Bell Theorem and probabilty theory

[mn4j]

Second point is that you also say that de Raedt model is not BLR because it violates Bell's inequalities. Here I disagree strongly, and I think that this shows that you don't really understand what a "coincidence loophole" is.

I think maybe you are the one who is not quite sure what it means. Do you deny the fact that de Raedt's model reproduces the QM result? Doesn't that mean it also violates the inequality. Please answer this question.

Here's again, how I see it: 1. de Raedt's simulation does not deviate from real experiment of Weihs et al (you asked me, how it deviated; well, it doesn't).

Does the real experiment deviate from Bell's inequality? Does the real experiment agree with QM? Does QM violate also violate the coincidence time loophole? (see http://arxiv.org/abs/0801.1776)

2. The coincidence loophole (which this experiment did not avoid) means that it's possible to explain the apparent Bell's inequality violation by the fact that events are post-selected, and because of this post-selection the correlation is created out of nothing.

Again you focus only on the "dead men bleed" part and completely ignore the fact that the coincidence time loophole can also mean Bell's inequality does not model the behaviour of all real local systems.

3. This is exactly what de Raedt is exploiting. 4. Bottomline of this analysis: de Raedt's model is BLR, it obeys Bell's inequalities but in the non-perfect loopholed experiment it can LOOK like it violates them. This is the view expressed here: http://arxiv.org/abs/quant-ph/0703120 (see also http://arxiv.org/abs/quant-ph/0312035 about coincidence loophole). Do you understand this argumentation? You may disagree (please tell where exactly), but do you understand it?

http://arxiv.org/abs/quant-ph/0703120 has been refuted by de Raedt (see http://arxiv.org/abs/0706.2957).
The bottom line is this: de Raedt's model satisfies the Einstein's conditions of local causality and exactly reproduce the single particle and two-particle expectation values of the singlet state.

Let me also ask for a clarification of your point of view. Do you think that Bell's inequalities are in reality NOT violated (and all experimental violations are only due to loopholes)? Or do you think that his inequalities in reality ARE violated (so that even ideal perfect experiment will find violations), but these violations can be explained by some LR theory which is not accounted by Bell's theorem?

My viewpoint is that it is possible to find a model that satisfies the Einstein's conditions of local causality and exactly reproduce the expectation values of the singlet state, contrary to Bell's claims. My viewpoint is that constructing Bell's inequalities in a manner which accounts for all possible real experiments like the ones performed so far will result in inequalities that are never violated. My viewpoint is that no experiment has ever been performed exactly as Bell modeled in his equations. Therefore Bell's theorem is currently an untested theorem, and when such such an experiment is performed, it will not violated the inequalities.

Which means that even a perfect loophole-free experiment will not prove anything to you, right?

Show me a loophole free experiment which violates Bell's inequalities and I will concede. My view is there will never be a loophole free experiment because the problem is not with the experiments but with the inequality. If a theory is so restrictive in scope that it has taken many talented experimentalists several decades to test it in vain, then maybe the answer is not that "dead men bleed afterall" or rather, that "we need more perfect experiments". The answer is that "the bleeding man is alive" or rather, that Bell's inequalities do not accurately represent real experiments that can be performed.

I don't proclaim that his model is wrong on this basis, I'm proposing a bet (let's put it that way). Imagine this experiment is done exactly as de Raedt himself proposed it (screen is jittered from left to right parallel to itself). Question: what will happen? My bet: interference pattern doesn't change. "de Raedt's" bet: interference pattern gets smeared, because "detectors" on the screen won't have enough time to "learn". Your bet?

Isn't it common sense that a moving camera takes smeared images? In case you did not know, this experiment has been performed many times over by lay people and you lost the bet already.

 

[kobak]

Well, I fear that I don't see how this discussion can lead to anything more. From what you've just said, I'm now sure that you don't understand what a coincidence time loophole is. This is crucial, so there's unfortunately no point in this discussion anymore.

