Spin difference between entangled and non-entangled

[Alien8]

No. This and similar experiments are done with individual photons, while Malus's law describes the intensity of the classical electromagnetic waves, when there are a very large number of photons.

Are you saying Malus' law starts to deviate from cos^2 with lower light intensities?

 

[Nugatory]

Are you saying Malus' law starts to deviate from cos^2 with lower light intensities?

No. I am saying, as others have said about 48,786 times already in this thread, that photons are not classical electromagnetic waves so there is no particular reason to expect them to act according to an empirical observation about classical electromagnetic waves.

Light obeys Malus's law, photons don't, and there is no contradiction here.

 

[Alien8]

Light obeys Malus's law, photons don't, and there is no contradiction here.

Photons are quanta of light like H2O molecules are quanta of water. Light ought to do what photons do just like water does what H2O molecules do. It's the same phenomena, one macroscopic and the other microscopic description of the same thing.

Malus' law works out light intensity percentage. Intensity is energy per unit time per unit area, so given the same energy (frequency) of each photon QM translates this directly to amount of photons. I don't know what is intensity supposed to be in terms of classical physics, but "number of EM waves" sounds kind of awkward. It seems to me it's fair to say Malus' law, and "light intensity" in general, is much closer to QM than is to classical physics.

 

[Nugatory]

Photons are quanta of light like H2O molecules are quanta of water. Light ought to do what photons do just like water does what H2O molecules do. It's the same phenomena, one macroscopic and the other microscopic description of the same thing.

You'll see that analogy occasionally in the pop-sci press, but as with all such analogies it is terribly misleading if you take it too literally. You will have to put it out of your mind and replace it with something better before you will be able to understand QM as deeply as you clearly want to.

Give Feynmann's "QED: The strange theory of light and matter" a try; it also uses analogies, but at least they're good ones.

 

[Nugatory]

I don't know what is intensity supposed to be in terms of classical physics...

It's the amplitude of the wave, which is to say the magnitude of the electrical and magnetic fields at their peaks. If the electrical field of a given wave at the point $x$ and time $t$ is $sin(kt-vx)$, then $2sin(kx-vt)$ is a wave with the same frequency and wavelength but greater intensity.

It would be easy to think that "there are more photons in the second wave", but that would be a mistake. The second wave is more likely to deliver more energy when it interacts with matter (no surprise, as the fields involved are stronger). This energy will appear in fixed size amounts at single points within the area exposed to the radiation, and when this happens we say "a photon hit there". We would be better off saying and thinking "a photon appeared there".

 

[miosim]

No. This and similar experiments are done with individual photons, while Malus's law describes the intensity of the classical electromagnetic waves, when there are a very large number of photons.

It seems that the large number of photons should be identical to result of the large statistical sample of individual photons.
The article in wiki (see below) talks about Malus' Law and Bell theorem, but I am not sure if this article conforms or rejects this link.

http://en.wikipedia.org/wiki/Local_hidden_variable_theory

"Optical models deviating from Malus' Law
If we make realistic (wave-based) assumptions regarding the behavior of light on encountering polarisers and photodetectors, we find that we are not compelled to accept that the probability of detection will reflect Malus' Law exactly."

However this isn't my main concern. My biggest problem with the Bell theorem is that I don't understand the origin of the red line: how the unlimited scenarios of possible complex hidden variables is transformed into two simple strait lines. Unfortunately I wasn't able to find the comprehensive for a layman description of this transformation.
http://upload.wikimedia.org/wikipedia/en/thumb/e/e2/Bell.svg/600px-Bell.svg.png

 

[Nugatory]

My biggest problem with the Bell theorem is that I don't understand the origin of the red line: how the unlimited scenarios of possible complex hidden variables is transformed into two simple straight lines. Unfortunately I wasn't able to find the comprehensive for a layman description of this transformation.

If you haven't already found DrChinese's web page, give it a try: http://www.drchinese.com/Bells_Theorem.htm

 

[DrChinese]

Miosim,

http://www.drchinese.com/David/Bell_Theorem_Negative_Probabilities.htm

A. Figure 3 at the above link is very similar to the graph you presented, but instead for photons. The analogy is as follows: what does QM predict for matches for an angle difference between Alice and Bob's setting (blue line); what realistic prediction comes closest to QM (red line). The Red Line must not lead to logical contradictions such as negative probabilities, probabilities in excess of 100%, violations of CHSH etc.

