Bell's Theorem with Easy Math - Stuck

[Ilja]

We have to remember that the output from a single entangled photon is always 100% random, no matter what you do or how the polarizers are set. Only when you bring the measured data together you will see that there is indeed a connection between them in form of correlations. Thus there seems to be a non-local causal ‘link’ between the two entangled photons, and this non-local causality is suppose to be independent of distance. The ‘link’ is today interpreted as the shared wavefunction between the entangled photons, but no one knows exactly how this mechanism works (yet).

This argument is misleading, and here is why: Imagine I claim to have an FTL phone. So we test it. I talk something, you hear something. It is quite clear: If we compare this later, and it is the same, we can be sure that the FTL phone works.

Really? We have to remember that the output from the phone is always 100% random. Only when you bring the measured data about my input with these output data together, you will see that there is indeed a connection between them in form of correlations. Thus there seems to be a non-local causal ‘link’ between the two parts of the phone, and this non-local causality is suppose to be independent of distance. The ‘link’ is today interpreted as the shared wavefunction between the entangled photons, but no one knows exactly how this mechanism works (yet).

The same excuse works nicely. Even a working FTL phone can no longer falsify Einstein causality.

The bottom-line is: The old classical Local Realism has retired for good, and will not return...

Nor locality (in a meaningful definition of locality, which does not name a causal interaction with 0.99c local but with 1.001c nonlocal) nor realism are in any danger. Only the modern notion of causality invented by Einstein, Einstein causality, should be rejected. No problem, because all we have to do is to go back to pre-Einsteinian, classical causality.

 

[Badvok]

If you want a fuller picture on Bell’s view, I can warmly recommend this paper:

J.S. Bell's Concept of Local Causality

Many thanks, that is definitely something I need to read and I will, especially after a quick glance and seeing this bit: "Typically, for example, one encounters the claim that Bell’s inequality follows not from local causality alone, but from the conjunction of local causality with some additional premises; some of the usual suspects here include “hidden variables,” “determinism,” “realism,” “counter-factual definiteness”, or an improper insistence on a vaguely-defined “classical” way of thinking. One or more of these (rather than relativistic local causality) is then invariably blamed for the inconsistency with experiment."
So I think it is going to answer a lot of my questions.

 

[DrChinese]

Many thanks, that is definitely something I need to read and I will, especially after a quick glance and seeing this bit: "Typically, for example, one encounters the claim that Bell’s inequality follows not from local causality alone, but from the conjunction of local causality with some additional premises; some of the usual suspects here include “hidden variables,” “determinism,” “realism,” “counter-factual definiteness”, or an improper insistence on a vaguely-defined “classical” way of thinking. One or more of these (rather than relativistic local causality) is then invariably blamed for the inconsistency with experiment."
So I think it is going to answer a lot of my questions.

The EPR Paper specifies most of what one might look for regarding the Bell assumptions. Bell assumed familiarity with that paper and wasn't as explicit as he could have been.

1. EPR says there are "elements of reality" that one must accept if the result of an experiment can be predicted without disturbing the system in advance in any way. This is reasonable and justified.
2. EPR says that if there are "spooky" influences that are faster than c, then that is a loophole to their conclusion.
3. EPR says that if you require the "elements of reality" to be simultaneously observable, then that too is a loophole to their conclusion. See the last
4. Given the above, EPR concludes QM is incomplete. This is correct IF you agree to 1, 2, 3.

And finally:
5. EPR speculates that anything other than 3. is unreasonable. This (1+3) is the realism assumption, and this is actively exploited by Bell. If you agree with EPR that any other view (of 3.) is unreasonable, then the Bell realism assumption is both well-defined and acceptable to you. You cannot get the Bell result without the EPR assumption. Here is a quote from EPR, 2nd to last paragraph:

"One could object to this conclusion on the grounds that our criterion of reality is not sufficiently restrictive. Indeed, one would not arrive at our conclusion if one insisted that two or more physical quantities can be regarded as simultaneous elements of reality only when they can be simultaneously measured or predicted. On this point of view, since either one or the other, but no both simultaneously, of the quantities P and Q can be predicted, they are not simultaneously real. This makes the reality of P and Q depends upon the process of measurement carried out on the first system, which does not disturb the second system in any way. No reasonable definition of reality could be expected to permit this".

 

[Badvok]

Thanks DrChinese, I still haven't managed to fully read the paper DA linked but I think what I'm struggling with is this: EPR uses momentum and position as examples, but all the other examples seem to relate to polarisation or spin. Now momentum and position can be directly measured and quantified, however, polarisation and spin cannot. For polarisation and spin we only get a one-way-or-the-other result and which result we get is probabilistic and not a direct measurement of an actual property of the particle. I have a feeling that this difference should affect the maths some how but I'm not exactly sure how.

