Bell & CHSH type inequalities and experiments

[minkwe]

[

Michel, everyone knows this! What you don't realize, is that this is not an issue. If you want to apply the CHSH bound to an experiment, you have to take two steps. The first step is, you assume that local realism is true. The second step, is that you take account of statistical error. The bound is no longer a deterministic bound. It's a statistical bound. It's a bound on the expectation value. Which we don't know. But sample averages are close to population mean values, if the sample is large and random. The N/4 runs on which a particular correlation is based, are a random sample of the N runs of the whole experiment, which is a random sample of the infinite ensemble of runs which we obtain when we imagine an experiment with infinitely many runs. (One run = one pair of particles).

Richard, you agreed with my answers to the three question, which means you agree that the terms in the CHSH are mutually dependent while those in the experiment are independent. Not only that, you also agree that the upper bound for independent terms is 4 not 2. You are then just confusing you self by talking about random sampling.

[gill1109]

S = E(a,b) - E(a,b') + E(a,b') + E(a',b')

It is truly astonishing that a bunch of mathematicians are having difficulty seeing clearly that if each of the terms in the above expression has bounds [-1,+1], then the expression has an upper bound of 4 when the terms are all independent. But if and only if the terms are mutually dependent, can the upper bound be reduced below 4.

Astonishing indeed.

Nobody has any problems with seeing this at all.

It's quite simply *irrelevant*.

Experimenters want to say something about population mean values ("ensemble averages"). They observe sample averages. They do statistics. A random subsample of a random sample is a random sample. It's a perfectly adequate basis for doing statistical inference. One must allow for statistical error, and one must allow for the small chance that the statistical error might be very large. So one cannot say anything for certain. But one can be bloody confident.

[gill1109]

Richard, you agreed with my answers to the three question, which means you agree that the terms in the CHSH are mutually dependent while those in the experiment are independent. Not only that, you also agree that the upper bound for independent terms is 4 not 2. You are then just confusing yourself by talking about random sampling.

No, I am confusing *you* by talking about random sampling.

But we have to talk about random sampling; otherwise, we can't do any experimental science at all.

Does the CERN experiment which found the Higgs boson actually prove nothing at all? Should we stop doing experiments in physics?

Well, the answer is that the CERN experiment proved nothing for sure, since the observed data could have occurred even if the Higgs boson didn't exist. The point is, that the observed data would have been extremely unlikely if the Higgs boson didn't exist, while it would be quite "normal" if it did exist.

[minkwe]

Richard, are the terms from experiment independent? Yes.
Richard, what is the correct upper bound for independent terms? 4

Now why would any reasonable person expect the experiment to have an upper bound of 2, and proclaim violation when they get 2 root 2???

Contrary to your claims, almost everybody has problems with that. It is so simple it is actually just as silly as von Neumann's error.

Random sampling makes sure your experimental terms are independent, guaranteeing your upper bound of 4. The suggestion that random sampling reduces the upper bound from 4 to 2 is just baloney.

[gill1109]

Richard, are the terms from experiment independent? Yes.
Richard, what is the correct upper bound for independent terms? 4

Now why would any reasonable person expect the experiment to have an upper bound of 2, and proclaim violation when they get 2 root 2???

Contrary to your claims, almost everybody has problems with that. It is so simple it is actually just as silly as von Neumann's error.

I already answered this question.

Reasonable people do not expect the experiment to have an upper bound of 2. I don't know where you got this crazy idea.

Reasonable people (excepting Joy Christian) expect that, were LHV actually true, then the observed CHSH based on four separate large samples, one for each correlation, would, with large probability, not be much larger than 2.

[minkwe]

Reasonable people do not expect the experiment to have an upper bound of 2. I don't know where you got this crazy idea.

Reasonable people know that if LHV were true, the observed CHSH would with large probability not be much larger than 2.

Reasonable people understand the meaning of upper bound. Reasonable people will not even have 2 in the above sentence. They would have 4 instead. Upper bounds are just that. They don't care about statistics and error. They are mathematically impossible to exceed. You are confusing mean values with upper bounds.

[gill1109]

Reasonable people do not expect the experiment to have an upper bound of 2. I don't know where you got this crazy idea.

Reasonable people believe that if LHV were true, the observed CHSH would with large probability not be much larger than 2.

Reasonable people understand the meaning of upper bound. Reasonable people will not even have 2 in the above sentence. They would have 4 instead.

You are the person who is repeatedly using the word "upper bound". I do not use that word in contexts where it does not hold.

Reasonable people believe that if LHV were true, the observed CHSH would with large probability not be much larger than 2. Where did I say anything about an upper bound?

In my paper I put a bound on the probability that the experimentally observed value of CHSH would be larger than 2 + epsilon, for any given epsilon > 0. e.g., epsilon = 0.1.

I did not put an upper bound on the experimentally observed value of CHSH. I put a bound on the probability that the experimentally observed value would be larger than 2.1, assuming that LHV is true. When N gets large that probability gets very small. Comprende?