There is no guarantee that the Red Line is the ONLY possible line to compare to the Blue line. But at least it is "sorta close" to the QM prediction. And you can easily see that it does NOT match the QM expectation except at zero and a few other special cases.

-------

B. In my example at the above link, I choose A=0, B=67.5, C=45 degrees. Keeping A and C constant, and varying B only, gives the following function:

f(B)=(cos^2(B-0)+sin^(45-0)-cos^2(B-45))/2 which corresponds to the sum of the likelihood of 2 particular realistic cases occurring. These 2 cases have NO relevance in QM (they don't exist), and are simply an arbitrary formula otherwise - like the CHSH inequality. But f(B) is negative for any value between 45 degrees and 90 degrees exclusive, which means the QM prediction (if matched by a local realistic theory somehow) would predict a negative likelihood for those cases (which is obviously absurd).

C. On the other hand, if you plugged the red line prediction in instead, you would NOT get negative values. But it would not agree with experiment, because it is different than the QM expectation (because the Blue line and Red line are different).

-------

To summarize: if the Blue line is the LR prediction, you get a variance with experiment. If the Blue line were to be the LR prediction, you get absurd results (negative probabilities). But again, agreeing with what you are saying, the Blue line is only ONE of the "unlimited scenarios of possible complex hidden variables". But every one of those will reduce to either B. or C. eventually, so you always return to the same point.

 

[miosim]

DrChinese,

I read your site and the links you provided, but I am still didn't find the answer I am looking for: correspondence between realistic prediction and the red line. I am looking for an explanation that I can accept but after critical evaluation. However if this explanation requires mathematical background I would accept my limitations and gave up.

Thankscorresponds

 

[stevendaryl]

DrChinese,

I read your site and the links you provided, but I am still didn't find the answer I am looking for: correspondence between realistic prediction and the red line. I am looking for an explanation that I can accept but after critical evaluation. However if this explanation requires mathematical background I would accept my limitations and gave up.

Thanks
corresponds

The red line is not the only possibility for a local realistic theory. It's just that it is the prediction for a very specific locally realistic model that agrees with QM at the points \theta = 0^o, 90^o, 180^o, 270^o, 360^o.

As DrChinese pointed out this particular model is not very plausible, for other reasons.

The model is this:

• When an electron/positron pair is produced, there's an associated spin-vector \vec{S_e} associated with the electron, and the opposite spin-vector \vec{S_p} associated with the positron, where \vec{S_p} = -\vec{S_e}
• \vec{S_e} for each electron is chosen completely randomly; it is just as likely to point in any direction.
  
• When the spin of a particle with spin-vector \vec{S} is measured relative to an axis \vec{X}, the result is +1 if the angle between \vec{S} and \vec{X} is less than 90^o, and -1 if the angle is more than 90^o.

That's all there is to the model. It's a deterministic model, in the sense that the outcome for any measurement is a deterministic function of the spin-vector of the particle being measured. If one experimenter, Bob, measures the spin of his particle in direction \vec{b}, and a second experimenter, Alice, measures the spin of her particle in direction \vec{a}, then:

• If \vec{a} = \vec{b} (so the angle between them is 0^o), then they always get opposite results, because \vec{S_e} is always opposite \vec{S_p}. So the correlation is -1.
  
• If \vec{b} = -\vec{b} (so the angle between them is 180^o), then they always get the same results, because \vec{S_e} is always opposite \vec{S_p}, and so are their measurement orientations. So the correlation is +1.
• Halfway between, when the angle is 90^o or 270^o, there is no correlation at all, so the correlation is 0.

This model fails at other angles, but it works for the 5 easy angles: 0, 90, 180, 270, 360.

 

[miosim]

This model fails at other angles, but it works for the 5 easy angles: 0, 90, 180, 270, 360.

stevendaryl,

Your explanation helped me better understand the corresponding sections of the "BELL’S THEOREM : THE NAIVE VIEW OF AN EXPERIMENTALIST" by Aspect. I still not fully understand the math leading to the "red line" but now I understand a concept.

Just looking in this paper at QM model (6) and hidden variable model (16) one can tell that (6) may be link with Malus law

while (16) couldn't.

Question: Does the hidden variable model (16) contradicts with Malus law.Thanks

 

[Alien8]

The red line is not the only possibility for a local realistic theory. It's just that it is the prediction for a very specific locally realistic model that agrees with QM at the points \theta = 0^o, 90^o, 180^o, 270^o, 360^o.