 

[Nugatory]

TNow momentum and position can be directly measured and quantified, however, polarisation and spin cannot.

You sure about that? You may want to spend some time thinking about exactly how a momentum or a position measurement is made before you attach the word "directly" to these measurements while withholding it from the spin and polarization measurements.

For polarisation and spin we only get a one-way-or-the-other result and which result we get is probabilistic and not a direct measurement of an actual property of the particle.

The measurements of position and momentum are also probabilistic. And "actual property" might be among the slipperiest words you will encounter in a QM discussion.

 

[stevendaryl]

You sure about that? You may want to spend some time thinking about exactly how a momentum or a position measurement is made before you attach the word "directly" to these measurements while withholding it from the spin and polarization measurements.


The measurements of position and momentum are also probabilistic. And "actual property" might be among the slipperiest words you will encounter in a QM discussion.

Some people claim that the only thing that's measurable is position. Everything else is done by setting up a correlation between the quantity that you want to measure and a particle's position. For example, you measure spin by deflecting a particle by a magnetic field, and you measure the deflection by the position. You measure velocity by measuring two different positions. Etc.

I don't know if that's really true, but it has been claimed. I think that advocates of the Bohm model say this, because the Bohm model doesn't really say anything about any state variable other than position, but that's considered good enough.

 

[DevilsAvocado]

Many thanks, that is definitely something I need to read and I will

Glad you liked it, I think it’s very useful – especially the quotes from Bell.

However, there is one little “caveat” (isn’t there always in this stuff? ), Travis Norsen is a supporter of the de Broglie Bohm pilot wave theory, and this is good to know (can’t someone ban these darned interpretations! ). I think he balances this fact pretty well, but to be 100% sure you have to be a professional expert (which I’m not)...


Of course, I have absolutely no idea, but if you google: Speakable and Unspeakable in Quantum Mechanics pdf, there might turn up more interesting reading... seek, and ye shall find.

 

[DevilsAvocado]

Thanks DrChinese, I still haven't managed to fully read the paper DA linked but I think what I'm struggling with is this: EPR uses momentum and position as examples, but all the other examples seem to relate to polarisation or spin. Now momentum and position can be directly measured and quantified, however, polarisation and spin cannot. For polarisation and spin we only get a one-way-or-the-other result and which result we get is probabilistic and not a direct measurement of an actual property of the particle. I have a feeling that this difference should affect the maths some how but I'm not exactly sure how.

No, the Heisenberg uncertainty principle states that one cannot simultaneously know, with arbitrarily high precision, both the position and momentum of a particle.

Counterfactual definiteness (CFD) is closely related to Heisenberg uncertainty principle (HUP), and many scratches their head and wonder what ‘mystical QM thing’ prevent us from getting the data we want. If you think about it, it’s not weird at all. All this ‘commotion’ is due to fact that we automatically think of electrons and photons etc as particles, which they are not (until we measure them). If you instead think of waves it becomes much clearer. Assume we want to measure the exact frequency of a ‘normal’ sound wave – at an exact location in space. Can we do this? Nope, it’s impossible! To get the exact frequency you have to measure the sound wave for some time (at least one cycle), and there goes your exact location down the drain.

Not weird at all, is it?

DrC’s summary of EPR/CFD is great, but it’s good to know the historical facts behind the 1935 EPR paper. Einstein was obviously not particularly interested in CFD:

http://plato.stanford.edu/entries/qt-epr/]...[/PLAIN] as early as June 19, 1935 Einstein makes it plain that he is not especially interested in the question of simultaneous values for incompatible quantities like position and momentum. Just as in Solvay 1927, the concern that he expresses to Schrödinger is with the question of completeness, given the resources of the quantum theory, in describing the situation concerning a single variable (maybe position, maybe momentum). With respect to the treatment of an incompatible pair he tells Schrödinger “ist mir wurst”—literally, it's sausage to me; i.e., he couldn't care less. (Fine 1996, p. 38). In his writings subsequent to EPR, Einstein probes an incompatibility between affirming locality and separability, on the one hand, and completeness in the description of individual systems by means of state functions, on the other. His argument is that we can have at most one of these but never both. He frequently refers to this dilemma as a “paradox”.