[gill1109]

Reasonable people understand the meaning of upper bound. Reasonable people will not even have 2 in the above sentence. They would have 4 instead. Upper bounds are just that. They don't care about statistics and error. They are mathematically impossible to exceed. You are confusing mean values with upper bounds.

I am not interested in upper bounds. I do care about statistics and error.

Consider the following LHV model. lambda is chosen uniformly at random from the unit circle. The measurement outcomes for Alice are +/-1 depending on whether or not alpha and lambda fit in one semi-circle, and for Bob is -/+1 depending on whether beta and lambda fit in one semi-circle, respectively. This gives us the wonderful triangle-wave correlation function.

When we simulate a standard CHSH experiment (for the spin half case) on this model, we will see the "bound" 2 exceeded in about 50% of our experiments. Half the experiments will satisfy CHSH, half will violate it.

Agree?

But how many of those experiments will get a CHSH value bigger, say, than 2.4? Well, that depends on N. But if N is really huge the chance your experiment will give a value above 2.4 is bloody small.

Agree? You can run "epr-simple" with no rejections (no detection loophole) to verify my claim.

[minkwe]

Reasonable people do not expect the experiment to have an upper bound of 2. I don't know where you got this crazy idea.

Reasonable people believe that if LHV were true, the observed CHSH would with large probability not be much larger than 2.

Reasonable people understand the meaning of upper bound. Reasonable people will not even have 2 in the above sentence. They would have 4 instead.

You are the person who is repeatedly using the word "upper bound". I do not use that word in contexts where it does not hold.

Reasonable people believe that if LHV were true, the observed CHSH would with large probability not be much larger than 2. Where did I say anything about an upper bound?

The CHSH inequality is a statement about the upper bound of a relationship between 4 mutually dependent expectation values. It does not apply to independent terms. The inequality relating mutually independent expectation values has an upper bound of 4 and has never been violated by QM or any experiment, nor will it ever be.

This is the point I've been making in this thread. It now appears you actually agree with this. Again, you actually agree with the argument. You only think it is unimportant and irrelevant and everyone already knows and accepts it. I disagree with that view. I think it sinks Bells ship. Others can make up their own minds.

[gill1109]

Consider the following LHV model. lambda is chosen uniformly at random from the unit circle. The measurement outcomes for Alice and Bob are sign(cos(lambda - alpha)) and - sign(cos(lambda - beta)) respectively. This gives us the wonderful triangle-wave correlation function.

When we simulate a standard CHSH experiment (for the spin half case) on this model, we will see the "bound" 2 exceeded in about 50% of our experiments. Half the experiments will satisfy CHSH, half will violate it.

Agree?

But how many of those experiments will get a CHSH value bigger, say, than 2.4? Well, that depends on N. But if N is really huge the chance your experiment will give a value above 2.4 is bloody small.

Agree?

You can run "epr-simple" with no rejections (no detection loophole) to verify my claim.

That's why your problems do not sink Bell's ship. Experimenters are not interested in deterministic (certain) bounds on experimental averages. They are interested in bounds on expectation values. They need and use statistics and probability.

It's good that things are this way, or there would be no experimental science at all.

[gill1109]

Please copy and paste the following code into the R console window a number of times.

Code: Select all

N <- 10000

lambda <- runif(N, 0, 360)
alpha <- 0
beta <- 45
A <- sign(cos((alpha - lambda)*pi/180))
B <- - sign(cos((beta - lambda)*pi/180))
E11 <- mean(A * B)

lambda <- runif(N, 0, 360)
alpha <- 0
beta <- 135
A <- sign(cos((alpha - lambda)*pi/180))
B <- - sign(cos((beta - lambda)*pi/180))
E12 <- mean(A * B)

lambda <- runif(N, 0, 360)
alpha <- 90
beta <- 45
A <- sign(cos((alpha - lambda)*pi/180))
B <- - sign(cos((beta - lambda)*pi/180))
E21 <- mean(A * B)

lambda <- runif(N, 0, 360)
alpha <- 90
beta <- 135
A <- sign(cos((alpha - lambda)*pi/180))
B <- - sign(cos((beta - lambda)*pi/180))
E22 <- mean(A * B)

E12 - E11 - E21 - E22


Tell me what you see, please.

Obviously, if you delete the three lines which make new values of the hidden variable lambda, only keeping the line doing that for the first correlation, so that the same hidden variables are used for each of the four correlations, then we will indeed have a deterministic bound of 2.

But with a new sample for each correlation, we are not far off. Almost every time we get a value less than 2.1

[minkwe]

Sorry for being dense, but I guess your answer to my question is that it is uncontroversial that none of the four correlations would change?

Heinera,
S = E(a,b) - E(a,b') + E(a,b') + E(a',b')
What in your opinion is the upper bound for S when the expectation values are independent and each has bounds [-1,+1]?

[minkwe]

Please copy and paste the following code into the R console window a number of times.

