A simple two-page proof of local realism

Foundations of physics and/or philosophy of physics, and in particular, posts on unresolved or controversial issues

Re: A simple two-page proof of local realism

Postby gill1109 » Mon Mar 03, 2014 3:42 am

Pearle's (1970) model:

http://rpubs.com/gill1109/Pearle

Image

This is the important part of the code

Code: Select all
U <- runif(M)
s <- (2/sqrt(3*U+1)) - 1  # Pearle's "r" is arc cosine of "s"


Or in one line

Code: Select all
s <- (2/sqrt(runif(M, 1, 4))) - 1  # Pearle's "r" is arc cosine of "s"


The sample size was 10^6. This plot shows the difference between simulated curve and cosine, together with error bars. I predict that at sample size 10^8 both difference and error bars will be 10 times smaller!


Image
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Re: A simple two-page proof of local realism

Postby Joy Christian » Mon Mar 03, 2014 4:37 am

gill1109 wrote:Pearle's (1970) model:

http://rpubs.com/gill1109/Pearle


Very nice indeed!

I would have been surpised if there was an error in Pearle's paper, since I know that at least Clauser, Shimony, and Wigner had checked Pearle's math (not to mention the editors and referees of PRD---yes, in the olden days referees did check all the math of the papers they were reviewing).

OK, so now I need to understand what the following prescription means in terms of S^3:

U <- runif(M)
s <- (2/sqrt(3*U+1)) - 1 # Pearle's "r" is arc cosine of "s"

Note that whatever this prescription is, it is already derived in this one-page document. All one has to do is set f(theta_o) = "this prescription" to specify the correct initial state (e_o, theta_o) of the physical system. In terms of the geometry of S^3, it specifies the initial length of the quaternion (p_o + q_o), as seen in my eq. (7).
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Re: A simple two-page proof of local realism

Postby gill1109 » Mon Mar 03, 2014 4:46 am

Joy Christian wrote:Very nice indeed!

;-)

Note that Pearle's solution is not unique. But yes you are right, this can now be translated back to the terms of your S^3 model.

Everyone is happy, I think.
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Re: A simple two-page proof of local realism

Postby Joy Christian » Mon Mar 03, 2014 6:11 am

gill1109 wrote:
Joy Christian wrote:Very nice indeed!

;-)

Note that Pearle's solution is not unique. But yes you are right, this can now be translated back to the terms of your S^3 model.

Everyone is happy, I think.


Actually, I am more than happy. In the original simulation by Michel there remained one unsatisfactory feature from the theoretical perspective: f(theta_o) fell within the range 0 to 1/2. But the triangle inequality for quaternions disccused here suggests that f(theta_o) should range from 0 to 1, not 0 to 1/2. The above prescription based on Pearle's solution resolves this last theoretical wrinkle as well. The prescription is now fully consistent with the triangle inequality for unit quaternions!
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Re: A simple two-page proof of local realism

Postby gill1109 » Mon Mar 03, 2014 6:24 am

Joy Christian wrote:Actually, I am more than happy.

I'm getting warm feelings, again, Joy. ;)

Many thanks to Fred for setting this forum up for us!!!!
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Re: A simple two-page proof of local realism

Postby Ben6993 » Mon Mar 03, 2014 7:26 am

Richard:

I think the Perle and ChaoticUnsharpeBall have similar 'corr-cos' charts (the last chart in the listings) but the former is better and only used 10^6 data pairs while the latter used 10^7. So the Perle is the winner at the moment?

The 'corr-cos' graphs are very roughly -0.002*sin(angle) and +0.0025*sin(angle), respectively. Don't those rough sine curves indicate that there is more pattern still to be extracted from the data as , ideally, there would only be random noise remaining?
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Re: A simple two-page proof of local realism

Postby gill1109 » Mon Mar 03, 2014 7:33 am

Ben6993 wrote:Richard:

I think the Perle and ChaoticUnsharpeBall have similar 'corr-cos' charts (the last chart in the listings) but the former is better and only used 10^6 data pairs while the latter used 10^7. So the Perle is the winner at the moment?

