Computer Simulation of EPR Scenarios

[Heinera]

Of course they can choose any a and b they want. My point was that x must be equal to Alice's (resp. Bob's) choice, and no other value. But these choices can not be known to the initial state in a local model.

Alice and Bob are completely free to choose any directions a and b they want. Their choices do not affect the initial state (e, t) in any way. In particular, the choices made by Alice and Bob can be changed at any time during the flights of the "photons" from the initial state to the respective choices of detectors by Alice and Bob.

But I am talking about Michel's implementation of your model. Try assigning any other values to x than Alice's resp. Bob's choices in his code, and see what happens to the output.

[gill1109]

You're absolutely right, Heinera.

The problem seems to be the ambiguous meaning of the English phrase "for any". In logic-101 and in mathematics we distinguish between "for all" (upside down A) and "there exists" (backwards E). Now Joy correctly points out that some people say "for any" when they mean "for all". But in English, "for any" might just as well mean "for all" as "for some" = "there exists at least one, such that".

I'm afraid the Jumbo-jet EPRB.pdf is never going to fly, because of a confusion between "for all" and "there exists".

[Joy Christian]

But I am talking about Michel's implementation of your model. Try assigning any other values to x than Alice's resp. Bob's choices in his code, and see what happens to the output.

I see no effect on the output when I change the value a for Alice from a to a'. I get the average of A equal to zero. And similarly for B, when I change b from b to b'.

Next I look at joint probabilities, say P++, for example. They produce different numbers, P++(a, b) and P++(a', b' ). So what? They are supposed to produce different numbers. They are supposed to vary sinusoidally as a function of the angel between a and b, for the same state (e, t). I think you and Gill are seeing things.

[Heinera]

But I am talking about Michel's implementation of your model. Try assigning any other values to x than Alice's resp. Bob's choices in his code, and see what happens to the output.

I see no effect on the output when I change the value a for Alice from a to a'. I get the average of A equal to zero. And similarly for B, when I change b from b to b'.

Next I look at joint probabilities, say P++, for example. They produce different numbers, P++(a, b) and P++(a', b' ). So what? They are supposed to produce different numbers. They are supposed to vary sinusoidally as a function of the angel between a and b, for the same state (e, t). I think you and Gill are seeing things.

But we are not talking about changing the value a for Alice's choice from a to a'. You must keep a, but change x, which you didn't do. Alice can pick any value she wants. We are talking about decoupling x from a, since according to your theory x can take any value whatsoever, independent of the settings of Alice and Bob. If x is coupled to the choices of Alice and Bob the theory is nonlocal.

Say, code x as a random variable, completely independet of the choices of either Alice or Bob. What kind of output do you get then? Linear?

[Joy Christian]

But we are not talking about changing the value a for Alice's choice from a to a'. You must keep a, but change x,...

This makes no sense. x is a dummy, or generic variable. When we choose a, what we are doing is setting x = a.

Alice can pick any value she wants. We are talking about decoupling x from a,...

This too makes no sense. Again, x is a dummy, or generic variable. When we choose a, what we are doing is setting x = a.

If x is coupled to the choices of Alice and Bob the theory is nonlocal.

First, x is NOT "coupled" to the choices of Alice and Bob. The choices of Alice and Bob are INSTANCES of x.

Second, you are free to make your own private definition of locality if you like, but your definition is not the definition of Einstein and Bell, which I am respecting.

According to Bell's definition a theory is local if the measurement results A(a, L) are functions only of a and L, and the measurement results B(b, L) are functions only of b and L, where L is the initial state, or hidden variable. In my model L is given by two numbers, e and t. So L, in my model, is L := (e, t). Thus my model is perfectly local according to the definition provided by Bell. This can be seen quite clearly from the definitions of A(a, L) and B(b, L) given in my eqs. (A.9.2) and (A.9.3).

It is very important to recognize that the numbers e and t do not change when a is changed from a to, say, a'. And similarly when b is changed from b to, say, b'.

[gill1109]

I see no effect on the output when I change the value a for Alice from a to a'. I get the average of A equal to zero. And similarly for B, when I change b from b to b'. Next I look at joint probabilities, say P++, for example. They produce different numbers, P++(a, b) and P++(a', b' ). So what? They are supposed to produce different numbers. They are supposed to vary sinusoidally as a function of the angel between a and b, for the same state (e, t). I think you and Gill are seeing things.