I will briefly react to some points you made.

And here is my main point: The above statement is False for the following reasons:
1) Bell's particular *physical* intuition does not account for the most interesting class of local hidden variables <...>
2) The technical assumptions he uses are not always true, for reasons I have explained here. The effect is that this introduces further hidden assumptions -- at the very least, the assumption that those technical assumptions are always true for local hidden variables <...>
3) Those technical assumptions are not uncontroversial in all areas of classical physics. In fact violation of Bell's inequalities is not limited to quantum systems. Take a look at de Raedt's recent paper for an example in which there is violation of Bell's inequality for a voting game with three human players.

My statement was this: "in classical physics Bell's technical assumption was always uncontroversial and generally considered true". You're saying that this is false and give as an example some theories of hidden variables. I'm sorry, I was talking about CLASSICAL physics. So I disregard your points (1) and (2) here. The only meaningful response is your point (3). Well, I guess I should take a look on this de Raedt's example. But what I did already see, is the "Bernoulli urn" example from Jaynes, also repeated by de Raedt, -- and this example totally and completely misses the point. So I'm quite fed up by his examples.

This is false. I have already explained in this thread that any two time varying harmonic systems are correlated and as such their probabilities are not disjoint. Unless you want to claim that two pendulums or clocks on opposite sites of the globe are not classical.

I have seen your example with pendula. I don't see how this relates to the situation when two space-like separated measurements are performed and one result is statistically dependent on the choice of other experiment. I'm talking about this very situation (as described by EPR and Bell), not about some abstract correlation between something.

I think maybe you are the one who is not quite sure what it means. Do you deny the fact that de Raedt's model reproduces the QM result? Doesn't that mean it also violates the inequality. Please answer this question.

This is exactly the point where it becomes clear that you don't properly understand this loophole issue. de Raedt's model reproduces the QM result for THIS PARTICULAR experiment (Weihs et al.) which has loopholes. Because of this experimental flaws, it's possible for a BLR theory to give the impression that it violates Bell's inequalities. In the loophole-free experiment de Raedt's model will obey Bell's inequalities, unlike QM.

I really can't express this clearer (without giving technical arguments, that are anyway contained in the articles I cited).

http://arxiv.org/abs/quant-ph/0703120 has been refuted by de Raedt (see http://arxiv.org/abs/0706.2957).

Sure, I've seen this. I think his reply misses the point completely. Let me ask you something: did you actually read this critique of de Raedt and his reply? Or are you saying that it has been successfully refuted just because there exists an article claiming to be a refutation? I'm just curious.

Isn't it common sense that a moving camera takes smeared images? In case you did not know, this experiment has been performed many times over by lay people and you lost the bet already.

It's amazing how you keep avoiding answering my simple question for several postings already. I asked you what do you think about the outcome of double-slit interference when the screen is moved from left to right, and you're telling me something about cameras.

What on Earth has it to do with my question? If I project a movie on a white (ideal) screen with a projector and then start moving the screen from left to right with any frequency I want, the picture won't change, and all the viewers can still enjoy the movie. Now we're talking about double-slit interferometer instead of beamer. Screen is moved. Question: will the interference picture get smeared? If you say "yes" (like de Raedt does), here's the second question: assume, just assume for the sake of argument, that it's found NOT to change, exactly like the movie in the example above. What would be your conclusion? I remind here, that de Raedt said that in this case he would "retire".

I'm wondering whether you are again going to skip answering these two direct questions.

 

[mn4j]

My statement was this: "in classical physics Bell's technical assumption was always uncontroversial and generally considered true". You're saying that this is false and give as an example some theories of hidden variables. I'm sorry, I was talking about CLASSICAL physics. So I disregard your points (1) and (2) here. The only meaningful response is your point (3). Well, I guess I should take a look on this de Raedt's example. But what I did already see, is the "Bernoulli urn" example from Jaynes, also repeated by de Raedt, -- and this example totally and completely misses the point. So I'm quite fed up by his examples.

