Evidence that QM does not violate Bell's inequalities

[minkwe]

Somebody badly needs to recover their lost memory viewtopic.php?f=6&t=53

[FrediFizzx]

You guys keep saying in this thread that some expectations are "statistically independent". That's mathematical non-sense. It's well known that two events A and B are statistically independent if and only if P(A and B) = P(A) P(B). Also, two random variables X and Y are statistically independent if and only if P(X in A, Y in B) = P(X in A) P(Y in B) for all A and B. But the expectation E[Z] of some random variable Z is a real number. It does not make sense to say that two expectations E[Z] and E[W] are statistically independent. If you don't believe what I'm saying, please post a link to a single probability book which gives a definition of "statistically independent expectations". C'mon, guys.

Why would you need a definition for something that is clearly self evident? I suspect you are talking about something other than Heine's simulation. If you are talking about Heine's simulation, then please reference the terms he uses. IOW, simply show us mathematically how his expectations E11 and E12 are dependent on each other.

[Joy Christian]

You guys keep saying in this thread that some expectations are "statistically independent". That's mathematical non-sense. It's well known that two events A and B are statistically independent if and only if P(A and B) = P(A) P(B). Also, two random variables X and Y are statistically independent if and only if P(X in A, Y in B) = P(X in A) P(Y in B) for all A and B. But the expectation E[Z] of some random variable Z is a real number. It does not make sense to say that two expectations E[Z] and E[W] are statistically independent. If you don't believe what I'm saying, please post a link to a single probability book which gives a definition of "statistically independent expectations". C'mon, guys.

The relationship between joint probabilities and expectation values E(a, b) is elementary, straightforward, and well known. See, for example, Eq. (42) of this paper.

[FrediFizzx]

So, you guys, I will be on an expedition for the next couple of days where I probably will not have much of an internet access. In the mean time, I suggest you look at my simple model http://rpubs.com/heinera/16727. This model reproduces QM exactly, and no loopholes are exploited. Unfortunately it is non-local. But, given its simplicity, I trust it will be an easy task to convert it into a local model. See you on Wednesday, and I expect that by then one of you have made a local version available. Sayonara!

I don't see any advantage in trying to make this simulation into a local model. But I did modify it a bit in R Studio to better show how the dependency of the inequalities work.

Code: Select all

set.seed(9875)

## For reproducibility.

M <- 10^5  ## Sample size.

## Use the same, single sample of 'M' realizations of hidden variables for
## all four correlations

## The hidden variable is just a random number t between -1 and 1.  We send t
## to Alice and -t to Bob.

t <- runif(M, -1, 1)

## At both Alice's and Bob's stations the same model is used to determine
## outcomes.  We put this into a function obs(hv), that takes the hidden
## variable as the argument and returns -1 or 1.

obs <- function(hv) {
  s <- sign(hv)
  hv <- abs(hv)
  ## Since the following variable depend on both settings a and b, the model is
  ## blatantly non-local:
  L <- (1 + sum(a * b))/4
 
  if (hv < L) {
    o <- s
  } else if (hv < 2 * L) {
    o <- -s
  } else if (hv < L + 0.5) {
    o <- -1
  } else {
    o <- 1
  }
  return(o)
}

## Now we compute the four correlations in the CHSH inequality:

alpha <- 0
beta <- 45
a <- c(cos(alpha * pi/180), sin(alpha * pi/180))
b <- c(cos(beta * pi/180), sin(beta * pi/180))

## We generate the list of observations by applying the obs function to each
## element in the list of hidden variables.  First for Alice:
ca1 <- sapply(t, obs)
## Then for Bob:
cb1 <- sapply(-t, obs)

E11 <- mean(ca1 * cb1)
E11
[1] -0.70528

alpha <- 0
beta <- 135
a <- c(cos(alpha * pi/180), sin(alpha * pi/180))
b <- c(cos(beta * pi/180), sin(beta * pi/180))

ca2 <- ca1  ## this is the same as A in <AB> being the same A in <AB'>
cb2 <- sapply(-t, obs)

