The simplest illustration of Bell's error

[Justo]

Please, you refuse to answer a simple question. Did Bell or did he not generate P(b,c) through what you call "elementary and valid mathematical operations " on ?? Did those mathematical operations generate P(b,c) from P(a,b)-P(a,c) or not?

Until you answer this question, the rest of what you write is unimportant.

I did not refuse to answer it. Of course, he does. That is what STEP 2 says.

So you agree that according to STEP 2, P(b,c) is constructed from part of P(a,b) and Part of P(a,c)?

Then what makes you think P(b,c) represents a singlet state at all?

Answer: The assumption made in STEP 1.

Now we already reach a rational disagreement with @Fred. He rejected STEP 1. It seems that it is hard to find a rational disagreement with you. Now is my turn to as a question: do you reject STEP 1 or STEP 2 or perhaps both?

[minkwe]

Before you answer my question above, remember what we discussed in the other thread about Scenario 4:

We have 2 pairs of experimenters, the first pair measure N particle pairs generated in the singlet state at settings and , while the second pair measure a different set of N particle pairs generated in the singlet state at settings and . We take all the data from those two experiments and place them on a 4xN spreadsheet with columns . We then use these columns of data calculate and . (assuming N is very large of course).

Now what Bell does to arrive at P(b,c) is essentially the same as calculating from our spreadsheet.

Please tell me what is the QM prediction for .

Your answer above implies that you think must represent a singlet state if and represent singlet states.

[minkwe]

Answer: The assumption made in STEP 1.

You really believe that? Step 1 is equation 14 which is simply the definition of P(a,b) as the average of the paired product of outcomes of measurements. Please could you spell out what assumption you think is embedded in this that mandates that P(b,c) is a singlet state.

[Justo]

Before you answer my question above, remember what we discussed in the other thread about Scenario 4:

We have 2 pairs of experimenters, the first pair measure N particle pairs generated in the singlet state at settings and , while the second pair measure a different set of N particle pairs generated in the singlet state at settings and . We take all the data from those two experiments and place them on a 4xN spreadsheet with columns . We then use these columns of data calculate and . (assuming N is very large of course).

Now what Bell does to arrive at P(b,c) is essentially the same as calculating from our spreadsheet.

Please tell me what is the QM prediction for .

According to STEP 1, the expression P(a,b) is the result of a large number of measurements jointly made by Alice and Bob on entangled particles. That is the only scenario that is supposed to exist, i.e., equation (15) contains the result of three different series of independent experiments.
Now, it seems that you reject that. So I will answer for you. You reject STEP 1. If you do that, then there is little else that can be discussed because I accept it.
Or perhaps you reject that mathemacial expressions can be manipulated according to laws of arithmetic. I do not know what you reject becouse you wouldn't say it.
My gues is that you reject that equation (14) can always be interpreted as representing a series of experiments performed on entanglet particles. That is why you want to reinterpet them differently once they appear in equation (15).

[minkwe]

According to STEP 1, the expression P(a,b) is the result of a large number of measurements jointly made by Alice and Bob on entangled particles.

So what?

That is the only scenario that is supposed to exist, i.e., equation (15) contains the result of three different series of independent experiments.

This is false. Please, I'm not stupid. Look at the arithmetic between equations 14 and 15. On the left-hand side you have P(a,b) - P(a,c). Those are different series of independent experiments. But P(b,c) is absolutely not. P(b,c) is not independent of those two. It was reassembled from the first two through factoring inside the integral on the RHS. This is blatantly obvious. Continuing to claim that equation 15 represents three different series of independent experiments is just wrong and you are smart enough to see that this is the case.

Now, it seems that you reject that. So I will answer for you. You reject STEP 1.

There is nothing in STEP 1 to reject. Step 1 is a simple definition of the correlation between particle pairs.

If you do that, then there is little else that can be discussed because I accept it.

What is it that you claim I've rejected. Spell it out.

Or perhaps you reject that mathemacial expressions can be manipulated according to laws of arithmetic.

Duh, I've done no such thing. Please read what I write carefully.

I do not know what you reject becouse you wouldn't say it.

Again you are not reading carefully, this is not the first time. I've been very clear about my claims. You don't have to agree but please don't misrepresent me, ..., again!

My gues is that you reject that equation (14) can always be interpreted as representing a series of experiments performed on entanglet particles.

Obviously, you haven't read anything I've written in this thread. I'm starting to see a pattern. I do not reject anything in Bell's paper until equation 22 where he makes the wrong substitution of QM predictions. You are not paying attention.

That is why you want to reinterpet them differently once they appear in equation (15).