Those are perfect correlation and perfect anti-correlation angles. How can any local theory predict 100% matching or 100% mismatching pairs?

 

[DrChinese]

Question: Does the hidden variable model (16) contradicts with Malus law.

Yes, (16) contradicts Malus (which applies to a single photon stream rather than entangled pairs) in the sense that the single photon analogy would also be a straight line relationship.

 

[DrChinese]

Those are perfect correlation and perfect anti-correlation angles. How can any local theory predict 100% matching or 100% mismatching pairs?

Sure. All you need is a lot of hidden variables. Something like:

Hidden polarization at 00 degrees= -
Hidden polarization at 01 degrees= +
Hidden polarization at 02 degrees= +
Hidden polarization at 03 degrees= +
...
Hidden polarization at 30 degrees= +
Hidden polarization at 31 degrees= -
Hidden polarization at 32 degrees= +
Hidden polarization at 33 degrees= -
...
Hidden polarization at 60 degrees= -
Hidden polarization at 61 degrees= +
Hidden polarization at 62 degrees= -
Hidden polarization at 63 degrees= -
...
etc.

You would get perfect correlations with the above. There is no requirement that it is some simple formula. It could be a bunch of values that average to some formula.

 

[miosim]

Question: Does the hidden variable model (16) contradicts with Malus law.

DrChinese,
I am retrieving my question because I realized how wrong it is. Indeed, the "the red line" is for spin particles and not for polarized photons.

Reading the wiki link http://en.wikipedia.org/wiki/Local_hidden_variable_theory provided on your page I found the following graph:

Fig. 2: The realist prediction (solid curve) for quantum correlation in an optical Bell test. The quantum-mechanical prediction is the dotted curveIn this article the relation between Malus Law and predictions are mentioned multiple times (see below), but I still can't grasp their exact relationship.

1). In optical experiments using polarisation, for instance, the natural assumption is that it is a cosine-squared function, corresponding to adherence to Malus' Law."

2). If we make realistic (wave-based) assumptions regarding the behaviour of light on encountering polarisers and photodetectors, we find that we are not compelled to accept that the probability of detection will reflect Malus' Law exactly

3). By varying our assumptions, it seems possible that the realist prediction could approach the quantum-mechanical one within the limits of
experimental error (Marshall, 1983), though clearly a compromise must be reached. We have to match both the behaviour of the individual light beam on passage through a polariser and the observed coincidence curves. The former would be expected to follow Malus' Law fairly closely, though experimental evidence here is not so easy to obtain. We are interested in the behaviour of very weak light and the law may be slightly different from that of stronger light.

My question is:
What is exact (theoretical) relation between QM predictions, realistic predictions and Malus law.

Thanks

 

[DrChinese]

DrChinese,
I am retrieving my question because I realized how wrong it is. Indeed, the "the red line" is for spin particles and not for polarized photons.

Reading the wiki link http://en.wikipedia.org/wiki/Local_hidden_variable_theory provided on your page I found the following graph:

Fig. 2: The realist prediction (solid curve) for quantum correlation in an optical Bell test. The quantum-mechanical prediction is the dotted curveIn this article the relation between Malus Law and predictions are mentioned multiple times (see below), but I still can't grasp their exact relationship.

...

My question is:
What is exact (theoretical) relation between QM predictions, realistic predictions and Malus law.

Thanks

What would happen if you measured pairs of photons that were NOT entangled but had the same, random polarizations? You would get the graph above (solid line) which is also a "realistic" scenario/hypothesis. That differs noticeably from the results you get from entangled pairs (dashed line). So you can reject the hypothesis.

 

[Alien8]

DrChinese,

Fig. 2: The realist prediction (solid curve) for quantum correlation in an optical Bell test. The quantum-mechanical prediction is the dotted curve

In this article the relation between Malus Law and predictions are mentioned multiple times (see below), but I still can't grasp their exact relationship.

My question is:
What is exact (theoretical) relation between QM predictions, realistic predictions and Malus law.

Solid curve is the prediction based on Malus' law. It's 1/2 from QM prediction. To get that I think you work out independent probabilities where Pa(+) = Pb(+) and Pab(+ and +) = Pa(+)Pb(+).