You did not define what you mean by classical physics.

I have seen your example with pendula. I don't see how this relates to the situation when two space-like separated measurements are performed and one result is statistically dependent on the choice of other experiment. I'm talking about this very situation (as described by EPR and Bell), not about some abstract correlation between something.

Is it unclassical for photons and electrons which make up all experimental apparatus to exhibit time-varying harmonic oscillation? I take it you do not understand the difference between logical dependence and physical causation. You see, when you calculate probabilities of systems which are known to be correlated, like harmonic systems, you MUST consider them to be logically dependent even if there is no physical effect transferred between them, otherwise you get paradoxical results.

This is exactly the point where it becomes clear that you don't properly understand this loophole issue. de Raedt's model reproduces the QM result for THIS PARTICULAR experiment (Weihs et al.) which has loopholes. Because of this experimental flaws, it's possible for a BLR theory to give the impression that it violates Bell's inequalities. In the loophole-free experiment de Raedt's model will obey Bell's inequalities, unlike QM.

You seem to have stuck on this one loophole and you can't get passed it. Your statement is wrong. For all your claims about talking to de Raedt, you have no idea what his model is all about evidently. In case you did not know, de Raedt's model agrees with QM in single-photon beam-splitter and Mach-Zehnder interferometer experiments, wheeler’s delayed choice experiment, quantum eraser, EPRB experiments with photons, EPRB experiments with non-orthogonal detection planes etc.

You did not tell me if you believe the QM formalism also suffers from the coincidence time loop-hole. I wonder why?

Sure, I've seen this. I think his reply misses the point completely. Let me ask you something: did you actually read this critique of de Raedt and his reply? Or are you saying that it has been successfully refuted just because there exists an article claiming to be a refutation? I'm just curious.

I have read every article I point you to. Have you read the ones you point me to? Did you bother reading de Raedt's reply? What exactly do you believe misses the point. Spell it out and I will explain why it is you who missed the point. We can go into technical detail if you prefer.

It's amazing how you keep avoiding answering my simple question for several postings already. I asked you what do you think about the outcome of double-slit interference when the screen is moved from left to right, and you're telling me something about cameras.

The answer is so blindingly obvious I did not expect that you were serious. De Raedt gave you the answer even. The pattern gets smeared! It is not an experiment you need to bother doing. It has been done many times! Do I really need to answer the second question? If you think I'm wrong, go do the experiment and bring your results, then we can talk.

What on Earth has it to do with my question? If I project a movie on a white (ideal) screen with a projector and then start moving the screen from left to right with any frequency I want, the picture won't change

That is a very naive look at the situation. You can't be serious? A movie screen is not a detector. A photographic film is. Try projecting a still image on a photographic film while moving the film from left to right and then develop the film and see if it is not smeared. Do you seriously believe the same will not happen if the detector in front of a double-slit is jiggled? Isn't the interference image whatever is recorded on the detector. Or maybe let's jiggle your head real fast, while you watch, since your eyes are the real detector in this case. Are you telling me you will enjoy the same quality of movie as an unjiggled pair of eyes? You can perform this experiment right now. Jiggle your head while you read this page and tell me whether the image does not get smeared.

I'm wondering whether you are again going to skip answering these two direct questions.

I can point to umpteen questions of mine you have not answered but then again I'm not keeping score.

Are you ready to concede that classical systems DO learn just like in de Raedt's model? Are you ready to concede that Bell's model does not account for systems which learn, like de Raedt's?

 

[kobak]

You did not define what you mean by classical physics.

That's easy: let's just say, that by classical physics in this case I mean all physics known before 1920. To make this comment self-contained, I repeat my claim: in classical physics Bell's technical assumption (that you and Jaynes and de Raedt keep saying is unjustified) is always true. You disagree. Well, it seems that to refute my claim one example would suffice. This example must be of the following (just to spell it out): two space-like separated experiments are done and the result of Alice's experiment "A" is NOT statistically independent from Bob's choice of experimental setup "b" or his outcome "B".

