The CHSH urn model

[gill1109]

Heinera proposed we look at the following toy problem:

The CHSH urn model is an urn with slips of paper, each containing four numbers -1/+1. Alice randomly (coin toss) picks one of two settings a1 or a2, and Bob randomly picks one of two settings b1 or b2. A slip is then drawn from the urn, and Alice records one of the first two numbers according to her setting a1 or a2, while Bob does the same with the last two numbers according to his setting b1 or b2. The slip is then put back into the urn. We also assume there is nothing spooky going on, so everything behaves according to the standard rules of chance.

When we perform this experiment, each trial results in a setting and an outcome for Alice and a setting and an outcome for Bob. After the experiment is done, we can split the collected data from the N trials into four groups according to the four possible pairs of settings chosen in each trial. For each group, one can then compute the correlation (meaning here: the average of the products).

This particular random experiment is called "the CHSH urn model" because many people believe that the results will satisfy the Bell-CHSH inequality.

One can of course easily simulate the model, too.

Let's discuss whether or not the Bell-CHSH inequality will hold, and let's do some situation experiments to test any claims which people make.

Let's also discuss whether or not this model has anything to do with real Bell experiments and with simulations of Bell experiments.

[FrediFizzx]

Irrelevant piece of junk.
.

[gill1109]

Here's a quickly written piece of R code which creates an urn by filling it with slips of paper each containing a number from 1 to 16. A sample is taken of size 1000, with replacement, from the urn. Alice and Bob each perform 1000 fair coin tosses. The sample of 1000 slips is read one by one. The number on each slip has 1 subtracted from it, it is then converted into binary. Alice's coin toss determines which of the first two bits go to make her outcome, Bob's coin toss determines which of the last two bits go to make his outcome. At this point the outcomes are 0 or 1. They are converted to -1 or +1 and then four correlations are computed.

Code: Select all

set.seed(110951)
urn <- sample(10:100, 16, replace = TRUE)
urn
slips <- sample(1:16, 1000, replace = TRUE, prob = urn)
library(MASS)
truehist(slips, h = 1)
barplot(urn)
a <- sample(c(1, 2), size = 1000, replace = TRUE)
head(a)
b <- sample(c(1, 2), size = 1000, replace = TRUE)
head(b)
library(R.utils)
slips.string <- intToBin(slips - 1)
head(slips.string)
x <- rep(0, 1000)
y <- rep(0, 1000)
for (i in 1:1000) {x[i] <- as.numeric(substr(slips.string[i], a[i], a[i]));
                  y[i] <- as.numeric(substr(slips.string[i], b[i]+2, b[i]+2))}
head(x)
head(y)
x <- 2 * x - 1
y <- 2 * y - 1
mean((x*y)[a == 1 & b == 1])
mean((x*y)[a == 1 & b == 2])
mean((x*y)[a == 2 & b == 1])
mean((x*y)[a == 2 & b == 2])


I published this on RPubs, https://rpubs.com/gill1109/CHSH_urn.
This particular time, the four correlations were

Code: Select all

> mean((x*y)[a == 1 & b == 1])
[1] 0.06463878
> mean((x*y)[a == 1 & b == 2])
[1] 0.1111111
> mean((x*y)[a == 2 & b == 1])
[1] -0.05791506
> mean((x*y)[a == 2 & b == 2])
[1] -0.2252964

I need to clean up the code and add a lot of comments.

[FrediFizzx]

@gill1109 Irrelevant junk!
.

[gill1109]

@gill1109 Irrelevant junk!
.

Fred, you did already say that quite a lot of times.

Michel suggested to start a new topic on this theme, which was brought up by Heinera. So I started a new topic.

[FrediFizzx]

@gill1109 Irrelevant junk!
.

Fred, you did already say that quite a lot of times.

Michel suggested to start a new topic on this theme, which was brought up by Heinera. So I started a new topic.

So what? It is still irrelevant junk.
.

[gill1109]

@gill1109 Irrelevant junk!
.