E12 <- mean(ca2 * cb2)
E12
[1] -0.00296

alpha <- 90
beta <- 45
a <- c(cos(alpha * pi/180), sin(alpha * pi/180))
b <- c(cos(beta * pi/180), sin(beta * pi/180))

ca3 <- sapply(t, obs)
cb3 <- cb1  ## This is the same as the B in <AB> being the same as the B in <A'B>

E21 <- mean(ca3 * cb3)
E21
[1] -0.70528

alpha <- 90
beta <- 135
a <- c(cos(alpha * pi/180), sin(alpha * pi/180))
b <- c(cos(beta * pi/180), sin(beta * pi/180))

ca4 <- ca3  ## The A' from <A'B>
cb4 <- cb2  ## The B' from <AB'>

E22 <- mean(ca4 * cb4)
E22
[1] -0.00296

## CHSH expression
-E11 + E12 - E21 - E22
[1] 1.41056


From this, it is obvious that dependent expectation terms are not going to work to violate CHSH.

[minkwe]

I don't see any advantage in trying to make this simulation into a local model. But I did modify it a bit in R Studio to better show how the dependency of the inequalities work.

Code: Select all

set.seed(9875)

## For reproducibility.

M <- 10^5  ## Sample size.

## Use the same, single sample of 'M' realizations of hidden variables for
## all four correlations

## The hidden variable is just a random number t between -1 and 1.  We send t
## to Alice and -t to Bob.

t <- runif(M, -1, 1)

## At both Alice's and Bob's stations the same model is used to determine
## outcomes.  We put this into a function obs(hv), that takes the hidden
## variable as the argument and returns -1 or 1.

obs <- function(hv) {
  s <- sign(hv)
  hv <- abs(hv)
  ## Since the following variable depend on both settings a and b, the model is
  ## blatantly non-local:
  L <- (1 + sum(a * b))/4
 
  if (hv < L) {
    o <- s
  } else if (hv < 2 * L) {
    o <- -s
  } else if (hv < L + 0.5) {
    o <- -1
  } else {
    o <- 1
  }
  return(o)
}

## Now we compute the four correlations in the CHSH inequality:

alpha <- 0
beta <- 45
a <- c(cos(alpha * pi/180), sin(alpha * pi/180))
b <- c(cos(beta * pi/180), sin(beta * pi/180))

## We generate the list of observations by applying the obs function to each
## element in the list of hidden variables.  First for Alice:
ca1 <- sapply(t, obs)
## Then for Bob:
cb1 <- sapply(-t, obs)

E11 <- mean(ca1 * cb1)
E11
[1] -0.70528

alpha <- 0
beta <- 135
a <- c(cos(alpha * pi/180), sin(alpha * pi/180))
b <- c(cos(beta * pi/180), sin(beta * pi/180))

ca2 <- ca1  ## this is the same as A in <AB> being the same A in <AB'>
cb2 <- sapply(-t, obs)

E12 <- mean(ca2 * cb2)
E12
[1] -0.00296

alpha <- 90
beta <- 45
a <- c(cos(alpha * pi/180), sin(alpha * pi/180))
b <- c(cos(beta * pi/180), sin(beta * pi/180))

ca3 <- sapply(t, obs)
cb3 <- cb1  ## This is the same as the B in <AB> being the same as the B in <A'B>

E21 <- mean(ca3 * cb3)
E21
[1] -0.70528

alpha <- 90
beta <- 135
a <- c(cos(alpha * pi/180), sin(alpha * pi/180))
b <- c(cos(beta * pi/180), sin(beta * pi/180))

ca4 <- ca3  ## The A' from <A'B>
cb4 <- cb2  ## The B' from <AB'>

E22 <- mean(ca4 * cb4)
E22
[1] -0.00296

## CHSH expression
-E11 + E12 - E21 - E22
[1] 1.41056


From this, it is obvious that dependent expectation terms are not going to work to violate CHSH.