The output Heinera and I are talking about, Joy, is the collection of outcomes of N measurements performed by Alice and Bob, generated by Minkwe's simulation program. We are not talking about the summary statistics printed out at the end.

In each run, Alice and Bob pick settings, and they each get to see +1, -1, or 0. Change the value of a for Alice in run "n", and her output in run "n" changes. In particular, it can change from "0" to not-zero. i.e., depending on what setting she chooses, sometimes there is a state, sometimes there isn't! In other words, the population of states we are choosing from when Alice chooses setting a and Bob chooses setting b depends on both a and b.

A different issue is what is this "x" in the documents EPRB.pdf and complete.pdf. A dummy variable, you say? Good, then give it a different name, "elephant" for instance. When you do that you have to be sure what is the "scope" of the original variable. We come across formulas in which two x's are involved, one implicitly, one explicitly. Which one is the elephant or are they both?

One has to carefully distinguish "for all" and "there exists". Some people say "for any" instead of "for all" but that is dangerous since it is ambiguous. I am afraid an exactly similar ambiguity in scope of dummy variables, and using the same name several times when different instances were meant, is precisely what led to the catastropher of the one page paper. I fear that a similar catastrophe has occured here: you think Minkwe simulated your model but your model is ambiguously defined and the results you claim for it are actually false since they exploit the ambiguity to shift meaning.

Take an example from Minkwe and write out brief but unambiguous instructions to a programmer (a brief description of an algorithm). Minkwe's description is a beautiful model for you to imitate. I'll program your instructions in R, my program will be the same length as your list of instructions and it will run in a flash. It would be convenient if you could use the language of R^3. It is very simple to generate uniform random vectors of length one: create a vector of independent random standard normals, divide by its length.

[Heinera]

But we are not talking about changing the value a for Alice's choice from a to a'. You must keep a, but change x,...

This makes no sense. x is a dummy, or generic variable. When we choose a, what we are doing is setting x = a.

Ah, so this x has no connection to the x in your paper. Right. Apologize. Right. Ok, so where is the x in your paper reflected in Michel's code? And by the way, I think the best approach for the two of you would be to figure out a way to get rid of the 0 outcomes in the detection events in his code, since this is a pedagocical red herring to the Bell mafioso. Given your claim that these zeros don't really exist, this should be manageable.

[gill1109]

Ah, so this x has no connection to the x in your paper. Right. Apologize. Right. Ok, so where is the x in your paper reflected in Michel's code? And by the way, I think the best approach for the two of you would be to figure out a way to get rid of the 0 outcomes in the detection events in his code, since this is a pedagocical red herring to the Bell mafioso. Given your claim that these zeros don't really exist, this should be manageable.

Well said, Heinera! I think I can answer your question. Where is x in Michel's code? Well, Michel's t = theta = eta(x, g0), the angle between x and g0. See "complete.pdf". We start with e0 and g0 and we are supposed to restrict attention to e0 such that |cos(eta(x,e0))| >= 1/2 sin^2(eta(x,g0)) for all x. And now we define theta = eta(x,g0). Michel's hidden state is the pair (e0,theta).

I think there is a problem here (scope? circularity? "for all", or "for some"?)

[Joy Christian]

Ah, so this x has no connection to the x in your paper. Right. Apologize. Right. Ok, so where is the x in your paper reflected in Michel's code? And by the way, I think the best approach for the two of you would be to figure out a way to get rid of the 0 outcomes in the detection events in his code, since this is a pedagocical red herring to the Bell mafioso. Given your claim that these zeros don't really exist, this should be manageable.

Well, Michel has streamlined my notation somewhat, which is inevitable for a simulation. His x is a scalar, whereas my x in the EPRB.pdf is a vector. If you wish to know exactly where my x comes from, then have a look at this one-page derivation: http://libertesphilosophica.info/blog/w ... mplete.pdf

I appreciate that the so-called 0 outcomes can be confusing. From my perspective there are no such outcomes, as you can see from the above page. I know very little (i.e., almost nothing) about programming, so it is up to Michel to see whether this is a problem or not. From my perspective it is not, but I appreciate your suggestion.

[gill1109]

Joy, is this your model?