The 'corr-cos' graphs are very roughly -0.002*sin(angle) and +0.0025*sin(angle), respectively. Don't those rough sine curves indicate that there is more pattern still to be extracted from the data as , ideally, there would only be random noise remaining?


Pearle is exact. Take the sample size bigger and bigger (and increase the numerical precision of your computer ...) and you will converge exactly to the cosine.

The same is true for Gisin and Gisin.

None of the others is exact.

Pearle's model *is* a particular chaotic unsharp ball model. It's one with a spot-on choice for the probability distribution of the radius of the circular caps.

From Pearle's proof of this we can see that other choices are possible (his solution is not unique). But his is very simple, and as I said, very very exact.

The rough sinusoidal shape of the error curves is because the *same* sample of 10^6 hidden variables is being used for all possible measurement angles. That saves a heap of time, but creates correlation. Which wouldn't be there, of course, if we used a new sample to calculate each separate point on the curve.

Multiply the sample size 10^6 by 100, and the simulation error will get 10 times smaller.
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Re: A simple two-page proof of local realism

Postby Heinera » Mon Mar 03, 2014 2:05 pm

Joy Christian wrote:
gill1109 wrote:
Joy Christian wrote:Very nice indeed!

;-)

Note that Pearle's solution is not unique. But yes you are right, this can now be translated back to the terms of your S^3 model.

Everyone is happy, I think.


Actually, I am more than happy. In the original simulation by Michel there remained one unsatisfactory feature from the theoretical perspective: f(theta_o) fell within the range 0 to 1/2. But the triangle inequality for quaternions disccused here suggests that f(theta_o) should range from 0 to 1, not 0 to 1/2. The above prescription based on Pearle's solution resolves this last theoretical wrinkle as well. The prescription is now fully consistent with the triangle inequality for unit quaternions!


But the fact that theta_o ranges from 0 to 1 in Richard's implemetation of Pearle's model has only a technical reason. One standard method of sampling a non-uniform random variable is to derive its cdf, then invert the cdf (call this inverse f(x)), then sample a uniform random variable x from 0 to 1. f(x) will then have the required non-uniform distribution we were after in the first place. And this is what is employed here. So at best, the range 0 to 1 is a consequence of the convention that we measure probabilities between 0 and 1 (the convention could just as well be between 0 and 100, and lots of people do use that in ordinary language). What this has to do with the triangle inequality for unit quaternions is completely beyond me, I am afraid.
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Re: A simple two-page proof of local realism

Postby Joy Christian » Mon Mar 03, 2014 4:18 pm

Heinera wrote:But the fact that theta_o ranges from 0 to 1 in Richard's implemetation of Pearle's model has only a technical reason. One standard method of sampling a non-uniform random variable is to derive its cdf, then invert the cdf (call this inverse f(x)), then sample a uniform random variable x from 0 to 1. f(x) will then have the required non-uniform distribution we were after in the first place. And this is what is employed here. So at best, the range 0 to 1 is a consequence of the convention that we measure probabilities between 0 and 1 (the convention could just as well be between 0 and 100, and lots of people do use that in ordinary language). What this has to do with the triangle inequality for unit quaternions is completely beyond me, I am afraid.


