Bell's Theorem with Easy Math - Stuck

[Badvok]

Entangled photons are not polarized before measurement.
If you mean “result X” = prefect correlations at aligned (the same) angles, then yes.
And remember, this “result X” can look very different over a collection of say 10 pairs (A & B):

[B]Measurement 1[/B]
[B]A[/B] = 10101 01010
[B]B[/B] = 10101 01010

[B]Measurement 2[/B]
[B]A[/B] = 11001 10011
[B]B[/B] = 11001 10011
 
[B]Measurement 3[/B]
[B]A [/B]= 01000 10111
[B]B[/B] = 01000 10111

ect...

All these 3 measurements show prefect correlations (for Bell state Type I).

Entangled photons aren't polarised before measurement, but 'locally real' photons would be, wouldn't they? Therefore how can an assumption of perfect correlation be used in the maths for the 'local realism' side of the proof?

 

[Badvok]

So the best thing is to focus on those cases where there is a difference. 1) Try putting down a set of values for 3 angles. 2) Then go down the list and randomly select 2 of the 3 (imagine this is done without knowledge of the results of 1). The match percentage, for a sufficiently large set, cannot be less than 33% (1/3).

But doesn't the 33% only come about if there is an equal probability of each value being a + or a -, which is based on the assumption that when you test the polarisation of a photon you will always get the same result?

 

[stevendaryl]

Many thanks for answering, I think I get that, the local realism that EPR assumes is incorrect because it assumes that A, B and C have definite values.

I guess my thinking was more that local realism meant that any individual photon would have a 'reality' that is actually its polarization angle, and so couldn't have a definite value for being detected by an analyzer set at any given angle. I'm left with being unsure why EPR makes the assumption it does.

There is a good reason for assuming that A, B, and C have definite values. An irrefutable argument. Too bad the conclusion is false.

If you create a correlated pair of photons and you have two detectors, then what you find is that if detector #1 detects a photon at angle A, then detector #2 will DEFINITELY detect a photon at angle A. (Ignoring detector inefficiency, I guess). Einstein reasoned that what happens at detector #1 can't affect what happens at detector #2 (if they are far enough apart). So if the probability of detector #2 finding a photon at angle A was 100% afterward, it must have been 100% beforehand, as well.

 

[DevilsAvocado]

Entangled photons aren't polarised before measurement, but 'locally real' photons would be, wouldn't they?

Well, not necessary. Bell’s theorem doesn’t stipulate how Local Realism should be ‘constructed’ (if I mentioned “pre-polarization” it was just because this is the simplest/most natural assumption, but it could be anything, like a local pre-existing ‘function’ that would try to violate Bell’s inequality). The thing is – it doesn’t matter what you do – it can not work.

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

J.S. Bell's Concept of Local Causality

Therefore how can an assumption of perfect correlation be used in the maths for the 'local realism' side of the proof?

I don’t see the problem... perfect correlations was used in 1935 EPR paper, Bell went beyond that:

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

 

[DrChinese]

But doesn't the 33% only come about if there is an equal probability of each value being a + or a -, which is based on the assumption that when you test the polarisation of a photon you will always get the same result?

The 33% minimum is regardless of what value sets you pick, as long as you don't know which 2 of the 3 angles are going to be selected for observation. So there does NOT need to be an equal distribution, but obviously that would not match observation which does deliver a very nearly equal distribution. But it is not a requirement for this particular test.

Remember that the local realistic photons have values for all 3 angles even if only 2 are measured (I believe you already said this above). So they are non-contextual. While the QM photons only have values when measured (so here the context is a part of the equation from the beginning).

 

[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...?? ()

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.