Thanks Fred, for doing Heine's homework , the calculation confirms that the sets of outcome pairs are statistically independent. Despite Guest's confusion, the "statistically independent" description is apt.

[FrediFizzx]

Thanks Fred, for doing Heine's homework , the calculation confirms that the sets of outcome pairs are statistically independent. Despite Guest's confusion, the "statistically independent" description is apt.

You're welcome. It is pretty mind boggling that some people can't see that the Bell inequalities can't be violated by anything. What is up with that? It is so freakin' simple.

[Heinera]

So, you guys, I will be on an expedition for the next couple of days where I probably will not have much of an internet access. In the mean time, I suggest you look at my simple model http://rpubs.com/heinera/16727. This model reproduces QM exactly, and no loopholes are exploited. Unfortunately it is non-local. But, given its simplicity, I trust it will be an easy task to convert it into a local model. See you on Wednesday, and I expect that by then one of you have made a local version available. Sayonara!

I don't see any advantage in trying to make this simulation into a local model.

Of course it would be an advantage. You would then have produced a local loophole-free simulation that exactly reproduces QM. It would get you the Nobel Prize.

By the way, your modifications did turn it into a local model; unfortuantely, it also changed the correlations so they no longer agree with QM.

[FrediFizzx]

By the way, your modifications did turn it into a local model; unfortuantely, it also changed the correlations so they no longer agree with QM.

It still is using your non-local hidden variable and it is just a "proof" by negation that your simulation doesn't violate CHSH when the expectation terms are dependent. IOW, your expectation terms are independent from each other so you have shifted to an inequality with a bound of 4 instead of 2. CHSH has a bound of 2. As I said... it is pretty mind boggling.

[Heinera]

By the way, your modifications did turn it into a local model; unfortuantely, it also changed the correlations so they no longer agree with QM.

It still is using your non-local hidden variable and it is just a "proof" by negation that your simulation doesn't violate CHSH when the expectation terms are dependent. IOW, your expectation terms are independent from each other so you have shifted to an inequality with a bound of 4 instead of 2. CHSH has a bound of 2. As I said... it is pretty mind boggling.

The hidden variable is local. The nonlocality enters because outcomes in either wing depend directly on settings from both wings of the simulations. Your modifications broke that dependence, so the model is now local.

[FrediFizzx]

The hidden variable is local. The nonlocality enters because outcomes in either wing depend directly on settings from both wings of the simulations. Your modifications broke that dependence, so the model is now local.

Take another better look. The a in E11 is the same as the a in E12, etc.

[Heinera]

The hidden variable is local. The nonlocality enters because outcomes in either wing depend directly on settings from both wings of the simulations. Your modifications broke that dependence, so the model is now local.

Take another better look. The a in E11 is the same as the a in E12, etc.

Yes, that's what I'm saying. The outcomes for Alice are no longer influenced by Bob's setting and vice versa, so you turned the model local.

[FrediFizzx]

Take another better look. The a in E11 is the same as the a in E12, etc.

Yes, that's what I'm saying. The outcomes for Alice are no longer influenced by Bob's setting and vice versa, so you turned the model local.

Ok, let's say that is true. It doesn't matter for what I was showing about the dependency of expectation terms. What you are saying is that only a local model can have dependent terms and a non-local model will have independent terms. If that is the case, then it proves our point that nothing can violate CHSH with a bound of 2 since with independent terms the bound is 4.

Does anyone have a non-local model with dependent expectation terms?

[Heinera]

The whole difference between a local and a non-local model is that for a local model the a in E11 must be the same as the a in E12, while in a non-local model this need not be the case, since Bob's different setting in E12 can give rise to a different outcome for a.