In a nutshell, EPR experiment:

Code: Select all

set.seed(1234)  ## For reproducibility
s <- 1/2       ## spin 1 for photons, spin 1/2 for electrons
N <- 10^6
e <- runif(N, 0, 2*pi)    ## Hidden variable "e"
p <- (sin(runif(N, 0, pi/2))^2)/2   ## Hidden variable "p"
n <- 2*s
a <- runif(N, 0, 2*pi)    ## Alice's settings
b <- runif(N, 0, 2*pi)    ## Bob's settings
ep <- e + 2*pi*s          ## Hidden variable "e-prime"
c <- ((-1)*n)*cos(n*(a-e))
cp <- ((-1)*n)*cos(n*(b-ep))
A <- ifelse(abs(c)>p, sign(c), 0)          ## Alice outcome
B <- ifelse(abs(cp)>p, sign(cp), 0)        ## Bob outcome
theta <- ifelse( a-b > 0, a-b, a-b+2*pi )  ## Differences between settings
bin <- floor(theta*360/(7.5*2*pi))*7.5     ## Binned, bin-width 7.5 degrees
sample <- A!=0 & B!= 0                     ## Which runs both particles detected

corrs <- tapply(A[sample]*B[sample], bin[sample], mean)
bins <- sort(unique(bin[sample]))
plot(bins, corrs, xlab = "Theta", ylab = "Correlation",
    main = "Minkwe's Simulation, R version by R.D. Gill, Spin 1/2")


Output:



CHSH type experiment:

Code: Select all

set.seed(2345)  ## For reproducibility
s <- 1/2       ## spin 1 for photons, spin 1/2 for electrons
N <- 10^6
e <- runif(N, 0, 2*pi)    ## Hidden variable "e"
p <- (sin(runif(N, 0, pi/2))^2)/2   ## Hidden variable "p"
n <- 2*s

aLab <- sample(c(1, 2), N, replace = TRUE) ## Alice setting label
bLab <- sample(c(1, 2), N, replace = TRUE) ## Bob setting label
aDeg <- (aLab-1)*90
bDeg <- 135 + (bLab-1)*90
a <- aDeg*2*pi/360
b <- bDeg*2*pi/360

ep <- e + 2*pi*s          ## Hidden variable "e-prime"
c <- ((-1)*n)*cos(n*(a-e))
cp <- ((-1)*n)*cos(n*(b-ep))
A <- ifelse(abs(c)>p, sign(c), 0)          ## Alice outcome
B <- ifelse(abs(cp)>p, sign(cp), 0)        ## Bob outcome

combo <- 2*(aLab-1) + (bLab-1)
 ## (1,1) becomes 0
 ## (1,2) becomes 1
 ## (2,1) becomes 2
 ## (2,2) becomes 3

sample <- A!=0 & B!= 0                     ## Which runs both particles detected

corrs <- tapply(A[sample]*B[sample], combo[sample], mean)
names(corrs) <- c("(1,1)","(1,2)","(2,1)", "(2,2)")
corrs


Output:

Code: Select all

> corrs
     (1,1)      (1,2)      (2,1)      (2,2)
 0.7002947  0.6973906 -0.6982893  0.6984241


[Joy Christian]

Joy, is this your model?

My model is summarized in these two pages and references provided therein: http://libertesphilosophica.info/blog/w ... 1/EPRB.pdf

Further, extensive, details of my model can be found on my blog: http://libertesphilosophica.info/blog/

[Ben6993]

Hi Fred & Joy

Thank you for the replies. As you know, I support Joy's model but I have for a long time thought that the simulations would not resolve the issue. The simulations have not yet convinced the flatlanders, who see the simulation outcomes as exploiting loopholes in a test under flatland conditions. Whereas 3spherians see those loopholes as problems only for an incorrect R3 model and absolutely no problem for Joy's model. Impasse.

I understood Chantal's simulation in javascript only after learning a little javascript which was new to me. But I managed an approximate transcription of her simulation into Excel VB. I looked at Michel's simulation's code in Python and Paul's in Mathematica but found those languages too alien for me (and I don't have Mathematica). I note that Richard is using R language software and has given reference to freeware. I have spent a short time investigating that and it seems more promising for me. (Excel2007 VB seems to have a problem with its random number generator when generating millions of numbers in quick succession.)

Using Joy's two-page paper, I made a simulation using Excel VB which produces a curved rather than saw-toothed graph, but I would not call it a cosine curve as it is not curvy enough (see graph below, if the forum accepts the graph formatting). However, it does exclude 21% of pairs of particles which contain a zero somewhere in their results. That is in a run of 1 million pairs. I am classing that as a failure of mine to reproduce Michel's simulation curve, and part of the problem may lie in the Excel VB random number generator. But also I may be coding incorrectly.