Actually, theta_o still ranges from 0 to pi. The translation between Richard's code and my model, in the language of R, is as follows:

theta <- runif(M, 0, pi) ## My initial theta_o, or Michel Fodje’s “t”
s <- (2/sqrt(1+((3*theta)/pi))) - 1 ## Pearle's "r" is arc cosine of "s"

In other words, the initial function, as defined in eq. (7) of this document, is simply



In geometrical terms the range 0 to 1 is a consequence of the fact that my 3-sphere is normalized to have unit radius. If you want to use the convention 0 to 100 for probabilities, then the corresponding convention in geometry would be a 3-sphere of radius 100. No big deal. We just have to use consistent conventions all around.
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Re: A simple two-page proof of local realism

Postby Joy Christian » Tue Mar 04, 2014 11:08 am

Joy Christian wrote:Actually, theta_o still ranges from 0 to pi. The translation between Richard's code and my model, in the language of R, is as follows:

theta <- runif(M, 0, pi) ## My initial theta_o, or Michel Fodje’s “t”
s <- (2/sqrt(1+((3*theta)/pi))) - 1 ## Pearle's "r" is arc cosine of "s"

In other words, the initial function, as defined in eq. (7) of this document, is simply



I have now published the 3-sphere version of the simulation by Richard Gill: http://rpubs.com/jjc/13965

With sample size 10^7, the resulting plots are spectacular---especially the last one. Just have a look at it.
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Re: A simple two-page proof of local realism

Postby Heinera » Tue Mar 04, 2014 11:34 am

That is to be expected, given that Pearle's solution is analytically exact. I think that the only spectacular issue in this case is Richard Gill's persistance and success, where he quickly demonstrated that the original numerical model was just a crude and incorrect implemetation of Pearle's paper of 1970 - the "father" of the detection loophole. Lesson to be learned for the day: It sure pays off to know one's field.
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Re: A simple two-page proof of local realism

Postby Joy Christian » Tue Mar 04, 2014 12:09 pm

Heinera wrote:That is to be expected, given that Pearle's solution is analytically exact. I think that the only spectacular issue in this case is Richard Gill's persitance and success, where he quickly demonstrated that the original model was just a crude and incorrect implemetation of Pearle's paper of 1970 - the "father" of the detection loophole. Lesson to be learned for the day: It sure pays off to know one's field.


It surely does pay off to know one's field. The ultimate field is of course Nature, and how she works, not statistics or programming. What the above plot shows is that Nature respects the geometry and topology of the 3-sphere, and does so in a spectacular manner. From your flatland perspective one may get excited about statistics, programming, and loopholes hunting, but from my topological perspective what is spectacular is the unmitigated profundity with which Nature reveals herself to us.

If you wish to learn more about the inner workings of Nature and how she reveals her sublime beauty to us, then do read at least some introductory parts of my book:

Image
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Re: A simple two-page proof of local realism

Postby gill1109 » Thu Mar 06, 2014 2:36 am

Joy: I suggest you add to the last "plot()" command in http://rpubs.com/jjc/13965 the extra optional argument "ylim = c(-0.0005, +0.0005)" so that the range of the y axis is extended enough to see the red curves. (Hope I got the number of zero's right). That will slightly extend the range of the y-axis so we can see all of the red curves, too.
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Re: A simple two-page proof of local realism

Postby Joy Christian » Thu Mar 06, 2014 3:21 am

gill1109 wrote:Joy: I suggest you add to the last "plot()" command in http://rpubs.com/jjc/13965 the extra optional argument "ylim = c(-0.0005, +0.0005)" so that the range of the y axis is extended enough to see the red curves. (Hope I got the number of zero's right). That will slightly extend the range of the y-axis so we can see all of the red curves, too.


Thanks. Good suggestion. I have replaced the rpub with the above line added to the last plot: http://rpubs.com/jjc/13965
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Re: A simple two-page proof of local realism