3). By varying our assumptions, it seems possible that the realist prediction could approach the quantum-mechanical one within the limits of
experimental error (Marshall, 1983), though clearly a compromise must be reached. We have to match both the behaviour of the individual light beam on passage through a polariser and the observed coincidence curves. The former would be expected to follow Malus' Law fairly closely, though experimental evidence here is not so easy to obtain. We are interested in the behaviour of very weak light and the law may be slightly different from that of stronger light.

I think to check Malus' law all we need is to look at Alice and Bob's readings individually. Total number of "+" should be about the same as "-" no matter what angle settings is at either Alice or Bob's analyzer. I don't see why would that not be easy to verify, but I also don't see that would change anything because it says nothing about how the two readings are supposed to match against each other. In local theory there is no any connection between the two events so the rest is in the hands of probability theory, and because the two events are supposed to be independent we use the equation for independent probabilities: P(A and B) = P(A)P(B), which leads to that solid cure. Or something like that, I'm not quite sure how to work out the integral.

 

[miosim]

In local theory there is no any connection between the two events so the rest is in the hands of probability theory, and because the two events are supposed to be independent we use the equation for independent probabilities: P(A and B) = P(A)P(B), which leads to that solid cure.

It starts making sense to me. I need to study this statistical approach in more details. I expect that this statistical approach doesn't treat a pair of particle as classical objects. These particles still should exhibit "weird" quantum behavior cased by some "weird" hidden variables.

I should take a brake until I better understand the statistical approach.

Thank you for the help

 

[DrChinese]

I think to check Malus' law all we need is to look at Alice and Bob's readings individually. Total number of "+" should be about the same as "-" no matter what angle settings is at either Alice or Bob's analyzer. I don't see why would that not be easy to verify, but I also don't see that would change anything because it says nothing about how the two readings are supposed to match against each other. In local theory there is no any connection between the two events so the rest is in the hands of probability theory, and because the two events are supposed to be independent we use the equation for independent probabilities: P(A and B) = P(A)P(B), ...

You are correct. The sentence you highlighted in the wiki article and the sentence that follows, these should not appear in the article and reflect a bias by the writer (who is a local realist). I can tell from the reference to Marshall (1983). Local realists love to add stuff into the wiki pages and it is a lot of work to keep it out. I occasionally police the Bell page for that. :)

 

[moriheru]

$\psi(x) = \sum\limits_{n} \phi_{n}(x)$.

Not very helpfull but anyway...one usually replaces the summs with integrals or sums stepse of lim a-->0

 

[stevendaryl]

Solid curve is the prediction based on Malus' law. It's 1/2 from QM prediction. To get that I think you work out independent probabilities where Pa(+) = Pb(+) and Pab(+ and +) = Pa(+)Pb(+).
I think to check Malus' law all we need is to look at Alice and Bob's readings individually. Total number of "+" should be about the same as "-" no matter what angle settings is at either Alice or Bob's analyzer. I don't see why would that not be easy to verify, but I also don't see that would change anything because it says nothing about how the two readings are supposed to match against each other. In local theory there is no any connection between the two events so the rest is in the hands of probability theory, and because the two events are supposed to be independent we use the equation for independent probabilities: P(A and B) = P(A)P(B), which leads to that solid cure. Or something like that, I'm not quite sure how to work out the integral.

No, a local theory doesn't imply independence of the results, and it does not imply P(A and B) = P(A)P(B). The reason why not is that even though A can't influence B, and B can't influence A, there might be a third cause that influences both. That's what the "local hidden variables" idea is all about: whether the correlations can be explained by assuming that there is a cause (the hidden variable) that influences both measurements.

A locally realistic model based on Malus' law is this: assume that in the twin-photon version of EPR, two photons are created with the same random polarization angle \phi. If Alice's filter is at angle \alpha then she detects a photon with probability cos^2(\alpha - \phi). Similarly, if Bob's filter is at angle \beta, then he detects a photon with probability cos^2(\alpha - \phi). The correlation E(\alpha, \beta) would then be:

E(\alpha, \beta) = P_{++} + P_{--} - P_{+-} - P_{-+}

where P_{++} is the probability both Alice and Bob detect a photon, P_{+-} is the probability Alice detects one and Bob doesn't, etc.