[Joy Christian]

The whole difference between a local and a non-local model is that for a local model the a in E11 must be the same as the a in E12, while in a non-local model this need not be the case, since Bob's different setting in E12 can give rise to a different outcome for a.

Your mistake in this deceptively innocuous looking statement is the same as the one Gill keeps making (not to mention other Bell believers living in the flatland R^3).
I have exposed Gill's mistake systematically in Eqs. (10) to (26) of this paper. Its recognition leads to the strong correlation derived in Eq. (B10) of the paper.

[FrediFizzx]

The whole difference between a local and a non-local model is that for a local model the a in E11 must be the same as the a in E12, while in a non-local model this need not be the case, since Bob's different setting in E12 can give rise to a different outcome for a.

I think you found your problem. Or maybe you meant something different? To properly adhere to Bell-CHSH with a bound of 2, all models must have the same a in E11 and E12. Even the quantum experiments adhere to that.

[Heinera]

The whole difference between a local and a non-local model is that for a local model the a in E11 must be the same as the a in E12, while in a non-local model this need not be the case, since Bob's different setting in E12 can give rise to a different outcome for a.

I think you found your problem. Or maybe you meant something different? To properly adhere to Bell-CHSH with a bound of 2, all models must have the same a in E11 and E12. Even the quantum experiments adhere to that.

There is no problem. My model does not respect the bound of 2, because of its non-locality. All local models wil have a bound of 2.

[Joy Christian]

All local models wil have a bound of 2.

Not all local models. Here is a local model that has bound of : http://arxiv.org/pdf/1501.03393.pdf [see Eq. (26)].

[minkwe]

There is no problem. My model does not respect the bound of 2, because of its non-locality. All local models wil have a bound of 2.

This is complete and utter garbage and if I did not suspect that Heine may really be completely clueless about this, I would have concluded that he was lying through his teeth.

The dependence and independence we are talking about, have no relationship to locality/non-locality. I have already shown since post #1 that it is impossible to satisfy the conditions required to compare the experimental and QM expectations with Bell's inequalities -- even if the model/experiment is fully local realistic. Heine simply ignores the argument.

He does not deny that Bell's inequality is the following:
,
which makes use of the four terms all defined for the same set of outcomes

He presents a simulation which is equivalent to the terms each calculated from 8 separate columns of outcomes: recombined into 4 independent paired spreadsheets, yielding the 4 terms . He then calculates the expression

and finds that the value is greater than 2. But this is to be expected because the terms in this expression are statistically independent. Note that there are 8 columns of data in 4 independent pairs in the data from his simulation, whereas there are 4 columns of data with cyclic dependent pairing in Bell's inequality. Note also that the origin of the upper bound is entirely due to this fact. It is the dependencies or lack thereof that determine the upper bound of the inequality, so a proper understanding of it's presence or absence is crucial to any understanding of Bell's theorem.

As you can see in every proof of the CHSH or Bell's inequality (https://en.wikipedia.org/wiki/Bell%27s_ ... inequality for example), There are only 4 columns of data , recombined into pairs such that the cyclic dependency of the paired terms is present. What is often missing is that the subscripts are left out which allows them to later confusingly or intentionally mislead by assuming there are only 4 columns of data in , since without subscripts, all the As, Bs, Cs and Ds look alike and you can simply say .


When presented with this obvious fact, Heine then proceeds first to make the incredible claim that he is indeed calculating

not

As shown in post #1, you are calculating the independent terms which are definitely not the terms in Bell's inequality.

My model does indeed produce . Read the code again.[/quote]
Obviously anyone with a shred of statistics training should be able to see that if Heine is right that he is calculating , then it means . Only then can his claim be true. An easy statistics test of this claim is to calculate the cross correlation between or or , or . If those columns of data are equivalent, then those correlations will be significantly different from zero. This is what Fred has shown in viewtopic.php?f=6&t=181&start=110#p5302 and found that the cross correlations if you substitute A_i for A_k becomes zero and the same for the rest of them. Therefore Heine is calculating not like he claims, and therefore the upper bound is 4 not 2.