Fred, you did already say that quite a lot of times.
Michel suggested to start a new topic on this theme, which was brought up by Heinera. So I started a new topic.

So what? It is still irrelevant junk.

You are welcome to your opinion, Fred!
I won’t say the same thing about your program, don’t want to get another ban!

Thanks for the forum!

@Heinera: how should I fill the urn so that we get three correlations of 0.5 and one of -0.5?

@Fred: can you fill the urn so as to get three correlations above 0.5 and one below -0.5?
Can you rewrite my program in Mathematica?

@minkwe: please give us a Python version, should be faster than R

[minkwe]

Heinera proposed we look at the following toy problem:

The CHSH urn model is an urn with slips of paper, each containing four numbers -1/+1. Alice randomly (coin toss) picks one of two settings a1 or a2, and Bob randomly picks one of two settings b1 or b2. A slip is then drawn from the urn, and Alice records one of the first two numbers according to her setting a1 or a2, while Bob does the same with the last two numbers according to his setting b1 or b2. The slip is then put back into the urn. We also assume there is nothing spooky going on, so everything behaves according to the standard rules of chance.

When we perform this experiment, each trial results in a setting and an outcome for Alice and a setting and an outcome for Bob. After the experiment is done, we can split the collected data from the N trials into four groups according to the four possible pairs of settings chosen in each trial. For each group, one can then compute the correlation (meaning here: the average of the products).

This particular random experiment is called "the CHSH urn model" because many people believe that the results will satisfy the Bell-CHSH inequality.

One can of course easily simulate the model, too.

Let's discuss whether or not the Bell-CHSH inequality will hold, and let's do some situation experiments to test any claims which people make.

Let's also discuss whether or not this model has anything to do with real Bell experiments and with simulations of Bell experiments.

This is fiction. You have a single urn. This experiment bears no relationship to a Bell test experiment. It doesn't matter what many people believe. Many people also believe the Vaccines are bad for them but Ivermectin can cure COVID.

[gill1109]

(1) Let's discuss whether or not the Bell-CHSH inequality will hold, and let's do some situation experiments to test any claims which people make.
(2) Let's also discuss whether or not this model has anything to do with real Bell experiments and with simulations of Bell experiments.

This is fiction. You have a single urn. This experiment bears no relationship to a Bell test experiment. It doesn't matter what many people believe. Many people also believe the Vaccines are bad for them but Ivermectin can cure COVID.

Of course it is fiction. It’s a mathematical model of a real experiment that you could imagine performing in a classroom. Call it a thought-experiment, if you like.

According to you, the Einstein-Podolsky-Rosen paper is also fiction. Are you saying that it should never have been published in a respectable physics journal?

I think you are mistaken, Michel, in your claim that my simulation bears no relationship to a Bell test experiment. I think this experiment does bear a useful relationship to the present day generation of Bell test experiments. It most certainly bears a relationship to many Bell test simulations.

I will discuss this later. In the meantime, I want to improve and decorate my R program. I would be so happy if Mathematica and Python versions were also forthcoming, from people on this forum with the necessary programming skills. Ask me questions if there is something you don’t understand about my R code.

[Heinera]

@Heinera: how should I fill the urn so that we get three correlations of 0.5 and one of -0.5?

Of the 16 different types of slips, half of them will give a CHSH value of 2 if the urn only contains that one type of slip. A uniform distribution over those eight types of slips gives 0.5, -0.5, 0.5, 0.5 for the terms in the CHSH expression.

[Heinera]

This is fiction. You have a single urn.

It doesn't matter how many urns you have, as long as the results of Alice and Bob's coin tosses don't influence what urn is chosen.

If they do, it's either "spooky action at a distance," or retrocausality.

[FrediFizzx]

This is fiction. You have a single urn.

It doesn't matter how many urns you have, as long as the results of Alice and Bob's coin tosses don't influence what urn is chosen.

If they do, it's either "spooky action at a distance," or retrocausality.

It is beyond fiction. It is completely irrelevant nonsense!
.

[minkwe]

Of course it is fiction. It’s a mathematical model of a real experiment that you could imagine performing in a classroom. Call it a thought-experiment, if you like.