Bah . You are the one re-interpreting equation (15). I do no such thing.

But we made some progress. You agreed that P(b,c) in equation 15 is a result of mathematical operations carried out on P(a,b) - P(a,c). I use the subscript "f" to distinguish the correlation measured on a distinct set of particles from the correlation obtained through mathematical operations carried out on P(a,b) - P(a,c). I call the former and then latter . The only disagreement I see remaining is that you believe:

according to QM

I claim that
according to QM

Are you able to at least acknowledge that this is the disagreement?

BTW you did not answer what you think the QM prediction for is. Note that P(b_1, c_2) is the result of elementary and valid mathematical operations on and . If you don't know the answer, maybe Richard can help. He gave the right answer in the other thread.

[Justo]

BTW you did not answer what you think the QM prediction for is. Note that P(b_1, c_2) is the result of elementary and valid mathematical operations on and . If you don't know the answer, maybe Richard can help. He gave the right answer in the other thread.

I did not answer because for me is irrelevant. You accuse me of not reading neither understanding what you are claiming. I could say exactly the same of you.
You do not interpret the mathematical operations that Bell made to obtain the equation (15). You are thinking of the values contained in P(a,b) and P(a,c) as directly representing experimental data. In that case you can not make P(b,c) appear from thin air, that is correct. I can understand that it may be misleading and confusing for someone that is not used to abstract thinking.
In that case and to avoid confusion, it is convenient to start with data of three different actual experiments from the beginning and use a different derivation from the one Bell used. I did just that in my paper but using the CHSH inequality, not the original inequality. The same method can be used with the original Bell inequality to avoid the "counterfactual claims", in case anybody is thinking of that and, on the other hand, to avoid the interpretation you are using.
Let me disagree with you. I do understand what you are claiming. It is you who do not understand what I am saying.

[minkwe]

BTW you did not answer what you think the QM prediction for is. Note that is the result of elementary and valid mathematical operations on and . If you don't know the answer, maybe Richard can help. He gave the right answer in the other thread.

I did not answer because for me is irrelevant. You accuse me of not reading neither understanding what you are claiming. I could say exactly the same of you.You do not interpret the mathematical operations that Bell made to obtain the equation (15).

It is relevant. And I interpret exactly the mathematical operations that Bell made to obtain equation (15). They are equivalent to the mathematical operations I used above to obtain .

You are thinking of the values contained in P(a,b) and P(a,c) as directly representing experimental data.

What else do they represent other than the average of the product of outcomes of measurements?

In that case you can not make P(b,c) appear from thin air, that is correct.

I did not claim that P(b,c) appeared from thin air. I said P(b,c) originated from P(a,b) - P(a,c). And you already agree with this point so I'm not sure why you are revisiting it.

I can understand that it may be misleading and confusing for someone that is not used to abstract thinking.

Are you talking about yourself here? Certainly, it's not me you are talking about.

In that case and to avoid confusion, it is convenient to start with data of three different actual experiments from the beginning and use a different derivation from the one Bell used.

You are dodging. I'm explaining to you an error in Bell's paper and you want to leave Bell's paper and talk about a different derivation? We can do that in a different thread once we are done with Bell's paper. And I will show you in that thread that the same error exists. Mark my words. For this thread, our focus is on Bell's paper. No dodging.

I do understand what you are claiming. It is you who do not understand what I am saying.

I doubt it. You just claimed a few posts ago that I reject Bell's equations 14 or 15. There's no way anyone who understands what I'm saying makes such a statement. But let us see if you now understand what I'm saying:

You agreed that P(b,c) in equation 15 is a result of mathematical operations carried out on P(a,b) - P(a,c). I use the subscript "f" to distinguish the correlation measured on a distinct set of particles from the correlation obtained through mathematical operations carried out on P(a,b) - P(a,c). I call the former and then latter . The only disagreement I see remaining is that

you believe:

according to QM

I claim that

according to QM

Are you able to at least acknowledge that this is the disagreement?

[FrediFizzx]

@minkwe Can you show why P_f(b, c) is zero according to QM?
.

[Justo]

@menkwe, that is why I don't want to discuss with you. It is impossible even to agree to disagree.

[minkwe]

@menkwe, that is why I don't want to discuss with you. It is impossible even to agree to disagree.

It's minkwe BTW. You've used every spelling on earth except the correct one . I'm asking you a simple question.