Maybe I miss something here, but I don't see how your harmonic oscillators can provide a counterexample to that. Please explain, if you think it can. And here's another consideration: if it were THAT easy to provide a counterexample to Bell (by just taking two harmonic oscillators), then why would all this story about deterministic learning machines be necessary? It seems to me that you make an oversimplification here. It's impossible to replicate QM correlations by just considering two oscillators or clocks, is it?

I take it you do not understand the difference between logical dependence and physical causation.

Just a short note: I think I do. It also seems to me that "logical dependence" is a term that you took from Jaynes (and de Raedt), because it's never being used in modern probability/statistics treatments. The precisely defined term that is usually used is "statistical dependence". Try googling this term, and your one. "Logical dependence" is used only in mathematical logic, but that's different.

You seem to have stuck on this one loophole and you can't get passed it.

Well, I'm sorry, but I believe that this is the most important and crucial point here.

You did not tell me if you believe the QM formalism also suffers from the coincidence time loop-hole. I wonder why?

Because I don't understand what your question means! How can a "formalism" suffer from a "loophole"? I think that this question gives another indication that you fail to understand the meaning of this loophole issue.

Let me give an analogy (it will be a rather silly one). Imagine we're trying to prove general relativity by observing gravitational waves. The opposing theory is Newtonian gravity, where there's no waves. So it the waves are observed, Newtonian theory is proved to be false. The experiment is done and experimental setup for some reason has to be located on the surface of the ocean. The gravitational waves are indeed observed. It may seem that GR is proven, but there's a subtlety: the whole apparatus was located in water and there were waves in water, so in principle it's possible to explain the apparent gravitational waves that were observed by just some influence from the water waves. So folks that don't believe in GR take this view. This experimental problem is called "water loophole" and experimenters are working hard to avoid it putting the setup on a hard ground, but they didn't succeed so far.

To spell it out: GR corresponds to QM, gravitational waves to the violation of Bell's inequalities, water loophole to coincidence-time loophole, explanation of how this faulty experiment can result in apparent detection of gravitational waves to de Raedt's explanation of how the faulty experiments can result in apparent violation of Bell's inequalities.

And now you come and ask, whether "the QM formalism also suffers from the coincidence time loop-hole". Analogy: does the general relativity suffer from the water loophole? This question just doesn't make any sense to me.

I have read every article I point you to. Have you read the ones you point me to? Did you bother reading de Raedt's reply? What exactly do you believe misses the point. Spell it out and I will explain why it is you who missed the point. We can go into technical detail if you prefer.

OK, I'm sorry for suspecting that you didn't read them. Yes, I also did. We could go into technical details, but it doesn't make any sense before we understand the issue of loophole in a similar way. Actually, when I read de Raedt's reply I was just lost several times. I think it misses the point, because it's unclear, and it is unclear because de Raedt apparently also does not understand this loophole issue. It's just that his reply doesn't really reply to what was said by Seevinck and Larsson. The critique is almost one page long and makes a clear point, while de Raedt writes a lengthy reply with all kind of beating around the bush.

Here's one particular example (maybe it's not the most important one. I don't remember any more, but I just have this written down anyway). Seevinck and Larsson say that in de Raedt's model \gamma is less then \gamma_0 which would be necessary to violate the inequality modified to take a loophole into account (and this equality is still violated by QM). de Raedt triumphally replies that they made a mistake in derivation, and then presents his own version of this derivation, which results in \gamma -> 0. Zero is clearly still less than \gamma_0, but he never comments on it.

The answer is so blindingly obvious I did not expect that you were serious. De Raedt gave you the answer even. The pattern gets smeared! <...> That is a very naive look at the situation. You can't be serious?

