## Evidence that QM does not violate Bell's inequalities

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

### Re: Evidence that QM does not violate Bell's inequalities

Guest wrote:You're right. The code must be slightly adapted for each LHV model. If each lambda is a pair (e,s), in which e=(e1,e2,e3) and s is a scalar, then we can use:

Code: Select all
rm(list = ls())N <- 10^4lambda <- matrix(nrow = N, ncol = 4)# for each line of the matrix lambda, the first three coordinates# will be e and the fourth coordinate will be sA <- function(a, lambda) { # angle a will be in degrees  e <- lambda[1:3]  s <- lambda[4]  # insert your code here}B <- function(b, lambda) { # angle b will be in degrees  e <- lambda[1:3]  s <- lambda[4]  # insert your code here}# don't change anything after this pointfor (a in runif(2, 0, 360)) {  for (b in runif(2, 0, 360)) {    cat("(a, b) = (", a, ", ", b, ")\n", sep = "")    cat("LHV: ",         mean(apply(lambda, 1, function(lambda) A(a, lambda)) *              apply(lambda, 1, function(lambda) B(b, lambda))), "\n")    cat("QM: ", -cos((a-b)*pi/180), "\n\n")  }}

This is still far too restrictive. It only proves that the limited (I would even say unimaginative) LHV model you have set up may not produce the QM result. So what?

There is also a further problem, although it may not be serious: a and b are not random variables. They are arbitrarily chosen settings, not randomly chosen settings.

You didn't answer my question. Why is the following simulation not a simulation of a LHV model?

Joy Christian wrote:
To see how this can be accomplished in a manifestly local-realistic manner, please take a look at this R code and tell us why it is not a simulation of a LHV model.

The essential part of the code is very simple, and it reproduces the strong correlation in full compliance with all of the requirements of Bell for a LHV model:

Code: Select all
A = +sign(g(a,e,s))  # Alice's measurement results A(a, e, s) = +/-1          B = -sign(g(b,e,s))  # Bob's measurement results B(b, e, s) = -/+1        N = length((A*B)[A & B]) # Number of all possible events observed in S^3        corrs[i,j] = sum(A*B)/N  # Product moment correlation coefficient E(a, b)

Here Bell's hidden variable "lambda" is a pair (e, s), where e is like your "vector" and s is another scalar parameter.
Joy Christian
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### Re: Evidence that QM does not violate Bell's inequalities

Code: Select all
rm(list = ls())N <- 10^4lambda <- matrix(nrow = N, ncol = 4) # lambda[i] = (e[i], s[i])r = runif(N, 0, 2*pi)z = runif(N, -1, +1)h = sqrt(1-z^2)x = h * cos(r)y = h * sin(r)e = cbind(x, y, z)s = runif(N, 0, pi)lambda <- cbind(e, s)A <- function(a, lambda) {  e <- lambda[1:3]  s <- lambda[4]  f <- -1+2/sqrt(1+3*s/pi)  sign(ifelse(abs(sum(a*e)) > f, sum(a*e), 0))}B <- function(b, lambda) {  e <- lambda[1:3]  s <- lambda[4]  f <- -1+2/sqrt(1+3*s/pi)  -sign(ifelse(abs(sum(b*e)) > f, sum(b*e), 0))}for (alpha in c(0, 90)) {  a <- c(cos(alpha), sin(alpha), 0)  for (beta in c(45, 135)) {    b = c(cos(beta), sin(beta), 0)    cat("(alpha, beta) = (", alpha, ", ", beta, ")\n", sep = "")    cat("LHV: ",         mean(apply(lambda, 1, function(lambda) A(a, lambda)) *              apply(lambda, 1, function(lambda) B(b, lambda))), "\n")    cat("QM: ", -cos((alpha-beta)*pi/180), "\n\n")  }}

But these functions A and B may return the value zero. If +1 is spin up and -1 is spin down, what does zero means?
Guest