You agreed that P(b,c) in equation 15 is a result of mathematical operations carried out on P(a,b) - P(a,c). I use the subscript "f" to distinguish the correlation measured on a distinct set of particles from the correlation obtained through mathematical operations carried out on P(a,b) - P(a,c). I call the former and then latter . The only disagreement I see remaining is that

you believe:

according to QM

I claim that

according to QM

Are you able to at least acknowledge that this is the disagreement?[/quote]

[minkwe]

@minkwe Can you show why P_f(b, c) is zero according to QM?
.

Fred, it is very easy. There is no correlation between the measurements of two particles that are not a singlet pair. If you take one particle from a singlet pair and another particle from a different singlet pair, the two particles are not in a singlet pair so no correlation. Bell's arithmetic between equations 14 and 15 amounts to dissecting P(a,b) and P(a,c) and using the parts to reconstruct P(b,c ). That is why the QM prediction is 0.

[gill1109]

Look at the arithmetic between equations 14 and 15. On the left-hand side you have P(a,b) - P(a,c). Those are different series of independent experiments. But P(b,c) is absolutely not. P(b,c) is not independent of those two. It was reassembled from the first two through factoring inside the integral on the RHS. This is blatantly obvious. Continuing to claim that equation 15 represents three different series of independent experiments is just wrong and you are smart enough to see that this is the case.

The mathematical expression P(a, b) in the context you mention is not a “series of independent experiments”.

A(a, b) is just a mathematical function of two arguments a and b. I am abusing notation here, if I was pedantic I would say that “A” is a function and that involves specifying its domain and its range. “A(a, b)” would then just stand for the result of evaluating the function “A” at a particular point “(a, b)”, and it would just be the number +1 or the number -1.

So “P” is another function and “P(a, b)” is just a number

Assuming local realism, it is computed from some functions A(a, lambda) and B(b, lambda) and a probability measure rho (not depending on a or b) on a set which one might call Lambda. Assuming quantum mechanics and a particular state and particular measurements it is computed using some standard computations involving the complex Hilbert space C^2 otimes C^2, ie from a bunch of complex vectors of length 4 and complex 4x4 matrices. Or, if you believe Joy Christian, it is computed by some calculations involving an 8 dimensional real unital algebra related to certain real Clifford algebras. Or, if you follow Tim Palmer, Sabine Hossenfelder and Jonte Hance, it follows from yet other computations on yet other mathematical structures involving fractal sets and unusual topologies.

All those *theories* are supposed to predict certain things which can be measured in experiments. When we talk about experiments, P(a, b) is the average of a product of measurement outcomes +/-1 registered by two pieces of lab equipment on which there are dials allowing the experimenters to choose settings a and b.

In local realism, A(a, lambda) is supposed to represent the measurement outcome which comes about when Alice’s setting is “a” and “lambda” stands for the initial microscopic state of everything relevant in one trial of the whole experiment (measurement devices, source, and everything in between) at an initial time point *before* settings a and b are delivered to the two sets of apparatus.

I can go to the market and buy 5 apples. I can go to another market stall and buy 6 oranges. I now have 5 + 6 = 11 pieces of fruit for my fruit bowl at home. You can’t say that I’m *not allowed* to add “5” and “6” because they came from different experiments.

I can take a sample of married men, and compute their average height. I can take another sample of different size of married women and do the same. The difference is a decent estimator of the average difference in height of members of a married couple, provided the two original samples can be thought of as defining decent samples from the *same* population of married couples.

@minkwe Can you show why P_f(b, c) is zero according to QM?
.

Fred, it is very easy. There is no correlation between the measurements of two particles that are not a singlet pair. If you take one particle from a singlet pair and another particle from a different singlet pair, the two particles are not in a singlet pair so no correlation. Bell's arithmetic between equations 14 and 15 amounts to dissecting P(a,b) and P(a,c) and using the parts to reconstruct P(b,c ). That is why the QM prediction is 0.

This true but irrelevant.

I claim that

according to QM[/i]
Are you able to at least acknowledge that this is the disagreement?

I guess we could say that. I know that Pf is not P. It is only that Pf does not exist for me. It is your way of understanding Bell's derivation. Equation 14 is a mathematical expression that has a physical interpretation and you are changing that interpretation.

Justo is right.

[Justo]

@menkwe, that is why I don't want to discuss with you. It is impossible even to agree to disagree.

It's minkwe BTW. You've used every spelling on earth except the correct one . I'm asking you a simple question.

You agreed that P(b,c) in equation 15 is a result of mathematical operations carried out on P(a,b) - P(a,c). I use the subscript "f" to distinguish the correlation measured on a distinct set of particles from the correlation obtained through mathematical operations carried out on P(a,b) - P(a,c). I call the former and then latter . The only disagreement I see remaining is that

you believe:

according to QM

I claim that

according to QM

Are you able to at least acknowledge that this is the disagreement?