Postby Joy Christian » Fri May 09, 2014 8:05 am

I am reproducing here what Michel Fodje wrote elsewhere, because (1) his observations are relevant for all realizable physical experiments, and (2) they beautifully spell out elementary facts of logic, arithmetic, and physics that the vast majority of the Bell-believers among us seem to be incapable of understanding:

minkwe wrote:1 - If you measure (A,B), (A',B), (A,B'), (A,B') on a different particle pair, the A in (A,B) can be different from the A in (A,B') without any mistake or cheating.
2 - If you measure the same particle at a (A,B), and exactly the same particle again at (A,B'), then A in (A,B) can be different from the A in (A,B') without any mistake or cheating.
3 - The only way to measure (A,B), (A',B), (A,B'), (A,B') on the same particle, and make sure the A in (A,B) and the A in (A,B') are the same (and each outcome is the same in each pair), is to measure the same particle pair, simultaneously at (A, A', B, B'), an impossibility. Therefore a genuine experiment testing S <= 2 is impossible.
4 - If the probability of obtaining H for a coin is 0.75, the probability of the counter-factual H outcome for the same coin cannot be 0.75 too. It must be 0.25.
5 - No 4xN spreadsheet can violate the S <= 2. It doesn't matter where you get your data to put in the spreadsheet, from LHV/QM/non-local model/non-real model/statistical error etc.
6 - The correct inequality for 4 different 2XN spreadsheets is S<= 4, it doesn't matter where you get your data to put in the spreadsheet, from LHV/QM/non-local model/non-real model/statistical error etc. 4 *different* 2xN spreadsheets can easily violate S <= 2, because that inequality does not apply to such data. It is a mathematical error to even compare them.
7 - It is utter nonsense to compare an inequality derived from a 4xN spreadsheet, with data in the form of 4 different 2xN spreadsheets, even if your 4 *different* 2xN spreadsheets are randomly sampled from a single 4xN spreadsheet. What determines the upper bound is the degrees of freedom in the data, not the degrees of freedom in the original spreadsheet you randomly sampled from.
8 - These inequalities have nothing to do with physics, they are mathematical tautologies about real numbers and degrees of freedom. Please read the Rosinger paper carefully. Their violation points to a mathematical error in their application. Nothing can violate them.
9 - No EPRB experiment will ever be done which produces a 4xN spreadsheet, as it must if it purports to *test* the S <= 2 relationship. As long as they keep producing 4 *different* 2XN spreadsheets, the appropriate inequality is S <= 4, and it will never be violated.
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Re: A simple two-page proof of local realism

Postby gill1109 » Fri May 09, 2014 8:29 am

A breakthrough. At last JJC listens to Michel Fodje. Perhaps he will now revise his two experimental pre-publications http://arxiv.org/abs/0806.3078, http://arxiv.org/abs/1211.0784, which both contain the same howler of a mistake because JJC was back in 2008 and 2012 apparently not aware of relevant elementary facts of logic and arithmetic.
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Re: A simple two-page proof of local realism

Postby Joy Christian » Fri May 09, 2014 8:46 am

gill1109 wrote:A breakthrough. At last JJC listens to Michel Fodje. Perhaps he will now revise his two experimental pre-publications http://arxiv.org/abs/0806.3078, http://arxiv.org/abs/1211.0784, which both contain the same howler of a mistake because JJC was back in 2008 and 2012 apparently not aware of relevant elementary facts of logic and arithmetic.


Your statement actually proves what I wrote above: "... Michel Fodje ... beautifully spells out elementary facts of logic, arithmetic, and physics that the vast majority of the Bell-believers among us seem to be incapable of understanding:"
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Re: A simple two-page proof of local realism

Postby gill1109 » Fri May 09, 2014 1:25 pm

Pity the vast majority of Bell deniers know next to nothing about probability or statistics. It's quite a handicap. (And explains quite a lot)
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Re: A simple two-page proof of local realism

Postby Joy Christian » Fri May 09, 2014 1:30 pm

gill1109 wrote:Pity the vast majority of Bell deniers know next to nothing about probability or statistics. It's quite a handicap. (And explains quite a lot)


Pity the vast majority of Bell believers know next to nothing about geometry, topology, and algebra. It's quite a handicap. (And explains quite a lot.)
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Re: A simple two-page proof of local realism

Postby FrediFizzx » Fri May 09, 2014 10:05 pm

OK, guys, get back on topic.
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