### Re: Evidence that QM does not violate Bell's inequalities

Guest wrote:
Code: Select all
rm(list = ls())N <- 10^4lambda <- matrix(nrow = N, ncol = 4) # lambda[i] = (e[i], s[i])r = runif(N, 0, 2*pi)z = runif(N, -1, +1)h = sqrt(1-z^2)x = h * cos(r)y = h * sin(r)e = cbind(x, y, z)s = runif(N, 0, pi)lambda <- cbind(e, s)A <- function(a, lambda) {  e <- lambda[1:3]  s <- lambda[4]  f <- -1+2/sqrt(1+3*s/pi)  sign(ifelse(abs(sum(a*e)) > f, sum(a*e), 0))}B <- function(b, lambda) {  e <- lambda[1:3]  s <- lambda[4]  f <- -1+2/sqrt(1+3*s/pi)  -sign(ifelse(abs(sum(b*e)) > f, sum(b*e), 0))}for (alpha in c(0, 90)) {  a <- c(cos(alpha), sin(alpha), 0)  for (beta in c(45, 135)) {    b = c(cos(beta), sin(beta), 0)    cat("(alpha, beta) = (", alpha, ", ", beta, ")\n", sep = "")    cat("LHV: ",         mean(apply(lambda, 1, function(lambda) A(a, lambda)) *              apply(lambda, 1, function(lambda) B(b, lambda))), "\n")    cat("QM: ", -cos((alpha-beta)*pi/180), "\n\n")  }}

But these functions A and B may return the value zero. If +1 is spin up and -1 is spin down, what does zero means?

The functions A and B do not return "zero values" within S^3, as explained in the code (see also viewtopic.php?f=6&t=188#p5129). There is nothing outside S^3.
Joy Christian
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### Re: Evidence that QM does not violate Bell's inequalities

Heinera wrote:
Joy Christian wrote:
Heinera wrote:This was why the CHSH-inequality was developed; the perfect anti-correlation for a=b is hard to demonstrate experimentally, so the CHSH-inequality doesn't use that pair of settings.

False!

CHSH allows any pair of settings, including a = b.

Yes, it allows any pair of settings, which means you can avoid a=b. In particular, you can choose settings where QM predicts a maximal violation of the inequality, and a=b is not one of them.

The settings actually used is completely irrelevant to the perfect-anticorrelation requirement. The requirement comes in because during the derivation, A(a,lambda)B(c, lambda) gets converted into A(a,lambda)[-A(c, lambda)] in order to derive the inequality. It does not matter what settings you use in the end if that substitution was used during the derivation. The assumption will still be present.

But as Fred says, it is all moot anyway, all experimenters have run away from the CHSH already, and are just about to run away from the CH and Eberhard inequalities given the recent developments. I wonder what the next "inequality"/"trick" will be.
minkwe

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### Re: Evidence that QM does not violate Bell's inequalities

minkwe wrote:The settings actually used is completely irrelevant to the perfect-anticorrelation requirement. The requirement comes in because during the derivation, A(a,lambda)B(c, lambda) gets converted into A(a,lambda)[-A(c, lambda)] in order to derive the inequality. It does not matter what settings you use in the end if that substitution was used during the derivation. The assumption will still be present.

That subtitution is only used in Bell's original proof of his inequality. It is not used in the derivation of the CHSH-inequality. The CHSH-inequality holds for any LHV model, including those where B(c, lambda) != -A(c, lambda).

(The usual caveats on detection and coincidence loopholes apply. You can circumvent the CHSH-inequality in those cases.)
Heinera

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### Re: Evidence that QM does not violate Bell's inequalities

Heinera wrote:
minkwe wrote:The settings actually used is completely irrelevant to the perfect-anticorrelation requirement. The requirement comes in because during the derivation, A(a,lambda)B(c, lambda) gets converted into A(a,lambda)[-A(c, lambda)] in order to derive the inequality. It does not matter what settings you use in the end if that substitution was used during the derivation. The assumption will still be present.

That subtitution is only used in Bell's original proof of his inequality. It is not used in the derivation of the CHSH-inequality. The CHSH-inequality holds for any LHV model, including those where B(c, lambda) != -A(c, lambda).

(The usual caveats on detection and coincidence loopholes apply. You can circumvent the CHSH-inequality in those cases.)