[my bolding]

And the paper was written by Podolsky:

http://arxiv.org/abs/quant-ph/0310010]Some[/PLAIN] time after that work, Rosen became a post-doc of Einstein at the Institute of Advanced Studies in Princeton. One day, at the traditional 3 o’clock tea, Rosen mentioned to Einstein a fundamental issue of interpretation related to entangled wave-functions. Einstein immediately saw the implications for his long standing disagreement with Bohr. As they discussed the problem, Boris Podolsky joined the conversation, and later proposed to write an article. Einstein acquiesced. When he later saw the text, he disliked the formal approach, but agreed to its publication. Then, as soon as the EPR article appeared, Podolsky released its contents to the New York Times (4 May 1935, page 11) in a way implying that the authors had found that quantum mechanics was faulty. This infuriated Einstein, who after that no longer spoke with Podolsky.

Entanglement was a newly discovered property of QM, and Einstein wanted to utilize this in his ‘fight’ with Bohr, thus the situation made CFD momentum/position the ‘main attraction’ in the 1935 EPR paper.

It could have been different if Einstein had been more skilled in the English language...

maybe

 

[Badvok]

No, the Heisenberg uncertainty principle states that one cannot simultaneously know, with arbitrarily high precision, both the position and momentum of a particle.

Sorry, I know that, that's not what I meant. I meant that if I choose to measure a particle's position I can do that fairly accurately, I don't limit the position measurement to a yes/no result (is it in one region or another?) And likewise, I can measure momentum fairly accurately without limiting it to a yes/no result. I know I can't measure both on the same particle.

 

[Nugatory]

Sorry, I know that, that's not what I meant. I meant that if I choose to measure a particle's position I can do that fairly accurately, I don't limit the position measurement to a yes/no result (is it in one region or another?) And likewise, I can measure momentum fairly accurately without limiting it to a yes/no result.

If I understand you properly, you're describing the difference between a continuous eigenvalue spectrum and a discrete one. Bell's proof is formulated in a way that covers both cases, but to see this you have to have to read the actual Bell paper and that takes us beyond "easy math".

 

[DrChinese]

If I understand you properly, you're describing the difference between a continuous eigenvalue spectrum and a discrete one. Bell's proof is formulated in a way that covers both cases, but to see this you have to have to read the actual Bell paper and that takes us beyond "easy math".

Badvok, I agree with Nugatory on this point. Once you understand the full implications of the Bell result, you can find ways to apply it to all kinds of observables. In fact, I would say there are perhaps hundreds of observables*, many continuous such as momentum or energy, which can be shown to violate Bell-type inequalities. But the math gets progressively more complex, and the experimental setups tax the noodle.

The point being that the Bell example I present using photon polarization is good as is. You don't get the classical result switching to a different basis.

*A recent paper of Zeilinger et al mentions well over 100 designer observables!
http://arxiv.org/abs/1306.0096

 

[stevendaryl]

Badvok, I agree with Nugatory on this point. Once you understand the full implications of the Bell result, you can find ways to apply it to all kinds of observables. In fact, I would say there are perhaps hundreds of observables*, many continuous such as momentum or energy, which can be shown to violate Bell-type inequalities. But the math gets progressively more complex, and the experimental setups tax the noodle.

The point being that the Bell example I present using photon polarization is good as is. You don't get the classical result switching to a different basis.

*A recent paper of Zeilinger et al mentions well over 100 designer observables!
http://arxiv.org/abs/1306.0096

I haven't spent much time thinking about it, but it seems to me that doing an EPR-type experiment for other observables, such as momentum and position, would be a lot more difficult. It's very simple to change from measuring spin along axis A to measuring spin along axis B, but to change from measuring position to measuring momentum is a big change.

 

[DrChinese]

I haven't spent much time thinking about it, but it seems to me that doing an EPR-type experiment for other observables, such as momentum and position, would be a lot more difficult. It's very simple to change from measuring spin along axis A to measuring spin along axis B, but to change from measuring position to measuring momentum is a big change.

In some cases it can very difficult. For example:

http://arxiv.org/abs/1206.2141

"We propose an experiment which can demonstrate quantum correlations in a physical scenario as discussed in the seminal work of Einstein, Podolsky and Rosen. Momentum-entangled massive particles are produced via the four-wave mixing process of two colliding Bose-Einstein condensates. The particles' quantum correlations can be shown in a double double-slit experiment or via ghost interference. "

 

[Badvok]

Thanks all for your assistance with this but it looks like I'm falling between two stools. On the one hand I see simplified explanations that make assumptions that I can't clearly see the validity of, on the other hand my mathematical ability is not up to the required level to read and fully understand the actual papers. So I have a lot more learning to do before I can get my head around these concepts.

