The CHSH urn model

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Re: The CHSH urn model

Post by gill1109 » Mon Sep 13, 2021 1:25 am

This RPubs document computes the optimal urn in the CHSH urn model.
https://rpubs.com/gill1109/optimalUrn

Just 8 slips of paper: "0000" "0100" "0110" "1110" "0001" "1001" "1011" "1111"; these are the outcomes of the semi-hidden variables x1, x2, y1, y2; outcomes coded 0/1 instead of -1/+1
The code *computes* those 8 four-bit strings.

The code is a bit clumsy because the easy way to create this is using the order x1, y1, x2, y2 (thought of as lying on the corners of a square starting at top left hand corner, and going clockwise). The urn is optimal for getting high correlations in the 21, 11, 12 settings; high and negative in the 22 setting. For large N those four correlations will be close to +/- 0.5.

Recall: in the model one picks one slip of paper. One independently picks a and b, = 1 or 2. One then reads off the chosen xa and yb.
You can think of half of the slip of paper going to Alice, the other half to Bob; and Alice and Bob can choose their settings independently of one another.
But they don't have to.

Put the slip back in the urn, repeat.

Re: The CHSH urn model

Post by Heinera » Fri Sep 10, 2021 2:16 am

I don't think it is important how people convinces themselves that a CHSH urn experiment has a long run upper bound of 2 for CHSH expression, and can thus not reproduce the expected results of a quantum experiment. One could be convinced by some mathematical argument, or alternatively by doing a lot of simulations with different distributions of the slips.

The important point is that if you are convinced the statistical upper bound is two for the CHSH urn experiment, you also believe Bell's theorem to be true, because this is Bell's theorem. Any LHV model can be reduced to a CHSH urn experiment, as explained here:

viewtopic.php?f=6&t=489

Re: The CHSH urn model

Post by gill1109 » Fri Sep 10, 2021 2:14 am

Heinera wrote:
gill1109 wrote:
Heinera wrote:
minkwe wrote: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.


Hm, maybe. I think that @minkwe is confused. I think he’s thinking of the Nx4 matrix of factual and counterfactual outcomes (no settings) which appears in my Statistical Science paper.

In my simulation there is, in effect,
- an N x 4 spreadsheet of sets of four counterfactual outcomes X1, X2, Y1, Y2
- an N x 2 spreadsheet of settings A, B
- an N x 4 spreadsheet of observed Dara (settings and outcomes) A, B, X, Y

The relationships between them are that X = X_A, Y = Y_B; and the settings are independent of the quadruplets of “potential outcomes”

Obviously, CHSH holds for the counterfactuals
The point is to show that it holds, in expectation value, for the observed data (conditional on the subsample sizes Nij)
That’s all very easy.
The tricky thing is to give probability bounds on “statistical fluctuations”. That’s what I did, in three different ways over the years, in three different papers; and my bounds have since been improved (Peter Bierhorst; and the Delft people; and others).

*****
Added later, response to *next* post:

Heinera wrote:I don't think it is important how people convinces themselves that a CHSH urn experiment has a long run upper bound of 2 for CHSH expression, and can thus not reproduce the expected results of a quantum experiment. One could be convinced by some mathematical argument, or alternatively by doing a lot of simulations with different distributions of the slips.

The important point is that if you are convinced the statistical upper bound is two for the CHSH urn experiment, you also believe Bell's theorem to be true, because this is Bell's theorem. Any LHV model can be reduced to a CHSH urn experiment, as explained here:

viewtopic.php?f=6&t=489


Agreed!

Re: The CHSH urn model

Post by Heinera » Fri Sep 10, 2021 1:58 am

gill1109 wrote:
Heinera wrote:
minkwe wrote: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.

Re: The CHSH urn model

Post by gill1109 » Fri Sep 10, 2021 1:49 am

Heinera wrote:
minkwe wrote: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.

Re: The CHSH urn model

Post by Joy Christian » Fri Sep 10, 2021 1:08 am

Heinera wrote:
Joy Christian wrote:
Heinera wrote:
minkwe wrote: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."
.

Re: The CHSH urn model

Post by Heinera » Fri Sep 10, 2021 12:24 am

Joy Christian wrote:
Heinera wrote:
minkwe wrote: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.

Re: The CHSH urn model

Post by Joy Christian » Fri Sep 10, 2021 12:07 am

Heinera wrote:
minkwe wrote: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).
.

Re: The CHSH urn model

Post by Heinera » Thu Sep 09, 2021 11:57 pm

minkwe wrote: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.

Re: The CHSH urn model

Post by gill1109 » Thu Sep 09, 2021 3:36 pm

minkwe wrote:
gill1109 wrote: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.

Re: The CHSH urn model

Post by minkwe » Thu Sep 09, 2021 3:03 pm

gill1109 wrote: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

Re: The CHSH urn model

Post by FrediFizzx » Thu Sep 09, 2021 8:27 am

Heinera wrote:
minkwe wrote: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!
.

Re: The CHSH urn model

Post by Heinera » Thu Sep 09, 2021 7:47 am

minkwe wrote: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.

Re: The CHSH urn model

Post by Heinera » Thu Sep 09, 2021 5:27 am

gill1109 wrote:@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.

Re: The CHSH urn model

Post by gill1109 » Thu Sep 09, 2021 12:22 am

minkwe wrote:
gill1109 wrote:(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.

Re: The CHSH urn model

Post by minkwe » Wed Sep 08, 2021 10:38 pm

gill1109 wrote: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.

Re: The CHSH urn model

Post by gill1109 » Wed Sep 08, 2021 8:21 pm

FrediFizzx wrote:
gill1109 wrote:
FrediFizzx wrote:@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. :mrgreen: :mrgreen: :mrgreen:

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! 8-) 8-) 8-)

@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

Re: The CHSH urn model

Post by FrediFizzx » Wed Sep 08, 2021 7:21 am

gill1109 wrote:
FrediFizzx wrote:@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. :mrgreen: :mrgreen: :mrgreen:
.

Re: The CHSH urn model

Post by gill1109 » Wed Sep 08, 2021 5:38 am

FrediFizzx wrote:@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.

Re: The CHSH urn model

Post by FrediFizzx » Wed Sep 08, 2021 4:37 am

@gill1109 Irrelevant junk!
.

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