I guess we could say that.
I know that Pf is not P. It is only that Pf does not exist for me. It is your way of understanding Bell's derivation.
Equation 14 is a mathematical expression that has a physical interpretation and you are changing that interpretation.

[gill1109]

Sorry for the misprints at the beginning of my last post. It's true, I'm getting old (I'll turn 70 in a few days - on 9-11, to be precise).

Here's another try at the first couple of sentences:

P(a, b) is just a mathematical function of two arguments a and b.

I am abusing notation here, if I was pedantic I would say that “P” is a function and that involves specifying its domain and its range. “P(a, b)” would then just stand for the result of evaluating the function “P” at a particular point “(a, b)”, and it would just be a number between -1 and +1.

“A” is just another function and “A(a, lambda)” is just a number -1 or +1.

The important thing is that P(a, b) is *not* a “series of independent experiments”!

One might like to get an estimate of a correlation by performing a series of experiments. Alternatively, one might use some mathematical physical theory to come up with a guess of what it might be.

NB, the series of experiments need not be *independent* experiments. It is sufficient that one chooses settings by *independent fair coin tosses*. That's enough independence and randomness to get valid p-values for testing local realism versus quantum mechanics.

[Mikko]

I'm amazed that a whole bunch of learned people can't follow a simple argument. Richard, Justo, Mikko etc please answer one question. And don't go off on a tangent talking about CHSH and Bell's other papers. Please focus on just Bell's 1964 paper and answer this one question:

In Equation 14a, Bell has P(a,b) and P(a,c). There is no P(b,c) anywhere in that expression. Then all of a sudden, P(b,c) appears in equation 15.

Where did P(b,c) come from?

The answer is very easy. Just look at the arithmetic between eq 14 and eq 15. The only thing in doubt is whether any of you are intellectually honest enough to admit it.

As stated on the line before equation 15, from the previous unnumbered equation, using the definition of P as rewritten in equation 14.

[gill1109]

I'm amazed that a whole bunch of learned people can't follow a simple argument. Richard, Justo, Mikko etc please answer one question. And don't go off on a tangent talking about CHSH and Bell's other papers. Please focus on just Bell's 1964 paper and answer this one question:

In Equation 14a, Bell has P(a,b) and P(a,c). There is no P(b,c) anywhere in that expression. Then all of a sudden, P(b,c) appears in equation 15.

Where did P(b,c) come from?

The answer is very easy. Just look at the arithmetic between eq 14 and eq 15. The only thing in doubt is whether any of you are intellectually honest enough to admit it.

As stated on the line before equation 15, from the previous unnumbered equation, using the definition of P as rewritten in equation 14.

Indeed Bell's equation (2) is the place where he originally defines a function P, I'll call it P_LHV, of two arguments. In the rest of the paper he is talking about what properties such a function would have.

If he had been a somewhat pedantic mathematician, he would have written next to equation (2): Let us define the function P_LHV from ... to ... by P_LHV(a, b) = ... for all a, b in ...

The paper explains what properties would necessarily hold for the correlations in the EPR-B experiment, if LHV was true. ie, if the actual correlations were equal to P_LHV for some functions A and B and probability measure rho.

[minkwe]

Sorry for the misprints at the beginning of my last post. It's true, I'm getting old (I'll turn 70 in a few days - on 9-11, to be precise).

Here's another try at the first couple of sentences:

P(a, b) is just a mathematical function of two arguments a and b.

I am abusing notation here, if I was pedantic I would say that “P” is a function and that involves specifying its domain and its range. “P(a, b)” would then just stand for the result of evaluating the function “P” at a particular point “(a, b)”, and it would just be a number between -1 and +1.

“A” is just another function and “A(a, lambda)” is just a number -1 or +1.

The important thing is that P(a, b) is *not* a “series of independent experiments”!

One might like to get an estimate of a correlation by performing a series of experiments. Alternatively, one might use some mathematical physical theory to come up with a guess of what it might be.

NB, the series of experiments need not be *independent* experiments. It is sufficient that one chooses settings by *independent fair coin tosses*. That's enough independence and randomness to get valid p-values for testing local realism versus quantum mechanics.

You are confusing unrelated issues. P(a,b) is one experiment, and P(a,c) is an independent experiment. We are not talking about the independence between particle pairs within the measurements carried out for the P(a,b) experiment. We are talking about the fact that all the particle pairs use to measure P(a,b) are independent of all the particles used to measure P(a,c). Therefore P(a,b) and P(a,c) are measured on independent series of particle pairs.

@menkwe, that is why I don't want to discuss with you. It is impossible even to agree to disagree.