 

[Ilja]

For a simple introduction into the violation of Bell's inequality see http://ilja-schmelzer.de/realism/game.php

 

[DevilsAvocado]

Thanks all for your assistance with this but it looks like I'm falling between two stools. On the one hand I see simplified explanations that make assumptions that I can't clearly see the validity of, on the other hand my mathematical ability is not up to the required level to read and fully understand the actual papers. So I have a lot more learning to do before I can get my head around these concepts.

Why do I get the feeling a non-local bullet just penetrated my little green heart...?? :uhh: (:rofl:)

Could this be a remedy?

Everything should be made as simple as possible, but not simpler.

If it worked for J.S. Bell, why shouldn’t it work for us? In lectures he used exactly the same example as I gave you:

N(+30°, -30°) ≤ N(+30°, 0°) + N(0°, -30°)​

Sorry, I know that, that's not what I meant. I meant that if I choose to measure a particle's position I can do that fairly accurately, I don't limit the position measurement to a yes/no result (is it in one region or another?) And likewise, I can measure momentum fairly accurately without limiting it to a yes/no result. I know I can't measure both on the same particle.

If this is what stopping you from proceeding, I don’t really understand why... on a normal macroscopic scale we can pin down objects to a precise position in continuous space, but can we really do this in the QM world?

Well, from the Stern–Gerlach we know that angular momentum takes only certain quantized values:

https://www.youtube.com/watch?v=rg4Fnag4V-E


Then the question arises - Is space[time] itself quantized?

According to Loop quantum gravity it is, consisting of an extremely fine fabric of finite loops. The size of this structure is the Planck length which is approximately 10⁻³⁵ meters.

If spacetime is quantized – and you want to measure position – you will get a 'quantized' Yes/No answer...

(Besides, all particles in QM are ‘wobbling around’ due to energy and virtual particles bumping in and out of the QM soup. And if you try to remove the heat/energy by freezing them near absolute zero to a Bose–Einstein condensate and lowest accessible quantum state, they become indistinguishable!)

If it helps, you can think of the entangled photon as having a superposition of all angles between 0-360°, and you can set the polarizer to any degree + arcminute + arcsecond and so on, for any ‘continues resolution’ you want. The answer will however be Yes/No for the measurement...


... I wish I could understand what the problem is ...

 

[DevilsAvocado]

If I understand you properly, you're describing the difference between a continuous eigenvalue spectrum and a discrete one. Bell's proof is formulated in a way that covers both cases, but to see this you have to have to read the actual Bell paper and that takes us beyond "easy math".

Please have patience with my ignorance – but are there continuous eigenvalues in QM? Where and what is it?


(... getting prepared for embarrassment ... )

 

[Nugatory]

Please have patience with my ignorance – but are there continuous eigenvalues in QM? Where and what is it?

x and p for unbound particles, for example. Eigenfunctions are delta functions, eigenvalues form a continuous spectrum, is what I'm thinking about.

 

[DevilsAvocado]

x and p for unbound particles, for example. Eigenfunctions are delta functions, eigenvalues form a continuous spectrum, is what I'm thinking about.

Well... I did warn for the embarrassment...

Still try to wrap my head around this, but could a continuous spectrum, from let’s say a free electron becoming bound to an hydrogen ion, be used as entanglement in EPR-Bell experiments?

I.e. two free electron meet and get entangle and are then sent to respective hydrogen ion. Could we somehow measure correlations from the spectrum?

(sounds hard)


EDIT:
This can’t work, it’s the electrons that are entangled not the spectrum, right?

 

[lugita15]

DevilsAvocado, any two objects can be entangled, with respect to any observable, whether the observable has discrete eigenvalues, like spin or polarization, or continuous eigenvalues, like momentum.

 

[DevilsAvocado]

I know lugita, the thing that has gotten my nut temporarily dizzy is the continuous spectrum. Could we measure correlations in the spectrum? What are we looking for? A smeared out signal?

 

[lugita15]

I know lugita, the thing that has gotten my nut temporally dizzy is the continuous spectrum. Could we measure correlations in the spectrum? What are we looking for? A smeared out signal?

What spectrum are you talking about? Spectrum refers to the set of eigenvalues of some observable. What observable are you talking about?

 

[DevilsAvocado]

What spectrum are you talking about?

In my example above that would be the light emitted by the two free electrons becoming bound to an hydrogen ion.

Could we see the correlations in that continuous spectrum?

(sounds impossible to me)

 

[DrChinese]

In my example above that would be the light emitted by the two free electrons becoming bound to an hydrogen ion.

Could we see the correlations in that continuous spectrum?

