Institutionalized Denial of the Disproof of Bell's Theorem

[Justo]

Sorry, I still do not understand. Alice tosses a fair coin and then chooses from an urn that contains two values, +1 and -1. After choosing her value she puts back the slip on the urn and now Bob does the same as Alice. Is that it?

No, they both toss fair coins (one for each), which tells them if they should record the first or the second number in their pair on the slip. Then they together draw a slip from the urn. If the slip contains e.g. [-1, -1, +1, -1], then Alice records one of -1 or -1 (the two first) depending on her coin toss (in this example the values are the same, so the coin toss doesn't matter), and Bob records one of +1 or -1 (the two last) depending on the outcome of his coin toss.

I suppose that all sexteen posibilities are equally probable in which I think that S=0. The general case for arbitrary probabilities would give but reduces to a tedious counting problem.

[Heinera]

Sorry, I still do not understand. Alice tosses a fair coin and then chooses from an urn that contains two values, +1 and -1. After choosing her value she puts back the slip on the urn and now Bob does the same as Alice. Is that it?

No, they both toss fair coins (one for each), which tells them if they should record the first or the second number in their pair on the slip. Then they together draw a slip from the urn. If the slip contains e.g. [-1, -1, +1, -1], then Alice records one of -1 or -1 (the two first) depending on her coin toss (in this example the values are the same, so the coin toss doesn't matter), and Bob records one of +1 or -1 (the two last) depending on the outcome of his coin toss.

I suppose that all sexteen posibilities are equally probable in which I think that S=0. The general case for arbitrary probabilities would give but reduces to a tedious counting problem.

Yes, any arbitrary distribution is allowed. And the exact same strategy used to prove this result can also be used to prove Bell's theorem¹. So the odd thing is that the Bell deniers usually think that this result holds true, but that Bell's theorem is false.

¹) Just replace the slips of paper with the vector [A(a1, lambda), A(a2, lambda), B(b1, lambda), B(b2, lambda)] and start drawing lambdas.

[Joy Christian]

Sorry, I still do not understand. Alice tosses a fair coin and then chooses from an urn that contains two values, +1 and -1. After choosing her value she puts back the slip on the urn and now Bob does the same as Alice. Is that it?

No, they both toss fair coins (one for each), which tells them if they should record the first or the second number in their pair on the slip. Then they together draw a slip from the urn. If the slip contains e.g. [-1, -1, +1, -1], then Alice records one of -1 or -1 (the two first) depending on her coin toss (in this example the values are the same, so the coin toss doesn't matter), and Bob records one of +1 or -1 (the two last) depending on the outcome of his coin toss.

I suppose that all sexteen posibilities are equally probable in which I think that S=0. The general case for arbitrary probabilities would give but reduces to a tedious counting problem.

Yes, any arbitrary distribution is allowed. And the exact same strategy used to prove this result can also be used to prove Bell's theorem. So the odd thing is that the Bell deniers usually think that this result holds true, but that Bell's theorem is false.

There is no such thing as "Bell's theorem." Bell attempted to make a physical argument based on a major physical blunder. Get over it and move on: https://arxiv.org/pdf/1704.02876.pdf.
.

[FrediFizzx]

There is no such thing as "Bell's theorem." Bell attempted to make a physical argument based on a major physical blunder. Get over it and move on: https://arxiv.org/pdf/1704.02876.pdf.
.

Yeah, Bell's theory was shot down to pieces in 2007 and the Bell fanatics try everything to justify it. Pretty absurd.

http://dx.doi.org/10.13140/RG.2.2.28311.91047/1
.

[minkwe]

In real present-day Bell experiments there are four streams of binary data, matched by belonging to matched pairs of time-slots. A binary input for Alice’s lab, a binary output from Alice’s lab, a binary input for Bob’s lab, and a binary output from Bob’s lab.

4 streams of binary data equals 8 streams of data = 8 random variables = 8 functions. Justo refuses to explain how his 4 functions originate from the 8.

[minkwe]

That is right, as written it does not directly represent any experiment but its values represent results of real experiments. Equation (23) directly represents the results of the experiments

Do you remember when I asked you if it represented measurements on the same set of particles or measurements on independent sets of particles? I think you are smart enough to not play games with me. I will also assume that you read what I explained in viewtopic.php?f=6&t=482&start=100#p13917.

In equation 23, you are essentially calculating the averages of the pairs of products. We could say you have four 2xN spreadsheets. One for each of the averages you calculate in equation (23). Now look closely at those spreadsheets, Column from the first spreadsheet is identical to Column from the second spreadsheet, and column is identical for the third and fourth spreadsheets. So when you answer that equation (23) represents a real experiment. You have to explain what type of experiment. Obviously, it is not a Bell test experiment, rather, it is the experiment described in my Scenario 3 & 4. Because what you have is a single 4xN spreadsheet that you have duplicated columns in order to arrive at (23). This is why you are able to tautologically arrive at an upper bound of 2.