It's minkwe BTW. You've used every spelling on earth except the correct one . I'm asking you a simple question.

You agreed that P(b,c) in equation 15 is a result of mathematical operations carried out on P(a,b) - P(a,c). I use the subscript "f" to distinguish the correlation measured on a distinct set of particles from the correlation obtained through mathematical operations carried out on P(a,b) - P(a,c). I call the former and then latter . The only disagreement I see remaining is that

you believe:

according to QM

I claim that

according to QM

Are you able to at least acknowledge that this is the disagreement?

I guess we could say that.
I know that Pf is not P. It is only that Pf does not exist for me. It is your way of understanding Bell's derivation.
Equation 14 is a mathematical expression that has a physical interpretation and you are changing that interpretation.

Justo, I can't take you seriously after this response. You are utterly able to understand a simple argument and to understand a simple discussion. If you are unable to restate my position accurately, then you can't be expected to reasonably respond to it.

[minkwe]

I'm amazed that a whole bunch of learned people can't follow a simple argument. Richard, Justo, Mikko etc please answer one question. And don't go off on a tangent talking about CHSH and Bell's other papers. Please focus on just Bell's 1964 paper and answer this one question:

In Equation 14a, Bell has P(a,b) and P(a,c). There is no P(b,c) anywhere in that expression. Then all of a sudden, P(b,c) appears in equation 15.

Where did P(b,c) come from?

The answer is very easy. Just look at the arithmetic between eq 14 and eq 15. The only thing in doubt is whether any of you are intellectually honest enough to admit it.

As stated on the line before equation 15, from the previous unnumbered equation, using the definition of P as rewritten in equation 14.

Duh?! and where do those terms under the integral come from? I'm sure you will say from the numbered equation just before that line, and we can keep going until we arrive at P(a,b) - P(a,c).

The term P(b,c) comes from what Justo calls "elementary and valid mathematical operations on P(a,b) - P(a,c)". This is uncontested. The only remaining disagreement here is that you guys believe, P(b,c) pertains to a measurement on an independent series of particle pairs with the same standing as P(a,b) and P(a,c) while I believe P(b,c), is not independent of P(a,b) and P(a,c), having been generated using "elementary and valid mathematical operations on P(a,b) - P(a,c)".

We just need intellectually honest individuals to accept that this is the core of the disagreement.

[Justo]

I'm amazed that a whole bunch of learned people can't follow a simple argument. Richard, Justo, Mikko etc please answer one question. And don't go off on a tangent talking about CHSH and Bell's other papers. Please focus on just Bell's 1964 paper and answer this one question:

In Equation 14a, Bell has P(a,b) and P(a,c). There is no P(b,c) anywhere in that expression. Then all of a sudden, P(b,c) appears in equation 15.

Where did P(b,c) come from?

The answer is very easy. Just look at the arithmetic between eq 14 and eq 15. The only thing in doubt is whether any of you are intellectually honest enough to admit it.

As stated on the line before equation 15, from the previous unnumbered equation, using the definition of P as rewritten in equation 14.

Duh?! and where do those terms under the integral come from? I'm sure you will say from the numbered equation just before that line, and we can keep going until we arrive at P(a,b) - P(a,c).

The term P(b,c) comes from what Justo calls "elementary and valid mathematical operations on P(a,b) - P(a,c)". This is uncontested. The only remaining disagreement here is that you guys believe, P(b,c) pertains to a measurement on an independent series of particle pairs with the same standing as P(a,b) and P(a,c) while I believe P(b,c), is not independent of P(a,b) and P(a,c), having been generated using "elementary and valid mathematical operations on P(a,b) - P(a,c)".

We just need intellectually honest individuals to accept that this is the core of the disagreement.

minkwe, you accused before, more than once, of not reading and not understanding what you say. I don't like to accuse people when discussing ideas, I only mention it because you use that technic of discussion that I don't like. Besides If I had assumed your intellectual dishonesty I wouldn't be having this discussion.
The point is what I mentioned in STPEP 1 before. It is about the interpretation of equation (14) in Bell's paper. If you accept that whenever you have that equation it represents P(a,b), for any "a" and "b", then you just have to admit that equation (15) contains the results of three different series of independently performed experiments.
The only coherent way to claim that (15) does not represent three different series of independent experiments is by rejecting the interpretation of eq. (14) or rejecting the mathematical steps the lead to it, i.e., rejecting the laws of arithmetic.
That is what is called logical inference and is essential in scientific reasoning.

[FrediFizzx]

@Justo Well heck, I think eq. (14) is nonsense and I can further prove it if necessary.
.