(sounds impossible to me)

When PDC creates 2 photons from 1, the pairs are entangled on the basis of their frequency/wavelength among other things. The frequency is a continuous spectrum of values, they are not limited to discrete values. On the other hand, their polarization is only either 1 or 0 (or +/- or whatever).

As to seeing correlations... absolutely! A lot of observables are available for correlation.

 

[DevilsAvocado]

The frequency is a continuous spectrum of values, they are not limited to discrete values. On the other hand, their polarization is only either 1 or 0 (or +/- or whatever).

As to seeing correlations... absolutely! A lot of observables are available for correlation.

Gosh, there must be some short circuit in the guacamole... I can’t think straight... it’s been a long day... (must blame something :shy:)

Let’s break it down (to the avocado level):

• [Light] frequency is a continuous spectrum of values [of course].
• Higher frequency = higher energy.
• Entangled photons can have any frequency.
• Polarization of photons is not coupled to frequency, or?
• In QM, photon polarization is calculated with the Jones vector and applied to the Poincaré sphere.

Question: Is the Jones vector continues or discrete?

[my guess is continues...]


EDIT:
Of course when we measure the polarization is either 1 or 0, or thru/stopped, etc.

 

[Badvok]

The answer will however be Yes/No for the measurement...

... I wish I could understand what the problem is ...

You actually hit the nail on the head with that statement about the measurement. In all the examples the possible outcomes of a measurement are taken to be the 'elements of reality', this is the same assumption I think Bell makes?

In the EPR paper it says: "If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." To me that doesn't imply that we can actually measure that physical quantity or that we should be able to predict the result of a measurement of that quantity with equal certainty.

From Nugatory's posts I see that EPR refers to correlations between continuous spectrum eigenvalues, i.e. x and p - is that right?

However, Bell refers to measurements of those values that result in Boolean results. Now I guess this is because spin and polarisation are considered discrete spectrum eigenvalues? However, since we can only ever measure spin and polarisation by apparatus that gives us a Boolean result, I don't see how Bell and EPR are actually talking about the same thing. If we could measure the actual spin/polarisation is it possible that we might find that there was an 'element of reality' that was a continuous spectrum eigenvalue? And therefore I don't see how Bell rules this type of LHV out. My guess is that the maths would still work and rule it out but I don't have the ability to do that sort of maths myself.

Next we have the concept of 'perfect correlation' used in yours and Nick Herbert's examples. Now I understand that in a ideal system 'perfect correlation' would exist and that it could be shown to exist in terms of conservation of momentum/energy and such but I don't get how this concept can be extrapolated to apply to the later local interaction of a particle with a local measurement device.

Lastly, on to the logic in DrC's and Ilja's examples and looking at the Scholarpedia article (these are all ones that talk about triple values). In the Scolarpedia article it appears to make the assumption that all three values can't be the same in one bit and then forgets this later (?). If we actually look at this assumption in the context of Ilja's cards then the set of cards from which the selection is made is reduced to just 4 cards so we can at most get two the same (instead of the infinite set used to get 50:50 probability for each selection). With just that limited set the probabilities change and I get a bit lost trying to get my head around them, i.e. the probability of selecting a red card and then another red card from a set of three cards that have in turn been selected from a set of four cards (= 0.25?). Now I don't know whether the assumption that the three values can't all be the same is correct or not, in DrC's example with photon polarisation and 0,120,240 test angles, there is a small but definitely non-zero probability that a photon polarised at angle θ will pass all three polarisers.

I hope I don't raise anyones ire with my language here, I'm not suggesting that I think anything or anyone is wrong, just that I don't understand it.

 

[stevendaryl]

However, Bell refers to measurements of those values that result in Boolean results. Now I guess this is because spin and polarisation are considered discrete spectrum eigenvalues? However, since we can only ever measure spin and polarisation by apparatus that gives us a Boolean result, I don't see how Bell and EPR are actually talking about the same thing. If we could measure the actual spin/polarisation is it possible that we might find that there was an 'element of reality' that was a continuous spectrum eigenvalue?

Bell's analysis does not assume that the hidden variables are discrete, but he assumes that the discrete results that one gets from a spin measurement must be a function of those hidden variables. He gives as a "toy" example of such a hidden variable:

Suppose that associated with every electron is a vector $\vec{S}$. When you measure the spin in direction $\vec{P}$, then you get +1 if $\vec{S} \cdot \vec{P} \geq 0$, and you get -1 if $\vec{S} \cdot \vec{P} \leq 0$.

So this model has a continuous "element of reality", since the vector $\vec{S}$ can point in any direction. It doesn't agree with the predictions of QM, though.

 

[harrylin]

Thanks all, I think I've got it now.