Your argument is just plain silly since nothing can violate the Bell-CHSH inequality. It is mathematically impossible. Loopholes just don't matter anymore.
FrediFizzx
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### Re: Evidence that QM does not violate Bell's inequalities

FrediFizzx wrote:Your argument is just plain silly since nothing can violate the Bell-CHSH inequality. It is mathematically impossible. Loopholes just don't matter anymore.

And in what way does this not violate the CHSH-inequlity:
http://rpubs.com/heinera/16727
Heinera

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### Re: Evidence that QM does not violate Bell's inequalities

Heinera wrote:
FrediFizzx wrote:Your argument is just plain silly since nothing can violate the Bell-CHSH inequality. It is mathematically impossible. Loopholes just don't matter anymore.

And in what way does this not violate the CHSH-inequlity:
http://rpubs.com/heinera/16727

Your terms in the CHSH expression are independent therefore the bound is 4 not 2 like it is in the Bell-CHSH inequality. You are using this inequality,

$\left |\sum_{i}E_{i} \right | \leq N$ (independent terms)

$\left |\sum_{i}E_{i} \right | \leq N -2$ (with dependent terms)

FrediFizzx
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### Re: Evidence that QM does not violate Bell's inequalities

FrediFizzx wrote:
Heinera wrote:
FrediFizzx wrote:Your argument is just plain silly since nothing can violate the Bell-CHSH inequality. It is mathematically impossible. Loopholes just don't matter anymore.

And in what way does this not violate the CHSH-inequlity:
http://rpubs.com/heinera/16727

Your terms in the CHSH expression are independent therefore the bound is 4 not 2 like it is in the Bell-CHSH inequality. You are using this inequality,

$\left |\sum_{i}E_{i} \right | \leq N$ (independent terms)

$\left |\sum_{i}E_{i} \right | \leq N -2$ (with dependent terms)

And in what way are they independent? All four terms are evaluated on the same set of hidden variables.
Heinera

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### Re: Evidence that QM does not violate Bell's inequalities

Heinera wrote:And in what way are they independent? All four terms are evaluated on the same set of hidden variables.

Your calculation for E11 in no way depends on E12, E21, nor E22, etc. for the other combinations.
FrediFizzx
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### Re: Evidence that QM does not violate Bell's inequalities

FrediFizzx wrote:
Heinera wrote:And in what way are they independent? All four terms are evaluated on the same set of hidden variables.

Your calculation for E11 in no way depends on E12, E21, nor E22, etc. for the other combinations.

Of course they depend. The calculations all use the same set of hidden variables; that is their common link. So obviously they are not completely independent.
Heinera

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### Re: Evidence that QM does not violate Bell's inequalities

Heinera wrote:
FrediFizzx wrote:
Heinera wrote:And in what way are they independent? All four terms are evaluated on the same set of hidden variables.

Your calculation for E11 in no way depends on E12, E21, nor E22, etc. for the other combinations.

Of course they depend. The calculations all use the same set of hidden variables; that is their common link. So obviously they are not completely independent.

It doesn't matter if they all depend on the hidden variables. You are running four different independent experiments. One for E11, one for E12, one for E21 and one for E22.
FrediFizzx
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### Re: Evidence that QM does not violate Bell's inequalities

FrediFizzx wrote:It doesn't matter if they all depend on the hidden variables. You are running four different independent experiments. One for E11, one for E12, one for E21 and one for E22.

But again, they are not independent. Why do you think they are independent?
Heinera

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### Re: Evidence that QM does not violate Bell's inequalities

Heinera wrote:
FrediFizzx wrote:It doesn't matter if they all depend on the hidden variables. You are running four different independent experiments. One for E11, one for E12, one for E21 and one for E22.

But again, they are not independent. Why do you think they are independent?