In order to correct my version, it would help me to know if there are zero results in Michel's simulation which do not appear in the counts. Richard says that zero results are being excluded. This is not an issue about the 'model' for me as I know that these zero results are real for flatlander and not real for the 3spherers. I simply want to know if every pair of particles generated in the simulation contributed a pair of non-zero results to the graph.

Best wishes

Ben

[Joy Christian]

As you know, I support Joy's model but I have for a long time thought that the simulations would not resolve the issue. The simulations have not yet convinced the flatlanders, who see the simulation outcomes as exploiting loopholes in a test under flatland conditions. Whereas 3spherians see those loopholes as problems only for an incorrect R3 model and absolutely no problem for Joy's model. Impasse.

...Richard says that zero results are being excluded. This is not an issue about the 'model' for me as I know that these zero results are real for flatlander and not real for the 3spherers. I simply want to know if every pair of particles generated in the simulation contributed a pair of non-zero results to the graph.

Hi Ben,

Thanks for your comments. You have put your finger on the true reason behind the impasse. Thank you also for the graph. 21% rejection rate, however, is extremely high. It is no surprise that you are able to reproduce the curves (at least to some extent) with such a high rejection rate. Even from the flatlanders' perspective Michel's simulation does far better, because his rejection rate is only 3.8%. From the 3-spherers' perspective of course there is no rejection of any kind whatsoever, because one cannot reject that which is not there in the first place.

The answer to your last question also partly depends on your perspective. From the 3-sphere perspective every pair of particles generated in the simulation is indeed contributing to a pair of non-zero results to Michel's graph. This is because a particle is defined by the state (e, t), not just by e. On the other hand, you can see from the probabilities defined in my eqs. (A.9.4) to (A.9.7) that they add up to 1. This means that even from the flatlanders' perspective every particle pair contributes to a pair of results, and moreover if one detector clicks then the other does not fail to click. This is a quantum mechanical prediction, which is reproduced by my model.

Best,

Joy

[gill1109]

Michel told me he hopes to have time to look into these questions, coming weekend.

But I can already tell you that Michel's rejection rate is much, much higher than 3.8%. Obviously, it is large enough in order to be able to reproduce the singlet correlation in a local realistic way! It is easy to modify my R code to find out what are the actual rejection rate (double zero's, or single zero's on either side) for any particular pair of settings. See the four 3x3 tables for the four CHSH pairs of settings, which I generated from running Michel's Python code and saving the output to text files which I could easily analyse in R. [One could also easily do this, remaining in Python. And I wouldn't have needed to modify Michel's code at all, if I could have figured out how to read Python binary data files into R. There is an R package for this but it isn't yet available in the newest version of R, 3.0.2].

Ben wants to know if "every pair of particles generated in the simulation contributed a pair of non-zero results to the graph. The answer to this question depends on what you call a pair of particles. If you run Minkwe's program with N = 10 million you do not end up with 10 million pairs of particles both of which have delivered a legitimate measurement outcome (+/-1).

[Joy Christian]

But I can already tell you that Michel's rejection rate is much, much higher than 3.8%.

The rejection rate in Michel's simulation is 0% in the real world S^3. The state, or the particle, in S^3, is defined by a pair (e, t), not just by e. Each particle (e, t) is detected, if emitted, by a detector, producing a measurement result A(a; e, t) = +1 or -1. Of course, the flatlanders see things differently in their fictitious world R^3.

[Ben6993]

First, I must apologise for an error in which I attributed the Mathematica simulation to "Paul", whereas it was produced by John Reed.

Second, removing the contentious word 'particles' from what I wrote about the Excel VB simulation, it can be rephrased as a generation of 1 million e, a and b values of which 212198 (21.2%) did not produce a pair of non-zero results.

Third, Richard reported (in the CHSH thread; by gill1109 » Sun Feb 09, 2014 6:06 am) that he ran Michel's code with N = 1 million and extracted a small subset of data, i.e. the runs with Alice's angle 0 or 45 degrees, Bob's angle 22.5 or 67.5 degrees and he also tabulated the outcomes. I have summarised Richard's tabulated outcomes as follows:
1757 pairs of data generated of which 496 pairs (28.2%) did not produce a pair of non-zero results.

[gill1109]

That's right, Ben. Note that the angles I generated were binned, with bin-width 7.5 degrees, so "Alice's angle 0 degrees" really means "Alice's angle between 0 and 7.5 degrees".