The bit I was stumbling over in Nick Herbert's proof is this: "Starting with two completely identical binary messages". Where do these messages come from? So far as I could see the system has a stream of randomly polarized photons, no actual binary message. Thus the only mismatch that could be measured is by comparing the results obtained at A & B, which obviously links the two detectors and makes the mismatch 75% based on the mutual misalignment angle.[..]

Herbert's proof was also discussed on this forum:

https://www.physicsforums.com/showthread.php?t=90770
and
https://www.physicsforums.com/showthread.php?t=589134

 

[Ilja]

In all the examples the possible outcomes of a measurement are taken to be the 'elements of reality', this is the same assumption I think Bell makes?

No. Bell assumes that there is some reality λ, and this reality, together with the decisions of the experimenters a,b what to measure, defines the outcomes A and B of the experiment: A=A(λ,a,b) and B=B(λ,a,b).

In the EPR paper it says: "If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." To me that doesn't imply that we can actually measure that physical quantity or that we should be able to predict the result of a measurement of that quantity with equal certainty.

In this case, we have a fortunate situation where we can measure it, and where we can predict it, and whenever we really measure it, we find that the prediction is correct.

What can be predicted is the result of A given a=b if we measure B. The EPR criterion can be applied if we assume that measuring B in direction b does not disturb the system at A, so A in direction a=b can be predicted without disturbing this system by measuring B in direction b.

From Nugatory's posts I see that EPR refers to correlations between continuous spectrum eigenvalues, i.e. x and p - is that right?

It is right but quite irrelevant. The discrete version is easier to understand, so there is no reason to consider this earlier variant.

However, Bell refers to measurements of those values that result in Boolean results. Now I guess this is because spin and polarisation are considered discrete spectrum eigenvalues? However, since we can only ever measure spin and polarisation by apparatus that gives us a Boolean result, I don't see how Bell and EPR are actually talking about the same thing.

Very simple: We consider the situation considered by Bell, with discrete results, but apply the argument (the criterion of reality) proposed by EPR for a different, otherwise irrelevant, continuous situation.

If we could measure the actual spin/polarisation is it possible that we might find that there was an 'element of reality' that was a continuous spectrum eigenvalue?

It doesn't matter. The 'elements of reality' themself, as used in the assumption, may be arbitrary - Bell's inequality follows for discrete as well as continuous hidden variables λ.

Next we have the concept of 'perfect correlation' used in yours and Nick Herbert's examples. Now I understand that in a ideal system 'perfect correlation' would exist and that it could be shown to exist in terms of conservation of momentum/energy and such but I don't get how this concept can be extrapolated to apply to the later local interaction of a particle with a local measurement device.

First, there is no need for any extrapolation. Once Bell's inequality is violated in one experiment, it is violated. Point.

Then, there is a minor problem with an argument based on an ideal assumption. The perfect correlation happens only in the ideal case if the direction is ideally the same. So in any real experiment the correlation will not be ideal.

But there are variants of the inequalities, slightly more complicated, which do not depend on this, so that one needs only approximate accuracy in the whole reasoning.

Lastly, on to the logic in DrC's and Ilja's examples and looking at the Scholarpedia article (these are all ones that talk about triple values). In the Scolarpedia article it appears to make the assumption that all three values can't be the same in one bit and then forgets this later (?). If we actually look at this assumption in the context of Ilja's cards then the set of cards from which the selection is made is reduced to just 4 cards so we can at most get two the same (instead of the infinite set used to get 50:50 probability for each selection). With just that limited set the probabilities change and I get a bit lost trying to get my head around them, i.e. the probability of selecting a red card and then another red card from a set of three cards that have in turn been selected from a set of four cards (= 0.25?). Now I don't know whether the assumption that the three values can't all be the same is correct or not,

I don't assume it. If I choose three times the same card, you win with certainty, this would be stupid for me. But I can do this. Therefore, one can prove only an inequality, >=1/3.

If I choose only different cards, and you choose the two cards by accident, without any possibility for me to predict your choice, you have a chance 1/3. If I use three cards of the same color, your chance is 1. So it is always >= 1/3.

 

[DrChinese]

Gosh, there must be some short circuit in the guacamole... I can’t think straight... it’s been a long day... (must blame something :shy:)

Let’s break it down (to the avocado level):

• [Light] frequency is a continuous spectrum of values [of course].
• Higher frequency = higher energy.
• Entangled photons can have any frequency.
• Polarization of photons is not coupled to frequency, or?
• In QM, photon polarization is calculated with the Jones vector and applied to the Poincaré sphere.