Ironically it is Gill's formulation of CHSH that easily shows how the dependency works. http://arxiv.org/abs/1207.5103

$ + + - \leq 2$

And that formulation is absolutely true and mathematically impossible to violate. We can see that the A in AB is the same A as the A in AB'; and so forth... Now, in your simulation the A for E11 is not necessarily the same as the A in E12, etc.
FrediFizzx
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### Re: Evidence that QM does not violate Bell's inequalities

FrediFizzx wrote:
Heinera wrote:
FrediFizzx wrote:It doesn't matter if they all depend on the hidden variables. You are running four different independent experiments. One for E11, one for E12, one for E21 and one for E22.

But again, they are not independent. Why do you think they are independent?

Ironically it is Gill's formulation of CHSH that easily shows how the dependency works. http://arxiv.org/abs/1207.5103

$ + + - \leq 2$

And that formulation is absolutely true and mathematically impossible to violate. We can see that the A in AB is the same A as the A in AB'; and so forth... Now, in your simulation the A for E11 is not necessarily the same as the A in E12, etc.

As you know, Gill makes the same mistake as Heinera. And we now see that Abramsky and Hardy also make the same mistake in their paper, albeit in a different form.

I have exposed this mistake in my own way in this paper. I show that the replacement of Eq. (12) with Eq. (13) in the manner of Gill et al. is illegitimate, because the variables A, A' etc. involved are not on equal statistical and geometrical footings. And consequently Eq. (13) is simply wrong. The replacement can be made, however, if A and A' are standardized and put on equal statistical and geometrical footings. But then the corresponding A and A' become bivectors, and must be analyzed using the rules of geometric algebra. When this is done correctly, as in Eq. (18) of my paper, then the correct physical bound of ${2\sqrt{2}}$ on the four CHSH expectation values follows at once, as demonstrated in the derivation of Eq. (26) in the paper. The torsion within S^3 plays the key physical role in this derivation, as shown in the paper.
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### Re: Evidence that QM does not violate Bell's inequalities

"And that formulation is absolutely true and mathematically impossible to violate. We can see that the A in AB is the same A as the A in AB'; and so forth... Now, in your simulation the A for E11 is not necessarily the same as the A in E12, etc."

That's what local hidden variables does for you. Without LHV there is no dependence whatsoever. In QM there is no dependence. But if LHV is true, there is dependence, at the "hidden" level.

In a computer simulation of a LHV model, A = A(a, lambda) etc, so A, A', B, B' do all four "exist" simultaneously, whether or not you actually evaluate all four in each trial of each of the four experiments. Doing the 11 experiment means that we choose only to look at A and B. But the 12, 21, and 22 experiments are being effectively done in the hidden layer, simultaneously.

Now you just need some basic statistical thinking to tie everything together. A sample average is close to a population mean value. Unfortunately, elementary statistical thinking is not taught to physicists nor, even, to most mathematicians. Rutherford saud "if you need statsistics, you did the wrong experiment". This is the kind of thinking which blocks understanding of Bell and all that.
Guest

### Re: Evidence that QM does not violate Bell's inequalities

Guest wrote:"And that formulation is absolutely true and mathematically impossible to violate. We can see that the A in AB is the same A as the A in AB'; and so forth... Now, in your simulation the A for E11 is not necessarily the same as the A in E12, etc."

That's what local hidden variables does for you. Without LHV there is no dependence whatsoever. In QM there is no dependence. But if LHV is true, there is dependence, at the "hidden" level.

In a computer simulation of a LHV model, A = A(a, lambda) etc, so A, A', B, B' do all four "exist" simultaneously, whether or not you actually evaluate all four in each trial of each of the four experiments. Doing the 11 experiment means that we choose only to look at A and B. But the 12, 21, and 22 experiments are being effectively done in the hidden layer, simultaneously.

Now you just need some basic statistical thinking to tie everything together. A sample average is close to a population mean value. Unfortunately, elementary statistical thinking is not taught to physicists nor, even, to most mathematicians. Rutherford saud "if you need statsistics, you did the wrong experiment". This is the kind of thinking which blocks understanding of Bell and all that.