Yes it is a real experiment. An urn experiment. What is fiction is any implication that this has anything to do with Einstein-Podolsky-Rosen's paper or an EPR-Bohm type experiment.

Of course Bell's inequality or the CHSH is a tautology for a 4xN spreadsheet of outcomes. The inequalities are irrelevant for physics http://dx.doi.org/10.1166/qm.2014.1153

[gill1109]

Of course it is fiction. It’s a mathematical model of a real experiment that you could imagine performing in a classroom. Call it a thought-experiment, if you like.

Yes it is a real experiment. An urn experiment. What is fiction is any implication that this has anything to do with Einstein-Podolsky-Rosen's paper or an EPR-Bohm type experiment.

Of course Bell's inequality or the CHSH is a tautology for a 4xN spreadsheet of outcomes. The inequalities are irrelevant for physics http://dx.doi.org/10.1166/qm.2014.1153

Michel, take a look at *my* model. There is no 4xN spreadsheet of outcomes. There is a big connection with real EPR-Bohm type experiments as they are performed nowadays. I know that this topic is your hobby, not your day job, but you really should read some of the recent pro-Bell literature as well as anti-Bell literature. Your picture of EPR-Bohm type experiments belongs to the last century.

I have improved my simulation in a number of ways. As well as a randomly constituted urn I also use a cleverly filled urn which gave some quite exciting violations of CHSH at sample size 1000 as well as dull non-violations.
https://rpubs.com/gill1109/CHSH_urn2

Michel: please write this in Python for us.

John? Fred? Can we have a Mathematica version?

Michel: your link doesn't work.

Anyone like to guess what was the trick I used to create an optimal urn?

BTW, it's good that you emphasize that CHSH is a tautology for a 4xN spreadsheet of outcomes. Yes, of course it is.

[Heinera]

Of course Bell's inequality or the CHSH is a tautology for a 4xN spreadsheet of outcomes. The inequalities are irrelevant for physics http://dx.doi.org/10.1166/qm.2014.1153

This is a straw man argument. In the CHSH urn experiment, there is no 4xN spreadsheet of outcomes. There are four 2xn spreadsheets. That's the whole point.

[Joy Christian]

Of course Bell's inequality or the CHSH is a tautology for a 4xN spreadsheet of outcomes. The inequalities are irrelevant for physics http://dx.doi.org/10.1166/qm.2014.1153

This is a straw man argument. In the CHSH urn experiment, there is no 4xN spreadsheet of outcomes. There are four 2xn spreadsheets. That's the whole point.

In the real experiments, the absolute bound on four 2 x n spreadsheets of outcomes is 4, not 2. The bound of 4 has never been violated in any experiment (of course, it cannot be).
.

[Heinera]

Of course Bell's inequality or the CHSH is a tautology for a 4xN spreadsheet of outcomes. The inequalities are irrelevant for physics http://dx.doi.org/10.1166/qm.2014.1153

This is a straw man argument. In the CHSH urn experiment, there is no 4xN spreadsheet of outcomes. There are four 2xn spreadsheets. That's the whole point.

In the real experiments, the absolute bound on four 2 x n spreadsheets of outcomes is 4, not 2. The bound of 4 has never been violated in any experiment (of course, it cannot be).
.

And this is true for the CHSH urn experiment as well. The algebraic upper bound for the CHSH urn experiment is 4, not 2. Can easily be achieved with small N.

The bound of 2 is a statistical bound, for large N. It is important to understand the difference.

[Joy Christian]

Of course Bell's inequality or the CHSH is a tautology for a 4xN spreadsheet of outcomes. The inequalities are irrelevant for physics http://dx.doi.org/10.1166/qm.2014.1153

This is a straw man argument. In the CHSH urn experiment, there is no 4xN spreadsheet of outcomes. There are four 2xn spreadsheets. That's the whole point.

In the real experiments, the absolute bound on four 2 x n spreadsheets of outcomes is 4, not 2. The bound of 4 has never been violated in any experiment (of course, it cannot be).
.