You haven't shown an alternative derivation, you've simply re-stated scenarios 3 & 4 that are absolutely not analogous to a proper Bell test. I do not see any derivation in your paper that is applicable to a Bell test experiment.

There should be 8 functions in equation (29) not 4.

Not after applying the arithmetic rules and properties. That is why mathematics is useful and engineers and physicists study mathematics.

Then apply the rules here let us verify them. You did not apply the rules in your paper, you just hand-waved it. Are you concerned that it won't survive the scrutiny?

Yes, I do. It is mathematically explained in equations in steps from eq. (23) through (28). It is very elementary mathematics that any first-year college student can deal with and interpret.

You did not. You provide absolutely no justification on how to go from 8 functions to the 4 identified in equations 23 and 29. Don't gaslight.

Dear @minkew let us, at least, agree on this: "If we do not accept that a physically meaningful mathematical expression like eq. (23) can be transformed according to mathematical rules and whatever it is that we do, an expression with only 4 values like eq. (29) is not acceptable, then we cannot derive the Bell inequality."

Equation (23) is physically meaningful for my scenario 3 & 4. It is absolutely not meaningful for my scenarios 1 & 2 (Bell tests).

[Justo]

That is right, as written it does not directly represent any experiment but its values represent results of real experiments. Equation (23) directly represents the results of the experiments

Do you remember when I asked you if it represented measurements on the same set of particles or measurements on independent sets of particles? I think you are smart enough to not play games with me. I will also assume that you read what I explained in viewtopic.php?f=6&t=482&start=100#p13917.

In equation 23, you are essentially calculating the averages of the pairs of products. We could say you have four 2xN spreadsheets. One for each of the averages you calculate in equation (23). Now look closely at those spreadsheets, Column from the first spreadsheet is identical to Column from the second spreadsheet, and column is identical for the third and fourth spreadsheets. So when you answer that equation (23) represents a real experiment. You have to explain what type of experiment. Obviously, it is not a Bell test experiment, rather, it is the experiment described in my Scenario 3 & 4. Because what you have is a single 4xN spreadsheet that you have duplicated columns in order to arrive at (23). This is why you are able to tautologically arrive at an upper bound of 2.

You haven't shown an alternative derivation, you've simply re-stated scenarios 3 & 4 that are absolutely not analogous to a proper Bell test. I do not see any derivation in your paper that is applicable to a Bell test experiment.

Congratulations! I must confess that you nicely surprised me. I wasn't planning to respond anymore but you just pointed out a possible theoretical loophole that De Baere observed in a 1984 paper. That is what I explain in the following subseccion 4.2

[gill1109]

In real present-day Bell experiments there are four streams of binary data, matched by belonging to matched pairs of time-slots. A binary input for Alice’s lab, a binary output from Alice’s lab, a binary input for Bob’s lab, and a binary output from Bob’s lab.

4 streams of binary data equals 8 streams of data = 8 random variables = 8 functions. Justo refuses to explain how his 4 functions originate from the 8.

In real present day Bell experiments, 4 streams of binary data are used to define 4 subsets of two streams, of lengths adding up to the total original length.
The two outputs for inputs 0, 0
The two outputs for inputs 0, 1
The two outputs for inputs 1, 0
The two outputs for inputs 1, 1
Justo's 4 *functions* do not "originate" from the data. Those functions are mathematical functions which are considered within a certain mathematical physical model, which is hoped to be able to reproduce certain expectation values (in fact: theoretical correlations) which are also derived in another mathematical physical model. Quantum mechanics has formulas for correlations too. They do not "originate" from any streams of data.

See for instance Appendix B of Joy Christian and Fred Diether's latest paper https://www.researchgate.net/publication/353795166_Event-by-Event_Numerical_Simulation_of_the_Strong_Singlet_Correlations. Actually, their Alice and Bob's settings are chosen at random repeatedly from 0, 1, .... up to 360 degrees. Afterwards, they select the cases for four pairs of angles, and do the usual statistics on each of four small substreams of random length of pairs of binary outcomes.

[minkwe]

In real present day Bell experiments, 4 streams of binary data are used to define 4 subsets of two streams, of lengths adding up to the total original length.
The two outputs for inputs 0, 0
The two outputs for inputs 0, 1
The two outputs for inputs 1, 0
The two outputs for inputs 1, 1
Justo's 4 *functions* do not "originate" from the data. Those functions are mathematical functions which are considered within a certain mathematical physical model, which is hoped to be able to reproduce certain expectation values (in fact: theoretical correlations) which are also derived in another mathematical physical model. Quantum mechanics has formulas for correlations too. They do not "originate" from any streams of data.

Richard, I don't know if you are serious. Let me ask you this simple question:

Consider an experiment in which Alice makes the measurement . Irrespective of what model is producing this result. Is a random variable?

[minkwe]

Justo's 4 *functions* do not "originate" from the data.

I think Justo can speak for himself. Here is what the paper says:

Consequently, assuming LHV, the experimental correlations can be expressed
as in the LHS of (21)11 and the RHS of (18) becomes



...


