## The simplest illustration of Bell's error

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

### The simplest illustration of Bell's error

In a recent thread with Justo, I mentioned this problem but only in passing. Here I want to highlight it because, by itself, it shows Bell's faulty logic right from his original paper. I don't know if anyone has described this problem before and I have not come across any attempt in Bell's other papers to address the problem. In any case, the fact that such a clear error exists in Bell's original paper should be a red flag for anyone who takes him seriously. Now unto the argument.

In his paper, Bell is dealing with correlations between pairs of particles formed in the singlet state and moving in opposite directions towards measurement stations and those correlations are specified as the average of the paired product of outcomes observed at those stations $P(a,b)$.

Note that $P(a,b)$ is the correlation between pairs of particles in the singlet state. According to QM, $P(a,b) = -a \cdot b$, and $P(a,c) = -a \cdot c$ and obviously $P(b,c) = -b \cdot c$. This is not controversial at all and perfectly above board.

But this is where Bell makes the mistake. He derives his inequality by starting with the expression:

$P(a,b) - P(a,c)$

Note that here we have two series of particles in the singlet state. The first series measured at $P(a,b)$ and the second measured at $P(a,c)$. Obviously, the QM prediction for this expression is $P(a,b) - P(a,c) = - a \cdot b + a \cdot c$. However, Bell then makes a massive blunder! He uses some arithmetic operations from equations 14a to 15 which effectively amount to calculating the correlation between one member of the singlet pairs used to obtain $P(a,b)$ and one member of the distinct set of singlet pairs used to obtain $P(b,c)$. This is obvious from his calculations. the $P(b,c)$ appearing in equation (15), is not simply another measurement performed on yet a different distinct series of particle pairs in the singlet state which should have a QM prediction of $P(b,c) = -b \cdot c$. Rather $P(b,c)$ in Bell's equation (15) is a "franken-correlation", reassembled using parts of $P(a,b)$ and parts of $P(a,c)$ as is obvious from the arithmetic preceding equation 15. For that reason, let us identify it as $P_f(a,b)$.

In other words, $P_f(a,b)$ is the correlation you would expect between one member of a singlet pair and one member of a different singlet pair. Yet, in equation (22), Bell uses the same QM prediction $P_f(b,c) = P(b,c) = -b \cdot c$, that would be expected for a distinct set of particle pairs in the singlet state.

It turns out QM has something to say about Bell's "franken-correlation" $P_f(b,c)$. According to QM, $P_f(b,c) = 0$.

Using the correct QM prediction for $P_f(b,c) = 0$. Bell's theorem can't be derived. Specifically, the proof of the contradiction fails and there is no "violation" by QM.
minkwe

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### Re: The simplest illustration of Bell's error

Here is a link to that paper by Bell.

https://journals.aps.org/ppf/abstract/1 ... zika.1.195
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FrediFizzx
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### Re: The simplest illustration of Bell's error

Let's just forget about QM for a while. The CHSH version of the theorem reads "for any LHV model, the CHSH expression has an upper bound of 2 in the limit as N goes to infinity." No mention of QM here. In fact the first version of the theorem stems from Boole, who obviously knew nothing about QM.

Let us propose a similar, but simpler theorem: "For the CHSH urn experiment¹, the CHSH expression has an upper bound of 2 in the limit as N goes to infinity."

True or false? If true, how do we prove it?

(The algebraic upper bound is of course 4, since we here have eight datasets grouped into four pairs.)

¹) Explained elsewhere on this forum
Heinera

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### Re: The simplest illustration of Bell's error

Heinera wrote:Let's just forget about QM for a while. The CHSH version of the theorem reads "for any LHV model, the CHSH expression has an upper bound of 2 in the limit as N goes to infinity." No mention of QM here. In fact the first version of the theorem stems from Boole, who obviously knew nothing about QM.

Let us propose a similar, but simpler theorem: "For the CHSH urn experiment¹, the CHSH expression has an upper bound of 2 in the limit as N goes to infinity."

True or false? If true, how do we prove it?

(The algebraic upper bound is of course 4, since we here have eight datasets grouped into four pairs.)

¹) Explained elsewhere on this forum

What is the hidden variable for the urn model? It doesn't really apply to local HV models, does it? Besides that, we have a LHV model that thoroughly shoots down Bell-CHSH.
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FrediFizzx
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### Re: The simplest illustration of Bell's error

FrediFizzx wrote:
Heinera wrote:Let's just forget about QM for a while. The CHSH version of the theorem reads "for any LHV model, the CHSH expression has an upper bound of 2 in the limit as N goes to infinity." No mention of QM here. In fact the first version of the theorem stems from Boole, who obviously knew nothing about QM.

Let us propose a similar, but simpler theorem: "For the CHSH urn experiment¹, the CHSH expression has an upper bound of 2 in the limit as N goes to infinity."

True or false? If true, how do we prove it?

(The algebraic upper bound is of course 4, since we here have eight datasets grouped into four pairs.)

¹) Explained elsewhere on this forum

What is the hidden variable for the urn model? It doesn't really apply to local HV models, does it? Besides that, we have a LHV model that thoroughly shoots down Bell-CHSH.
.

I would really appreciate it everyone just tried to answer the actual yes or no question The urn experiment is an experiment. It is described here.
Heinera

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### Re: The simplest illustration of Bell's error

The troll decides to change the topic. He has nothing to say about this topic as usual.
minkwe

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### Re: The simplest illustration of Bell's error

minkwe wrote::shock: The troll decides to change the topic. He has nothing to say about this topic as usual.