Oh, now I see that that's just a misunderstanding. Probably I didn't explain my experiment well enough, and now I checked and found that it's not described in arXiv version of the paper (http://arxiv.org/abs/0809.0616). It's just that when I asked de Raedt about this paper, he sent me a draft with a fuller version, and the experiment is described there. We're just talking about different experiments. You're right about yours, that's clear (and that "experiment" really is stupid and obvious, I agree).

Roughly speaking, what I meant is that screen is jittered, detection events are counted in individual detectors and then relocated with a computer program, that takes into account where the screen was at the moment of detection. Does it make more sense now? The crucial point is that de Raedt's detectors have first to learn before they start reproducing QM probabilities, and by jittering we prevent them from learning correctly (I quote de Raedt: "In other words, the experiment should address a time scale that is sufficiently short such that our detector models have not yet reached the stationary state").

He actually proposes a bit different setup, where only *single* detector is used, and it's slided back and forth along the "screen" line. The detection events are counted and then plotted versus the position of the detector at the moment of detection. If this detector is moved slowly (so that it reaches stationary state at every point) then the resulting plot will show the usual interference pattern. However if it's moved fast, then it would get smeared. Not because of this silly "moving camera" effect, but because detector doesn't have time to learn afresh at every single point.

Please tell me if this experiment now makes sense to you. If it does, then my questions remain the same. Do you think that interference pattern will change? What do you think is the prediction of the usual QM? What would you say if it actually doesn't change?

I can point to umpteen questions of mine you have not answered but then again I'm not keeping score. Are you ready to concede that classical systems DO learn just like in de Raedt's model? Are you ready to concede that Bell's model does not account for systems which learn, like de Raedt's?

First question: well, in the certain sense, yes. If the billiard ball getting kicked by another billiard ball counts for "learning", then yes. It's just that nobody in the times of classical physics would think that parts of the screen "learn" the phase of the incoming photons.

Second question: no, not yet. As I already said, here I should take a better look on the Popescu article. I didn't have time so far.

 

[mn4j]

Hi Kobak,
Sorry for the late response. I will try to be brief because every response just seems to be getting longer and longer. So I will not attempt to respond to every teeny-weenie point.

That's easy: let's just say, that by classical physics in this case I mean all physics known before 1920. To make this comment self-contained, I repeat my claim: in classical physics Bell's technical assumption (that you and Jaynes and de Raedt keep saying is unjustified) is always true. You disagree. Well, it seems that to refute my claim one example would suffice. This example must be of the following (just to spell it out): two space-like separated experiments are done and the result of Alice's experiment "A" is NOT statistically independent from Bob's choice of experimental setup "b" or his outcome "B".

I have given one example already in this thread. Bob and Alice each have a pendulum, and they are free to adjust the length of the string as they like. Coincidences are said to occur if both Bob and Alice's pendulum have swung to the same angular position at the same time. Bob's selection of the length is not ontologically dependent on Alice's length selection and they are free to select any length they want. In fact they are not even aware of the existence of each other until the experiment is complete and we are looking at the results. The results were recorded as a time tagged values of the deviation of the pendulum from the target position for a given duration after every change in settings.
1) Do you agree that these two experiments are local realist in the classical sense?
2) Do you agree that there will be a correlation between the results obtained by Bob and Alice?
3) If you agree to (2). Do you agree that this correlation or (statistical dependence as you call it) is not due to spooky-action at a distance or conspiracy?
4) Do you then agree that given a coincidence, if I (the external observer doing the calculations) were to know Bob's string length, the result Bob obtained and the result Alice obtained, I should be able to calculate a probability for Alice's string length which is higher than the maximum entropy value?

In case you don't understand the last point, let me explain: Given a fair coin, the maximum entropy probability for heads or tails is 0.5. This value tells you nothing about the ACTUAL result of the experiment you just performed by throwing a coin. However, if the coin throw was part of a bet in which you chose heads and I see you going of to buy a bear, I will be able to say the probability that the result was heads is higher than 0.5. Even though I did not see the actual result and I have no determinative proof that the coin used in the particular case was fair. As you can see from this example, probability is NOT JUST about frequencies of outcomes but about state of knowledge (Read up on Harold Jeffreys). Frequencies of outcomes is just a subset of the ways of updating the state of knowledge.