For this model,
P_{++} = \frac{1}{\pi} \int d\phi cos^2(\alpha - \phi) cos^2(\beta - \phi) = \frac{1}{8} + \frac{1}{4}cos^2(\alpha - \beta)
P_{+-} = \frac{1}{\pi} \int d\phi cos^2(\alpha - \phi) sin^2(\beta - \phi) = \frac{1}{8} + \frac{1}{4}sin^2(\alpha - \beta)
P_{-+} = \frac{1}{\pi} \int d\phi sin^2(\alpha - \phi) cos^2(\beta - \phi) = \frac{1}{8} + \frac{1}{4}sin^2(\alpha - \beta)
P_{--} = \frac{1}{\pi} \int d\phi sin^2(\alpha - \phi) sin^2(\beta - \phi) = \frac{1}{8} + \frac{1}{4}cos^2(\alpha - \beta)

So
E(\alpha, \beta) = \frac{1}{2}(cos^2(\alpha - \beta) - sin^2(\alpha - \beta)) = \frac{1}{2} cos(2 (\alpha - \beta))

That's exactly 1/2 of the QM prediction.

 

[Alien8]

Thanks for that. I'll take your answer to the other thread about CHSH derivation which is about photons and polarizers rather than magnetic moments and magnets this thread is about.

 

[miosim]

In local theory there is no any connection between the two events so the rest is in the hands of probability theory, and because the two events are supposed to be independent we use the equation for independent probabilities: P(A and B) = P(A)P(B), which leads to that solid cure.

As I understand, Einstein accepts the the prediction of quantum mechanics; he just disagree about causes.
So why the predicted probabilities in the Bell's theorem for the local realism and for QM are different? It seams that per Einstein they should be the same.

 

[stevendaryl]

Thanks for that. I'll take your answer to the other thread about CHSH derivation which is about photons and polarizers rather than magnetic moments and magnets this thread is about.

Some of the graphs posted are about the spin-1/2 experiment, and some are about the photon case. The graph in your message #77 is about photons, not electrons. The graph in #66 is about electrons.

The arguments are basically the same in either case, but the details are different, such as the fact that perfect anti-correlation is at 90 degrees for the photon case, but at 180 degrees for the electron case.

 

[miosim]

A locally realistic model based on Malus' law

If a locally realistic model is based on Malus' law, does it means that QM model is in conflict with the Malus' law because it has a different than realistic model predicted probability?

 

[stevendaryl]

If a locally realistic model is based on Malus' law, does it means that QM model is in conflict with the Malus' law because it has a different than realistic model predicted probability?

No, not really. Malus' law is about how the intensity of polarized light is attenuated by a polarizing filter. It doesn't say anything about photons. QM agrees with Malus' law in those cases where the number of photons is very large.

Malus' law says nothing about individual photons, or the probability that two different photons pass through two different filters. What I was saying is that you could come up with a law for photons inspired by Malus' law that would apply probabilistically to individual photons, but such a law doesn't agree with experiment.

Since Malus' law doesn't say anything about individual photons, no experiment involving individual photons can really contradict Malus' law.

 

[miosim]

Malus' law says nothing about individual photons, or the probability that two different photons pass through two different filters.

Is the Malus' law about intensity of polarized light attenuated by a polarizing filter could be derived from the QM of individual photons?

 

[stevendaryl]

Is the Malus' law about intensity of polarized light attenuated by a polarizing filter could be derived from the QM of individual photons?

Yes. QM predicts that if a photon passes through one filter, then it will pass through a second filter with probability cos^2(\theta), where \theta is the angle between the two filters. So if intensity is proportional to the number of photons that pass through, then this makes the same prediction as Malus' law.

 

[miosim]

Yes. QM predicts that if a photon passes through one filter, then it will pass through a second filter with probability cos 2 (θ) cos^2(\theta), where θ \theta is the angle between the two filters. So if intensity is proportional to the number of photons that pass through, then this makes the same prediction as Malus' law.

A locally realistic model based on Malus' law is this: assume that in the twin-photon version of EPR, two photons are created with the same random polarization angle ϕ \phi ...

Would be fair to say that the differences between realistic and QM models in the Bell's theorem is that realistic model is equivalent to the photon that already interacted with a polarizer while for the QM model photon didn't have any interaction yet?

 

[DrChinese]

So why the predicted probabilities in the Bell's theorem for the local realism and for QM are different? It seams that per Einstein they should be the same.

Einstein died before Bell published. He never knew, and would have been forced to re-assess his position had he known.