Question: Is the Jones vector continues or discrete?

[my guess is continues...]EDIT:
Of course when we measure the polarization is either 1 or 0, or thru/stopped, etc.

Debil,

Polarization of photons is not tied to frequency or wavelength, as you suppose.

Imagine input pump of 400nm wavelength, typically would get out a pair of photons both around 800nm each. But you could also get out one at 802nm, another at 798nm (values are approx.). Or a pair at 804.3nm and 795.7nm. There are no specific values that are prohibited as long as conservation is preserved.

 

[DrChinese]

1. You actually hit the nail on the head with that statement about the measurement. In all the examples the possible outcomes of a measurement are taken to be the 'elements of reality', this is the same assumption I think Bell makes? ... If we could measure the actual spin/polarisation is it possible that we might find that there was an 'element of reality' that was a continuous spectrum eigenvalue?

2. Now I don't know whether the assumption that the three values can't all be the same is correct or not, in DrC's example with photon polarisation and 0,120,240 test angles, there is a small but definitely non-zero probability that a photon polarised at angle θ will pass all three polarisers.

1. Yes, p and q can be predicted with certainty for entangled particles just as spin can.

2. A photon has a definite probability of passing three such aligned filters. All things being equal, that would be 1/2 * 1/4 * 1/4 or about 3%.

You will see the problem if you think of it this way:

a. If I can predict the result of any measurement on Bob by first performing the same measurement on Alice, then you might at first glance that Bob is essentially a clone of Alice. How else to explain the fact that you can predict one by measuring the other? This is the position of the local realist, and it is the position of EPR.

b. In fact, there are literally hundreds if not an infinite number of different measurements that can be performed on Bob and predicted in advance (by measuring Alice). You can do at 1 degree, 2 degrees, 3 degrees, etc, or 1.1 degrees, 1.2 degrees, 1.3 degrees. Gosh, Bob must be carrying around a LOT of hidden variables! Alice too! And that is just the spin degrees of freedom. There are many more.

c. Now try to map values to those. For example, make 1 degree be +, 2 degrees be +, etc until finally you find one where you decide to say it is -. Maybe that is at 115 degrees. Whatever you say it is. Do this for all 360 degrees. (Or for simplicity, every 10 degrees or something.) Keep in mind that these values are preset if you follow the EPR program. You don't know what they are, but they must be something!

d. Here is the Bell stumbling block: they not only have to be "something", but across a series of successive measurements on different entangled pairs, they must match the quantum (QM) predictions when the angles are NOT the same! And what is that predictions? It is cos^2(theta) where theta is the difference in the measurements on Alice and Bob. It doesn't really matter how he figured it out, but Bell found that this requirement "broke the bank" on the EPR argument, so to speak.

e. It turns out there are NO sets of values that BOTH support the EPR outcomes - the perfect correlations of b.) - AND the QM requirement of d. If you actually try to come up with such a set, you will see in short order what the problem is. Just do a handful of examples and you will quickly see that you can't make it work out. Try it! Really! Get out a piece of scratch paper for 15 minutes and you will see what is wrong with the EPR program. (Keep in mind that Bell was the first person in 30 years to discover this. So don't beat yourself up that you need to invest 15 more minutes!)

f. Bell's Inequality is simply a generalized proof of this same fact.

 

[DevilsAvocado]

Thanks DrC

 

[Badvok]

c. Now try to map values to those. For example, make 1 degree be +, 2 degrees be +, etc until finally you find one where you decide to say it is -. Maybe that is at 115 degrees. Whatever you say it is. Do this for all 360 degrees. (Or for simplicity, every 10 degrees or something.) Keep in mind that these values are preset if you follow the EPR program. You don't know what they are, but they must be something!

This is a step that I have trouble understanding. Why are we reducing it to + and -? Why is there a preset limit for saying it is + or -? I don't see how that comes from EPR. The + and - are simply limits imposed by the experimental apparatus and I can't see how they are in themselves elements of reality.
My current understanding of the LHV concept is that whether an experiment registers + or - will be dependent on a LHV λ, and as Bell says that can be one or any number of locally hidden values. For a photon polarisation experiment is it not reasonable to assume that part of λ is in the detectors and not just in the photon? Or in other words, we can't reduce it to an exact either/or situation, we can only get a probability for + and a probability for - for any given LHV that is defined only for the photon.

 

[DevilsAvocado]

You actually hit the nail on the head with that statement about the measurement. In all the examples the possible outcomes of a measurement are taken to be the 'elements of reality', this is the same assumption I think Bell makes?