So then going back to point (4), you see that it is possible to obtain a probability for Alice's string length that is higher than the maximum entropy value, if I know Bob's string length. This is logical dependence and exists even for situations in which there is no communication or physical influence between Alice and Bob. If this doesn't make sense to you, then you can not even begin to understand my arguments in this thread, or Jayne's for that matter.

Maybe I miss something here, but I don't see how your harmonic oscillators can provide a counterexample to that.Please explain, if you think it can.

Consider the following harmonic oscillator equation.
$$y(t) = A sin(\omega t + \theta)$$
in your favourite plotting package, generate a range of t values. Any range you choose. Using two random number generators, pick two sets of triplets (A, $$\omega$$, $$\theta$$)
use one set to calculate the corresponding $$y(t)_1$$ and the other set to calculate the corresponding $$y(t)_2$$ with the previously generate t values.
Plot $$y(t)_1$$ vs $$y(t)_2$$ and confirm that indeed there is a correlation between the two, no matter what values of (A, $$\omega$$, $$\theta$$) you used for each, however randomly you generated them.

In case you still think I did not understand what the coincidence time loophole is, let me point out to you that in this case, it has to do with which point in $$y(t)_1$$ is plotted against which point in $$y(t)_2$$. I won't go into much further detail than that here other than to say you can even introduce a constant offset (or time delay) between t values used in both cases without eliminating the correlation.

Because I don't understand what your question means! How can a "formalism" suffer from a "loophole"? I think that this question gives another indication that you fail to understand the meaning of this loophole issue.

Bell's inequality is a mathematical formulation also. And it does suffer from many loopholes. Your question tell's me you believe loopholes are problems in the experiments rather than Bell's inequalities. How come then that in every paper trying to address a loophole, they always derive new inequalities which account for the loophole cases? See the original paper about coincidence-time loophole which confirms this. The loophole is in Bell's inequality not the experiment and Quantum mechanics also makes assumptions about coincident events.

 

[kobak]

I will answer a bit later, but a quick question right away: are you going to answer my considerations about double-slit? Sorry for bothering if you were going to anyway. It's just that I don't want this topic to get lost.

And then from the technical side, there's also this \lambda issue.

 

[mn4j]

I will answer a bit later, but a quick question right away: are you going to answer my considerations about double-slit? Sorry for bothering if you were going to anyway. It's just that I don't want this topic to get lost.

And then from the technical side, there's also this \lambda issue.

I was in a hurry earlier so here is my response to your double-slit experiment. In de Raedt's model of the double slit experiment, the fact the detector learns is not crucial for the model. All that matters is that somewhere along the path of the photons, you have entities that learn. So just to be clear, it could be the slits that learn. He clearly explains this in one of the papers, I'm not sure which one but I can find it. I already mentioned in this thread, maybe you did not see it, that a good way to test this would be to use a different set of apparatus for each event. So again, I agree with de Raedt. If you use a different set of apparatus for each event you will not obtain the same results. You will not see a diffraction pattern if you combine the results later. Moving the detector alone while you have other DLM's in the path of the photons will not rule out de Raedt's model that DLM's are responsible for the result. The only way to rule out any learning is to use a completely different set of apparatus for each event.

The other test will be to use a single set of apparatus, but emit the photons one at a time at such long intervals between each other that the detector and slits are allowed to reach stationary state before the next event arrives. I bet you that there will be no interference pattern, even if everything is left in exactly the same position.

I am not sure what lambda issue you were referring to. If you are referring to this:

Here's one particular example (maybe it's not the most important one. I don't remember any more, but I just have this written down anyway). Seevinck and Larsson say that in de Raedt's model \gamma is less then \gamma_0 which would be necessary to violate the inequality modified to take a loophole into account (and this equality is still violated by QM). de Raedt triumphally replies that they made a mistake in derivation, and then presents his own version of this derivation, which results in \gamma -> 0. Zero is clearly still less than \gamma_0, but he never comments on it.