Oops, I forgot to use @Heine instead of quoting.
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FrediFizzx
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### Re: The simplest illustration of Bell's error

@Heine Start a new topic if you are changing the topic.
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FrediFizzx
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### Re: The simplest illustration of Bell's error

minkwe wrote::shock: The troll decides to change the topic. He has nothing to say about this topic as usual.

This has got everything to do with the topic (and you would know it if you could answer my question). What I'm doing is trying to elicit "The simplest illustration of Bell's error." And I start with a very simple experiment.
Heinera

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### Re: The simplest illustration of Bell's error

@Heine He's not going to see your post so might as well start a new topic or explain in detail how it relates to this topic.
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FrediFizzx
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### Re: The simplest illustration of Bell's error

FrediFizzx wrote:@Heine He's not going to see your post so might as well start a new topic or explain in detail how it relates to this topic.
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Well then, I'll just assume everyone agrees that the CHSH urn experiment has an upper bound of 2 when N goes to infinity.

But the algebraic upper bound is clearly 4, since we have eight datasets.

And the restrictions put on the experiment are very benign; the four numbers on the slips can be anything (as long as they are each in the set {-1, +1}). Of the 16 possible combinations, we put no restriction whatsoever on how they are distributed or if there is any serial correlation for them.

So the question is now, how do we prove an upper bound of 2? When we have a proof for that result, we can go through minkwe's argument point by point.
Heinera

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### Re: The simplest illustration of Bell's error

@Heinera may have shifted from the original Bell inequality to CHSH but he is right. Bringing in the QM prediction only complicates the issue, the problem will be more clearly discussed within the hidden variables theory.
Justo

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### Re: The simplest illustration of Bell's error

Now, Bell's eq. (14) seems really funny to me. It only for half of the events. The other half would have to have a + sign in front of the integral. Add them together for all the events you get 0. P(a, b) = 0. So, to me his formulation fails right at eq. (14). Joy's formulation is much better.

${\cal E}({\mathbf{a}},\,{\mathbf{b}})\,=\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,{A}({\mathbf{a}},\,{\lambda}_k)\;{B}({\mathbf{b}},\,{\lambda}_k)\right]$

It doesn't go to 0 right off the bat.
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FrediFizzx
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### Re: The simplest illustration of Bell's error

Justo wrote:@Heinera may have shifted from the original Bell inequality to CHSH but he is right. Bringing in the QM prediction only complicates the issue, the problem will be more clearly discussed within the hidden variables theory.

And what has that got to do with this topic? Bell did not derive the CHSH in his original paper. This topic is about an obvious error in Bell's original paper. Please start a new thread.

Unless you guys are admitting that the error I described is present in Bell's original paper.
minkwe

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### Re: The simplest illustration of Bell's error

minkwe wrote:And what has that got to do with this topic?

As I said, that you brought in quantum mechanics.
Justo

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### Re: The simplest illustration of Bell's error

Justo wrote:
minkwe wrote:And what has that got to do with this topic?

As I said, that you brought in quantum mechanics.

What the heck is wrong with bringing in QM? Bell's theory is about comparing local HV models to QM.
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FrediFizzx
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### Re: The simplest illustration of Bell's error

FrediFizzx wrote:Now, Bell's eq. (14) seems really funny to me. It only for half of the events. The other half would have to have a + sign in front of the integral. Add them together for all the events you get 0. P(a, b) = 0. So, to me his formulation fails right at eq. (14). Joy's formulation is much better.

${\cal E}({\mathbf{a}},\,{\mathbf{b}})\,=\lim_{\,n\,\gg\,1}\left[\frac{1}{n}\sum_{k\,=\,1}^{n}\,{A}({\mathbf{a}},\,{\lambda}_k)\;{B}({\mathbf{b}},\,{\lambda}_k)\right]$

It doesn't go to 0 right off the bat.

Another problem with eq. (14) is if you make the substitutions from eq. (1) you get,

$P({\bf a},\,{\bf b}) = -1 (\pm 1)(\pm 1)= \mp 1$

So, besides what Michel mentions there are a lot of booboo's in Bell's first paper.
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FrediFizzx
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### Re: The simplest illustration of Bell's error

FrediFizzx wrote:
Justo wrote:
minkwe wrote:And what has that got to do with this topic?

As I said, that you brought in quantum mechanics.

What the heck is wrong with bringing in QM? Bell's theory is about comparing local HV models to QM.
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That you claim that his hidden variable theory does not imply the inequality he derived.That has nothing to with QM and bringing in QM into the discussion is unnessesary. It only distracts the main point.
In fact, if you want to shoot down Bell's theorem you only need a simulation that violates his inequality. It is not nessesary for your model to reproduce the QM predictions.
Justo

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### Re: The simplest illustration of Bell's error

Justo wrote:
minkwe wrote:And what has that got to do with this topic?

As I said, that you brought in quantum mechanics.

That's the topic. Bell made a mistake in the QM prediction. What do you mean I brought it in? It's not like you guys were talking about CHSH and I suddenly "brought in" QM. I started the topic to discuss Bell's error in proving his theorem using the QM prediction.
minkwe

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### Re: The simplest illustration of Bell's error

Justo wrote:That you claim that his hidden variable theory does not imply the inequality he derived.That has nothing to with QM and bringing in QM into the discussion is unnessesary. It only distracts the main point.
In fact, if you want to shoot down Bell's theorem you only need a simulation that violates his inequality. It is not nessesary for your model to reproduce the QM predictions.

Are you serious? You don't care if Bell used the wrong QM prediction to claim violation of the inequality? The claim in the first post of this thread is that QM does not violate the inequality contrary to Bell's claims. Why don't you engage with that argument?
minkwe

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