[gill1109]

Apologies for posting this in two threads simultaneously:

Here are links to html notebooks of my implentation in R of the the two-dimensional (S^1, subset of R^2) version of Minkwe's implementation of Christian's model

http://rpubs.com/gill1109/13269

and of a new three-dimensional version (S^2, subset of R^3)

http://rpubs.com/gill1109/13270

The other posting is in the thread "A simple two-page proof of local realism" at

viewtopic.php?f=6&t=18#p162

[gill1109]

Comparison of minkwe's code, Christian's text, and my code shows we have come full circle: Christian's model is nothing else than Caroline Thompson's spinning ball model, see http://arxiv.org/pdf/quant-ph/0210150v3.pdf and http://freespace.virgin.net/ch.thompson ... Record.htm



That is why it works exactly in S^2 but only approximately in S^1.

Here is the core piece of code from my http://rpubs.com/gill1109/13270

Code: Select all

alpha <- 0                          ## CHSH: try 0 and 90
beta <- 45*2*pi/360                 ## CHSH: try 45 and 135
a <- c(cos(alpha), sin(alpha), 0)   ## S1 version: c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta), 0)     ## S1 version: c(cos(beta), sin(beta))

sumprod <- 0
N <- 0
for (i in 1:10^6) {
   e0 <- rnorm(3)                  ## S1 version: rnorm(2)
   e0 <- e0/sqrt(sum(e0^2))
   theta0 <- runif(1, 0, pi/2)
   ca <- sum(e0*a)
   cb <- sum(e0*b)
   s <- (sin(theta0)^2)/2
   if (abs(ca) > s & abs(cb) > s ) {
      sumprod <- sumprod + sign(ca)*sign(cb)
      N <- N+1
   }
}
sum(a*b)        ## Theoretical correlation
N               ## Number pairs of particles
sumprod/N       ## Observed correlation


Notice I have two variants: Minkwe's version (S^1) and a new version (S^2). The S^2 version corresponds exactly to Christian's theoretical specification, the S^1 version is a little bit off. For instance, when the correlation should be 1/sqrt 2 = 0.7071, this is exactly what the S^2 version delivers, the S^1 version gives 0.701. At the sample size I am using, the standard error is 0.001. So this is highly significant.

[gill1109]

Let me explain the connection between Caroline Thompson's spinning ball model and my present implementation of Joy Christian's model. The figure (from Caroline Thompson's web pages and publications) shows a unit sphere covered by four spherical caps, all of the same size. The caps come in two pairs: the two caps of one pair are situated exactly opposite to one another. One pair of caps is centred around the measurement directions -a and a, the other pair is centred around the measurement directions -b and b. The "diameter" of each cap is a section of a diameter of the sphere. The total length of the diameter of the sphere is of course 2 pi. The lengths of the diameters of the caps are all equal to 2 phi, where phi = arc cos(s), s = 1/2 sin^2(theta), and theta is drawn uniformly at random from uniform distribution on [0, pi/2]. Thus if theta=0, then s=0, phi = pi/2, and the two caps around -a and +a are in fact two complete hemispheres covering the whole sphere; similarly for those around -b and b. The white bands in the figure have vanished. If however theta=pi/2 then s=1/2, phi = pi/3, the white bands have maximal "width" pi/3.

The two "a" caps and the two "b" caps intersect in the four areas labelled SN, SS, NN, NS. Pick a point uniformly at random in the *union* of the four caps. This is the vector e0. If e0 lies in the cap centered +a the outcome of measurement of A is +1, if it lies in the cap centered at -a the outcome of measurement of A is -1. If e0 lies in the cap centered +b the outcome of measurement of B is +1, if it lies in the cap centered at -b the outcome of measurement of B is -1.

If we replace the sphere S^2 by the circle S^1 but keep everything else the same (spherical caps become arcs of the circle) then we get Minkwe's simulation model.

The two models exhibit very similar statistics but the S^2 model gives a better fit to the singlet correlations, in fact, it possibly reproduces them exactly. I have to do the trigonometry ... There is a very pretty formula for the area of the intersection of two equal spherical caps so now we just need to average this over a particular distribution of the radii of the caps.

If the angle between a and b is 2 alpha, and the radii of the spherical caps is beta, then the area of the intersection of the +a and +b caps is equal to . Sanity check: when alpha=0 and beta=pi/2 we obtain the maximal area of the overlap of two hemispheres = half area of sphere = 2 pi.