This is a mischaracterization of the response. Are you sure you read it? The short summary of the Seevinck and Larsson critique is that de Raedt's model can not reproduce the coincidences of many real experiments. De Raedt goes on to show in his reply how his model reproduces the coincidences. Nowhere in his reply did he derive $$\gamma \rightarrow 0$$. If you disagree, provide the page and equation number.

Here is what de Raedt says about this issue on page 3 of their short response which for some reason you claim it is too long (5 pages cf 3 pages for the critique):

By trying to put our work in the context of ”hidden
variable theories”, Seevinck and Larsson also made mis-
takes in elementary algebra. Seevinck and Larsson assume
that the probability of coincidences is given by the denom-
inator of Eq. (6) in Ref. [2] (see Appendix A of Ref. [1]),
from which they derive an expression for the probability of
coincidences $$\gamma$$ (see Eq. (8) in Ref. [1]). However, Seevinck
and Larsson apparently overlooked the fact that in going
from Eq. (3) to Eq. (6) (see Ref. [2]), we take the limit
W/T0 = τ /T0 → 0 and let the number of events N in
both the numerator and denominator go to infinity.Al-
though the ratio remains finite,...

Emphasis added.

Also note that a majority of the Seevinck and Larsson comment is a strawman because they are mostly responding to claims De Raedt never made.

 

[kobak]

Hi mn4j, I'm going to reply you later (there are actually several things I'd like to think about before I answer), but there's a technical moment that I'd like to clarify right away.

Consider the following harmonic oscillator equation.
$$y(t) = A sin(\omega t + \theta)$$
in your favourite plotting package, generate a range of t values. Any range you choose. Using two random number generators, pick two sets of triplets (A, $$\omega$$, $$\theta$$)
use one set to calculate the corresponding $$y(t)_1$$ and the other set to calculate the corresponding $$y(t)_2$$ with the previously generate t values.
Plot $$y(t)_1$$ vs $$y(t)_2$$ and confirm that indeed there is a correlation between the two, no matter what values of (A, $$\omega$$, $$\theta$$) you used for each, however randomly you generated them.

I don't need to wait until I come to the lab tomorrow and can use Matlab to see that what you are saying is just wrong (if I understood you correctly, which I'm not sure). Here's an example: take both amplitudes to be 1, both frequencies to be 1 as well, and phases to be 0 and \pi/2. Then we'll have: X = sin(t), Y = cos(t). The plot of this function in the XY plane is a circle, and the correlation between X and Y in this case is obviously 0.

It's also obvious how to get a perfect correlation of 1, just take X=Y=sin(t). By fiddling with the parameters it's possible to get any correlation coefficient between -1 and 1.

Since it's completely obvious, I'm not sure whether I understood you correctly. Could you clarify it please?

 

[mn4j]

I don't need to wait until I come to the lab tomorrow and can use Matlab to see that what you are saying is just wrong (if I understood you correctly, which I'm not sure). Here's an example: take both amplitudes to be 1, both frequencies to be 1 as well, and phases to be 0 and \pi/2. Then we'll have: X = sin(t), Y = cos(t). The plot of this function in the XY plane is a circle, and the correlation between X and Y in this case is obviously 0.

It's also obvious how to get a perfect correlation of 1, just take X=Y=sin(t). By fiddling with the parameters it's possible to get any correlation coefficient between -1 and 1.

Since it's completely obvious, I'm not sure whether I understood you correctly. Could you clarify it please?

I don't think you understand correlation correctly. The correlation coefficient is not a sufficient descriptor of correlation, that is why I say you should plot it and look at it. If you do not get a random distribution of points, it is correlated. The correlation coefficient is only useful when you are studying linear relationships, or you already know what function to use to convert your data such that any relationship if present will be linear.