And this is true for the CHSH urn experiment as well. The algebraic upper bound for the CHSH urn experiment is 4, not 2. Can easily be achieved with small N.

The bound of 2 is a statistical bound, for large N. It is important to understand the difference.

As if. There is no "large N" in experiments. The big claim made in the so-called "loophole-free" experiments in 2015, the number of events recorded was 256. So much for the "large N."
.

[gill1109]

Of course Bell's inequality or the CHSH is a tautology for a 4xN spreadsheet of outcomes. The inequalities are irrelevant for physics http://dx.doi.org/10.1166/qm.2014.1153

This is a straw man argument. In the CHSH urn experiment, there is no 4xN spreadsheet of outcomes. There are four 2xn spreadsheets. That's the whole point.

Heinera, you are not quite right. I would say it like this.

The data from the CHSH urn experiment can be placed in an (N rows) x (4 columns) spreadsheet. Each *row* contains Alice’s setting (“1” or “2”), Bob’s setting (“1” or ‘2”), Alice’s outcome (+/-1), Bob’s outcome (+/-1).
This can be split into four data-sets of sizes N11, N12, N21 and N22, each consisting of pairs of outcomes.
N11 + N12 + N21 + N22 = N

One computes 4 correlations. Each could in principle lie anywhere between -1 and +1. My simulations demonstrate that for large N, CHSH tends to hold though the errors are typically of the order of size 1 / sqrt N; quite rarely they can be pretty big.

There are theorems which express this rigorously and precisely. One such theorem was proven by me
https://arxiv.org/abs/1207.5103
Statistics, Causality and Bell's Theorem

Richard D. Gill
Bell's [Physics 1 (1964) 195-200] theorem is popularly supposed to establish the nonlocality of quantum physics. Violation of Bell's inequality in experiments such as that of Aspect, Dalibard and Roger [Phys. Rev. Lett. 49 (1982) 1804-1807] provides empirical proof of nonlocality in the real world. This paper reviews recent work on Bell's theorem, linking it to issues in causality as understood by statisticians. The paper starts with a proof of a strong, finite sample, version of Bell's inequality and thereby also of Bell's theorem, which states that quantum theory is incompatible with the conjunction of three formerly uncontroversial physical principles, here referred to as locality, realism and freedom.
https://projecteuclid.org/journals/stat ... TS490.full
Statistical Science 2014, Vol. 29, No. 4, 512-528
Google scholar: 66 citations

The Delft experiment had N = 245. That was not enough to provide convincing evidence. A 10 times larger sample could have got an approximately 5 sigma result, if they could have maintained those statistics in the long run. Square root of 10 is bigger than 3. Three times as small a standard error would give a pretty convincing result as long as the estimated correlations did not change much. But the true (large N) correlations could have resulted in S = 2, the experimental result could be (1 chance in 20) merely a statistical fluctuation.

[Heinera]

Of course Bell's inequality or the CHSH is a tautology for a 4xN spreadsheet of outcomes. The inequalities are irrelevant for physics http://dx.doi.org/10.1166/qm.2014.1153

This is a straw man argument. In the CHSH urn experiment, there is no 4xN spreadsheet of outcomes. There are four 2xn spreadsheets. That's the whole point.

Heinera, you are not quite right. I would say it like this.

The data from the CHSH urn experiment can be placed in an (N rows) x (4 columns) spreadsheet. Each *row* contains Alice’s setting (“1” or “2”), Bob’s setting (“1” or ‘2”), Alice’s outcome (+/-1), Bob’s outcome (+/-1).
This can be split into four data-sets of sizes N11, N12, N21 and N22, each consisting of pairs of outcomes.
N11 + N12 + N21 + N22 = N

I assume that by "4xN spreadsheet" @minkwe did not mean the 4xN spreadsheet you construct here, but a spreadsheet with four columns, two of which are actual outcomes, and the other two counterfactual outcomes. No need to construct such a spreadsheet when performing the CHSH urn experiment. Only actual outcomes need to be recorded, just as in any other physical experiment.