What you have said is the kind of thinking that does block the proper understanding of Bell. It is called "goal post shifting" when the QM experiments use this logical inequality,

$\left |\sum_{i}E_{i} \right | \leq N$ (independent terms)

Instead of the one imposed for LHV models,

$\left |\sum_{i}E_{i} \right | \leq N -2$ (with dependent terms)

See previous discussion about Hardy's logical Bell inequalities in this thread. It is impossible for anything to violate either inequality. Not really fair is it?
FrediFizzx
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### Re: Evidence that QM does not violate Bell's inequalities

Guest wrote:Now you just need some basic statistical thinking to tie everything together. A sample average is close to a population mean value. Unfortunately, elementary statistical thinking is not taught to physicists nor, even, to most mathematicians. Rutherford saud "if you need statsistics, you did the wrong experiment". This is the kind of thinking which blocks understanding of Bell and all that.

Indeed. Basic statistical thinking is all that is required to tie everything together. But even the so-called "mathematical statistician", as Richard Gill claims to be for example, is utterly incapable of such "elementary statistical thinking." He continues to make one silly mistake after another for years without understanding basic physics, before finally realizing his silly mistakes. This is just one example of someone who has maneuvered himself to political influence in the community by dirty tricks. To see how to apply statistics correctly to the physical problem at hand, see my comments above. One needs to understand basic physics to do "statistics":

Joy Christian wrote:I have exposed this mistake in my own way in this paper. I show that the replacement of Eq. (12) with Eq. (13) in the manner of Gill et al. is illegitimate, because the variables A, A' etc. involved are not on equal statistical and geometrical footings. And consequently Eq. (13) is simply wrong. The replacement can be made, however, if A and A' are standardized and put on equal statistical and geometrical footings. But then the corresponding A and A' become bivectors, and must be analyzed using the rules of geometric algebra. When this is done correctly, as in Eq. (18) of my paper, then the correct physical bound of ${2\sqrt{2}}$ on the four CHSH expectation values follows at once, as demonstrated in the derivation of Eq. (26) in the paper. The torsion within S^3 plays the key physical role in this derivation, as shown in the paper.
Joy Christian
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### Re: Evidence that QM does not violate Bell's inequalities

FrediFizzx wrote:
Heinera wrote:
FrediFizzx wrote:It doesn't matter if they all depend on the hidden variables. You are running four different independent experiments. One for E11, one for E12, one for E21 and one for E22.

But again, they are not independent. Why do you think they are independent?

Ironically it is Gill's formulation of CHSH that easily shows how the dependency works. http://arxiv.org/abs/1207.5103

$ + + - \leq 2$

And that formulation is absolutely true and mathematically impossible to violate. We can see that the A in AB is the same A as the A in AB'; and so forth... Now, in your simulation the A for E11 is not necessarily the same as the A in E12, etc.

Yes, but the fact that the the A for E11 is not necessarily the same as the A in E12 is due to the property that in my simulation, A depends on the settings in both wings of the experiment (nonlocality). If it didn't (i.e. the model was local), then the two A would be the same, and the inequality would apply.
Heinera

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### Re: Evidence that QM does not violate Bell's inequalities

Heinera wrote:If it didn't (i.e. the model was local), then the two A would be the same, and the inequality would apply.

Not without first making an ad hoc replacement of

< A B > + < A B' > + < A' B > - < A' B' >

with

< A B + A B' + A' B - A' B' >

in the middle of the derivation, which is an illegitimate mathematical step within the physical context of the EPR-B experiments, as I explained above:

Joy Christian wrote:I have exposed this mistake in my own way in this paper. I show that the replacement of Eq. (12) with Eq. (13) in the manner of Gill et al. is illegitimate, because the variables A, A' etc. involved are not on equal statistical and geometrical footings. And consequently Eq. (13) is simply wrong. The replacement can be made, however, if A and A' are standardized and put on equal statistical and geometrical footings. But then the corresponding A and A' become bivectors, and must be analyzed using the rules of geometric algebra. When this is done correctly, as in Eq. (18) of my paper, then the correct physical bound of ${2\sqrt{2}}$ on the four CHSH expectation values follows at once, as demonstrated in the derivation of Eq. (26) in the paper. The torsion within S^3 plays the key physical role in this derivation, as shown in the paper.
Joy Christian
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