Code: Select all

N <- 10000

lambda <- runif(N, 0, 360)
alpha <- 0
beta <- 45
A <- sign(cos((alpha - lambda)*pi/180))
B <- - sign(cos((beta - lambda)*pi/180))
E11 <- mean(A * B)

lambda <- runif(N, 0, 360)
alpha <- 0
beta <- 135
A <- sign(cos((alpha - lambda)*pi/180))
B <- - sign(cos((beta - lambda)*pi/180))
E12 <- mean(A * B)

lambda <- runif(N, 0, 360)
alpha <- 90
beta <- 45
A <- sign(cos((alpha - lambda)*pi/180))
B <- - sign(cos((beta - lambda)*pi/180))
E21 <- mean(A * B)

lambda <- runif(N, 0, 360)
alpha <- 90
beta <- 135
A <- sign(cos((alpha - lambda)*pi/180))
B <- - sign(cos((beta - lambda)*pi/180))
E22 <- mean(A * B)

E12 - E11 - E21 - E22


Tell me what you see, please.

Richard this is off topic and I'm ignoring it. Like I mentioned, this thread is not about simulations

[gill1109]

Richard this is off topic and I'm ignoring it. Like I mentioned, this thread is not about simulations

This thread is about CHSH type inequalities and experiments. If you want to talk about experiments, you'll need statistics.

I am right on topic, and you are wilfully blind.

I did not ask you to do a simulation. I asked you to do the experiment. Please do it a small number of times (say 20 times). Please observe the 20 experimental outcomes, and ponder on what you see.

It's called Science. Learning from observing the world. Learning about CHSH experiments. Why Bell's boat has not been sunk by your issue, after all.

You claim in this thread that Bell's boat is sunk by your issue. I am proving to you that you are wrong.

[Heinera]

The CHSH inequality is a statement about the upper bound of a relationship between 4 mutually dependent expectation values. It does not apply to independent terms.

But a LHV model can't have independent terms. That's Bell's theorem. It says that for such a model, the dependence must in fact be so strong that CHSH can't exceed 2. In quantum mechanics, we see results that exceed that, but there's obviously still some dependence, since we don't see 4.

Now, if you think QM results can be mimicked by a LHV model, why don't you try to construct a LHV model (without data rejection) that consistently produces a CHSH value larger than two?

[minkwe]

Richard,
The main point I have been making in this thread is the following:

The CHSH inequality is a statement about the upper bound of a relationship between 4 mutually dependent expectation values. It does not apply to independent terms. The inequality relating mutually independent expectation values has an upper bound of 4 and has never been violated by QM or any experiment, nor will it ever be.

I'm happy you agree with this argument. We can discuss simulations in another thread.

[minkwe]

But a LHV model can't have independent terms.

This is just rubbish.

[gill1109]

Heinera,
S = E(a,b) - E(a,b') + E(a,b') + E(a',b')
What in your opinion is the upper bound for S when the expectation values are independent and each has bounds [-1,+1]?

Heinera, the correct answer is 4.

But nobody is interested in the smallest deterministic upper bound to the experimentally observed E(a,b) - E(a,b') + E(a,b') + E(a',b').

The experiment gives us an *estimate*. It has some statistical error. It is just a decent guess of what we are truly interested in: namely what we would have seen if we had done infinitely many runs, not only 100 000 (or whatever). The guess will be off by some small amount, positive or negative, with large probability. There is a tiny probability that it is wildly wrong. The probability that it is wildly wrong is so small that we may neglect it.

That's why they say the Higgs boson was found. They trusted that the random error was not improbably huge. But the data is perfectly possible even if there is no Higgs boson. It's just extremely unlikely.

There is no physics without statistics.

[gill1109]

Richard,
The main point I have been making in this thread is the following:

The CHSH inequality is a statement about the upper bound of a relationship between 4 mutually dependent expectation values. It does not apply to independent terms. The inequality relating mutually independent expectation values has an upper bound of 4 and has never been violated by QM or any experiment, nor will it ever be.

I'm happy you agree with this argument. We can discuss simulations in another thread.

You claimed in this thread that this argument sank Bell's boat. I am proving to you, by actually doing the experiment we are talking about, that Bell's boat is not sunk. So I am not off topic. This is not something for a new thread. This is the punch-line for this thread. The news which you don't want to know.

I'm talking about CHSH type experiments. There are no experiments without statistics.

I asked you to do the CHSH experiment 20 times, and learn from what you see. Please do it and then start a new thread to discuss your observations.

This is not about a simulation. It's about the CHSH experiment. It's about what the CHSH inequality has to say about experiments.

[gill1109]

Let me quote from the initial posting in this thread on "Bell & CHSH type inequalities and experiments"

I would like to start a focused and detailed discussion of Bell & CHSH type inequalities and their relevance for Aspect-type experiments.
...
In a genuine CHSH experiment, the terms must be related to each other in the same way as the terms of the CHSH inequality are related to each other. No such experiment has ever been performed.

Your conclusion has just been disproved. The first part of your argument was correct, but the final deduction was false. You jumped to conclusions, because you are missing an essential part of all empirical science. It's also essential in drawing inference from CHSH experiments (which is what this thread is all about, right?).

Your conclusion is disproven by mathematics (in my paper, for instance), and it can also be proven experimentally. You are forgetting the absolutely essential role of statistics and probability in empirical science.