In the EPR paper it says: "If, without in any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity." To me that doesn't imply that we can actually measure that physical quantity or that we should be able to predict the result of a measurement of that quantity with equal certainty.

Now you’re thinking about Counterfactual Definiteness (CFD), right? It’s quite interesting how hard it is to pin down the most fundamental property of human life – reality!

Einstein and Bohr was very gifted and bright, still they argued over these matters for nearly 30 years without consensus, with famous arguments like:

Einstein: Do you really think the moon isn't there if you aren't looking at it?
Bohr: How do you know? How can you prove the moon exists, if you don't observe it?

My guess is that it would be a mistake trying to repeat this debate once again...

Einstein certainly had his flaws, but so did Bohr, and the true champ in this debate is Bell who settled the matter with an experiment. One can have any ‘philosophical’ objections one like, but if you question rigorous experiments then you’re on thin ice...

From Nugatory's posts I see that EPR refers to correlations between continuous spectrum eigenvalues, i.e. x and p - is that right?

It looks like it’s possible, though it must be harder to execute in a real experiment.

However, Bell refers to measurements of those values that result in Boolean results. Now I guess this is because spin and polarisation are considered discrete spectrum eigenvalues? However, since we can only ever measure spin and polarisation by apparatus that gives us a Boolean result, I don't see how Bell and EPR are actually talking about the same thing.

You mean that if EPR used ‘continuous values’ for defining local realism, Bell will be faulty testing this assumption with Boolean ‘discrete values’, right?

This is where you lost me because there’s no explanation how/why continuous values ever could violate Bell inequalities with local realism preserved. In fact, I think you would run into even greater difficulties...

AND EPR-Bell experiments is never performed with only one pair, but the correlations are measured over an ensemble of photons and using the CHSH inequality the upper bound for QM is 2.828427...

A very ‘continuous value’ indeed!

But space is free, if you could construct an EPR toy model with ‘continuous values’ that violates Bell inequalities, my guess is the whole physics community would read the paper (if proven correct).

If we could measure the actual spin/polarisation is it possible that we might find that there was an 'element of reality' that was a continuous spectrum eigenvalue? And therefore I don't see how Bell rules this type of LHV out.

I do not get this. The photon spin is continuous and when we measure it thru a polarizer we get 0/1 or thru/stopped. Exactly the same phenomena happen in your classical polarized sunglasses. There is no “QM cheating” here... you can’t be saying that polarization is not real because is not continuous??

Next we have the concept of 'perfect correlation' used in yours and Nick Herbert's examples. Now I understand that in a ideal system 'perfect correlation' would exist and that it could be shown to exist in terms of conservation of momentum/energy and such but I don't get how this concept can be extrapolated to apply to the later local interaction of a particle with a local measurement device.

Maybe it’s my ignorance, but I have never heard that conservation of energy has anything to do with Bell’s inequalities... in SPDC yes, but in Bell??

Lastly, on to the logic in DrC's and Ilja's examples and looking at the Scholarpedia article (these are all ones that talk about triple values).

I’ve seen many go nuts over “the triple values”, thinking “Aha! A mathematical trick!”. It’s not; it’s the simplest way demonstrating Bell’s inequalities. You can’t do it with only two values – then we’re back to “tossing coins/gloves in a box” -type of correlations, which could take another 30 years to sort out. Of course you can try 4, 5, 6, 7... values but that doesn’t change anything.

I hope I don't raise anyones ire with my language here,

Absolutely not (and you can ignore my silly jokes = genetic disease ;).

 

[Badvok]

OK, I still seem to be having a problem expressing the points that I'm stuck on, here's another attempt. (Please note that I have no doubts whatsoever that QM predictions tally with experimental results - that sort of thing is unquestionable for someone at my level.)

EPR talks about predicting values for a particle based on somehow knowing the same value on its paired/entangled particle. However, Bell's theorem talks about predicting measurements of a value for a particle based on the result of a measurement of the same value on its paired/entangled particle.

It is this leap from talking about theoretical real values to just the results of measurements of those values, which is probably intuitive for you guys, which I have difficulty understanding.

Likewise, with the triple value examples, it is not the examples themselves that I have an issue with it is how they could possibly relate to reality. DrC constantly suggesting I do some exercises on paper to show how it works is totally disingenuous, I can see how the examples work it is simply the assumptions the examples make that I don't understand. The examples all assume that the three options are all equally likely to be +/-, 1/0, red/black, i.e. the three selections are in no way related to each other. However, in reality (CFD?) this is not the case is it? If one is red then it is very much more likely that the other two are opposites than the same, so if I randomly pick two of the three my probability of getting red+red is about 1/4 not 1/3.