 

[kobak]

I don't think you understand correlation correctly. The correlation coefficient is not a sufficient descriptor of correlation, that is why I say you should plot it and look at it. If you do not get a random distribution of points, it is correlated. The correlation coefficient is only useful when you are studying linear relationships, or you already know what function to use to convert your data such that any relationship if present will be linear.

I'm sorry that I'm again not continuing our main discussion (I've been rather busy during these days), but instead answering just this minor point. I would appreciate if we could use the standard terminology, otherwise it's hard to understand each other.

Correct me if I'm wrong, but when people in science say "correlation" it means linear correlation. For example, the wikipedia article on correlation (http://en.wikipedia.org/wiki/Correlation) begins with saying: "In probability theory and statistics, correlation (often measured as a correlation coefficient) indicates the strength and direction of a linear relationship between two random variables. That is in contrast with the usage of the term in colloquial speech, denoting any relationship, not necessarily linear". It's clear that in my example X=cos(t) and Y=sin(t) are clearly _related_ because X^2+Y^2=1. But the correlation between them in the usual scientific meaning of "correlation" is 0. And with sufficiently complex nonlinear transformations anything can be transformed such, that it would be correlated with anything else.

I also note that in quantum singlet correlations (like in EPR) correlation is defined to be as the number of equal spin measurements minus the number of non-equal spin measurement over the total number of measurements. I.e. it's the normal linear correlation: results are fully correlated if they lie on a diagonal (where one axis meaning spin found at A, and the other -- spin found at B): both pluses, or both minuses.

All that probably doesn't undermine your main point at all, because our main issue is about statistical dependence. And it's possible that two values are statistically dependent, though correlation coefficient between them is 0. Like the already mentioned cos(t) and sin(t). In case you believe my understanding here is flawed, I would appreciate any corrections.

 

[mn4j]

I'm sorry that I'm again not continuing our main discussion (I've been rather busy during these days), but instead answering just this minor point. I would appreciate if we could use the standard terminology, otherwise it's hard to understand each other.

Correct me if I'm wrong, but when people in science say "correlation" it means linear correlation. For example, the wikipedia article on correlation (http://en.wikipedia.org/wiki/Correlation) begins with saying: "In probability theory and statistics, correlation (often measured as a correlation coefficient) indicates the strength and direction of a linear relationship between two random variables. That is in contrast with the usage of the term in colloquial speech, denoting any relationship, not necessarily linear". It's clear that in my example X=cos(t) and Y=sin(t) are clearly _related_ because X^2+Y^2=1. But the correlation between them in the usual scientific meaning of "correlation" is 0. And with sufficiently complex nonlinear transformations anything can be transformed such, that it would be correlated with anything else.

I also note that in quantum singlet correlations (like in EPR) correlation is defined to be as the number of equal spin measurements minus the number of non-equal spin measurement over the total number of measurements. I.e. it's the normal linear correlation: results are fully correlated if they lie on a diagonal (where one axis meaning spin found at A, and the other -- spin found at B): both pluses, or both minuses.

Hello Kobak,
A correlation is said to exist between two 'variables' if their values change together in a manner different from what would be expected on the basis of chance. In other words, a relationship exists between both variables (cf. co-relation). Like I mentioned before The correlation coefficient is a statistic used ONLY for measuring linear relationships. A more general statistic of correlation is the NCIE or 'non-linear correlation information entropy'.

The standard terminology is to use "correlation coefficient" and NOT "correlation" when you mean correlation coefficient. It doesn't make much sense to say the correlation is zero. But it makes sense to say the correlation coefficient is zero, which again does not mean a correlation is absent.

All that probably doesn't undermine your main point at all, because our main issue is about statistical dependence. And it's possible that two values are statistically dependent, though correlation coefficient between them is 0. Like the already mentioned cos(t) and sin(t).

You are right, my main point does not depend on whether we agree on the definition of correlation.