of a simulated particle passing its polarizing filter. You will find yourself unable to violate the Bell inequalities.

 

[wle]

I am still troubled with how Bell theorems treats the Local Hidden Variable (LHV) theories by reducing them to classical objects, like magnets, socks, boxes, etc.

J. S. Bell, “Bertlmann’s socks and the nature of reality” 1981
“.. We formulated these considerations first for pairs of socks, moving with considerable confidence in those familiar objects. But why not reason similarly for the pairs of particles of the EPR experiment? …”

My answer to this rhetorical question would be: - no, because in general we should not use the same reasoning for classical objects (socks, magnets, etc.) and for quantum entities (as LHV models are).

As you hint at just after this, Bell's reasoning isn't really about quantum entities at all. It's applied to what you actually notice in an experiment, like whether a detector ends up clicking or a certain light bulb switches on or however else it is you are recording the result. From the point of view of Bell's reasoning, any theory about such an experiment is in the end going to have to predict whether you hear that detector click or whether you see that light bulb go on. It might be the case in a certain Bell experiment that the detectors happens to look a lot like a Stern-Gerlach magnets if you open them up and it might be that you think the theoretical explanation should involve things called "spin 1/2 particles", but the way Bell gives his reasoning, those are irrelevant details.

 

[PeterDonis]

Eventually this “stripped to bones” LHV model was extended to All LHV models

No. What you are calling the "stripped to the bones" LHV model is not really a model at all; it's a set of properties that all possible LHV models satisfy. Hence, Bells' theorem applies to all possible LHV models. He wasn't reasoning about any specific LHV model; he was reasoning about all of them by reasoning only about properties that all of them must possess.

Could the correlation for such LHV model match with correlation for QM model?

No, because any such LHV model must have the properties Bell used to derive his theorem. That's the whole point of the theorem, that it must apply to any LHV model.

 

[PeterDonis]

in general we should not use the same reasoning for classical objects (socks, magnets, etc.) and for quantum entities

Yes.

(as LHV models are).

No; LHV models are not "quantum entities". That's the whole point of Bell's theorem, to show that they can't be.

 

[miosim]

he was reasoning about all of them by reasoning only about properties that all of them must possess.

No, because any such LHV model must have the properties Bell used to derive his theorem

So what are these key properties of HV theories? As I understand, they are just a fact that values of spin, polarization, etc., are predetermined prior interacting with detector.
As I understand, the final value of spin, polarization, etc. are affected by a position of detector but we don't know much about specific mechanisms of these interactions - wave function doesn't tell us much and the results we calculate is just a guess - no experimental conformation that our HV model is correct. It is why I am not easy about jumping to calculate correlation based on simplified assumption of what HV are. They could be as wired as QM itself.

However let’s try again a simple example of HV. Say, if particle has HV that measure position of detector that “helps particle to pass”. I think that correlation for this model could be as high as we need if we can plug this “helps particle to pass” algorithm into HV.

 

[miosim]

It's a fun exercise (I've done it just for fun - but also google for "quantum Randi challenge") to actually code up a computer simulation in which you can plug in any functions you please to assign any values you please to any variables associated with each particle; and use any functions you please to calculate the probability of a simulated particle passing its polarizing filter. You will find yourself unable to violate the Bell inequalities.

Is it web application? I didn't find it.

 

[Nugatory]

Is it web application? I didn't find it.

"Quantum Randi challenge" should find plenty of hits. If you mean my simulation I've never posted it, as it's only a few hundred lines of Java code and writing it is 90% of the value.

 

[PeterDonis]

what are these key properties of HV theories? As I understand, they are just a fact that values of spin, polarization, etc., are predetermined prior interacting with detector.

No, they are that the probability distribution of measurement results at detector A cannot depend on the settings of detector B, and vice versa. That was Bell's key assumption. It is usually referred to as "locality", and is why the conclusion of the theorem applies only to "local" hidden variable theories.

Given that assumption, it doesn't matter what else determines the results at detector A: they could be predetermined, they could be sampled from a random number generator, they could depend on the phase of the moon--as long as that information is available at detector A to be taken as input at the time of measurement. (Note that detector B's settings are not available at detector A at the time of measurement if the measurements are spacelike separated.)

Say, if particle has HV that measure position of detector that “helps particle to pass”. I think that correlation for this model could be as high as we need if we can plug this “helps particle to pass” algorithm into HV.

I don't understand what you mean by "helps particle to pass", but as long as the algorithm for detector A does not take detector B's settings as input, and vice versa, you will be unable to produce results that violate the Bell inequalities. If you think you can, then start programming.