So what has this got to do with non-locality or locality? Nothing whatsoever, as shown in the first post of this thread, if you start with 4 random statistically independent sets of outcome pairs from any source whatsoever (local or non-local), it is impossible to carry out the required row-permutations in order to demonstrate that all the equivalences are simultaneously true. Some of them might be the same but all of them can not be simultaneously the same as verly clearly explained in post #1. It does not matter whether we are dealing with a local or a non-local model. Therefore it is a mathematical error to assume that all the expectation values are simultaneously equal, ie

Some of them may be equal, but they can not all be simultaneously equal. This was clearly explained in post #1. Bell's believers simply ignore the argument.

You may have heard noises about sample of a population having the same value as population. You will note that when this bogus argument is presented, they always talk about a single value, not 4 values simultaneously. But they ignore one very crucial detail. The expectations in Bell's inequality are not equivalent to independent random samples. They are heavily dependent on each other. You can not replace them with independent random samples. Bell's followers repeatedly proclaim how Bell's theorem forces us to rethink the freedom of the experimenter, but it is the complete opposite. The problem is not that the experimenters are not free enough to measure what they want. The problem is that they are too free compared to the freedom assumed in Bell's inequality. Therefore the push to have even more perfectly random and perfectly free experimental situations is completely misguided as far as the inequalities are concerned.

Even worse, the problem described above is fatal. The only solution is to put Bell's theorem in the garbage bin where it belongs. See http://link.springer.com/article/10.100 ... 010-9461-z for a slightly different take of these ideas.

[Heinera]

He does not deny that Bell's inequality is the following:
,
which makes use of the four terms all defined for the same set of outcomes

Oh, I do. Bell's inequality is the following:

.

He presents a simulation which is equivalent to the terms each calculated from 8 separate columns of outcomes: recombined into 4 independent paired spreadsheets, yielding the 4 terms . He then calculates the expression

and finds that the value is greater than 2. But this is to be expected because the terms in this expression are statistically independent.

As you can see in every proof of the CHSH or Bell's inequality (https://en.wikipedia.org/wiki/Bell%27s_ ... inequality for example), There are only 4 columns of data , recombined into pairs such that the cyclic dependency of the paired terms is present. What is often missing is that the subscripts are left out which allows them to later confusingly or intentionally mislead by assuming there are only 4 columns of data in , since without subscripts, all the As, Bs, Cs and Ds look alike and you can simply say .

With my model, it is impossible to compute a 4xN matrix as defined in Richard Gill's paper.

I you were to come up with a local loophole-free model, such a 4xN matrix would be trivial to construct: For each i, I would simply compute your model four times, changing only the settings, and keeping everything else the same. Now I would have a matrix that perfectly replicates your model for those four settings. So I could then just draw a random pair of settings for each row, and read off the results. The correlations I get would be the exact same values your model would produce. And because the starting point was a 4xN matrix, the expectations for the CHSH expression would be 2 or lower.

And by the way, it is not very important to me if you understand this or not.

[minkwe]

With my model, it is impossible to compute a 4xN matrix as defined in Richard Gill's paper.

Are you referring to this paper https://pubpeer.com/publications/D985B4 ... 3E3A314522? I encourage anyone who believes a shred of anything in it to read that discussion. From appearances, even Gill himself has stopped promoting the paper. It is seriously flawed.

I you were to come up with a local loophole-free model, such a 4xN matrix would be trivial to construct: For each i, I would simply compute your model four times, changing only the settings, and keeping everything else the same.

You've not learned any physics all these years. If you had read and understood post #1, you will also see that for the random experiment described it is impossible to generate a 4xN spreadsheet as well, even for a local realistic model. So why have you conveniently avoided the argument?