None of the terms present in (29) are assumed to have originated from incompatible experiments, neither materialized out of counterfactual reasoning nor pre-existed before actually measured. emerges when we assume
the real experimental data have the form given in (23),

Therefore you are just wrong about what the paper says.

[gill1109]

In real present day Bell experiments, 4 streams of binary data are used to define 4 subsets of two streams, of lengths adding up to the total original length.
The two outputs for inputs 0, 0
The two outputs for inputs 0, 1
The two outputs for inputs 1, 0
The two outputs for inputs 1, 1
Justo's 4 *functions* do not "originate" from the data. Those functions are mathematical functions which are considered within a certain mathematical physical model, which is hoped to be able to reproduce certain expectation values (in fact: theoretical correlations) which are also derived in another mathematical physical model. Quantum mechanics has formulas for correlations too. They do not "originate" from any streams of data.

Richard, I don't know if you are serious. Let me ask you this simple question:

Consider an experiment in which Alice makes the measurement . Irrespective of what model is producing this result. Is a random variable?

I'm very serious. I do not agree to your terminology, so I can't answer "yes" or "no".

Consider a situation in which Alice chooses setting a and Bob chooses setting B. They each observe an outcome "up" or "down". We can obviously write down a probability space on which are defined two random variables, call them X_a and Y_b, taking values +/- 1. We can assign many different probability measures to this probability space. Essentially we must choose, fix or somehow determine, the probabilities of (X_a, Y_b) = (++), (+-), (-+), or (--). Four nonnegative numbers adding up to one. We could come up with reasonable guesses for those numbers by actually performing that same experiment many, many times, and just using the observed relative frequencies as estimates of the probabilities. We could also fit a quantum mechanics model, or a local realism model, to the data, and come up with perhaps different "guestimates".

In a Bell type experiment one essentially does four such experiments, with settings a1, b1; a1, b2; a2, b1; a2, b2. If we believe that each single experiment has independent realisations of a pair X_a and Y_b for appropriate a, b, we then have 16 probabilities to fit, adding up to 1 in sets of 4. Notice: the random variable X_a is a different random variable defined on a different probability space depending on what experiment we are modelling: the (a, b) experiment or the (a, b') experiment!

"Local realism" puts certain constraints on those 16 probabilities. "Quantum mechanics" puts other constraints on those same probabilities. For instance, according to QM, and by Tsirelson's inequality, "S" can't exceed 2 sqrt 2. According to LR, and by the Bell-CHSH inequality, "S": can't exceed 2. "S" is the same function of 16 probabilities in both cases, but the 16 probabilities are constrained to live in different subsets of R^16 by the two physical models.

In present-day mathematics, a real-valued "random variable" is conventionally taken to be a measurable function X from a probability space (Omega, F, P) to the real numbers endowed with the Borel sigma-algebra. One does therefore conventionally model (intuitively speaking) random outcomes of real life experiments or observational processes with probability spaces and random variables. Bell was a physicist and did not use even the standard *mathematical* language of his day, let alone of today. He used the language of physicists of his day. I think his *thinking* was very sophisticated and modern.

[gill1109]

Consider a situation in which Alice chooses setting a and Bob chooses setting B. They each observe an outcome "up" or "down". We can obviously write down a probability space on which are defined two random variables, call them X_a and Y_b, taking values +/- 1. We can assign many different probability measures to this probability space. Essentially we must choose, fix or somehow determine, the probabilities of (X_a, Y_b) = (++), (+-), (-+), or (--). Four nonnegative numbers adding up to one. We could come up with reasonable guesses for those numbers by actually performing that same experiment many, many times, and just using the observed relative frequencies as estimates of the probabilities. We could also fit a quantum mechanics model, or a local realism model, to the data, and come up with perhaps different "guestimates".

In a Bell type experiment one essentially does four such experiments, with settings a1, b1; a1, b2; a2, b1; a2, b2. If we believe that each single experiment has independent realisations of a pair X_a and Y_b for appropriate a, b, we then have 16 probabilities to fit, adding up to 1 in sets of 4. Notice: the random variable X_a is a different random variable defined on a different probability space depending on what experiment we are modelling: the (a, b) experiment or the (a, b') experiment!

"Local realism" puts certain constraints on those 16 probabilities. "Quantum mechanics" puts other constraints on those same probabilities. For instance, according to QM, and by Tsirelson's inequality, "S" can't exceed 2 sqrt 2. According to LR, and by the Bell-CHSH inequality, "S": can't exceed 2. "S" is the same function of 16 probabilities in both cases, but the 16 probabilities are constrained to live in different subsets of R^16 by the two physical models.

Actually, there is another interesting model: it's called "no signalling". According to that model, the marginal distribution of the random variable called X_a in models for different experiments, namely those using settings (a, b) and those using (a, b'), are the same. It is not difficult to prove that the set of models allowed by no signalling strictly contains the set of models allowed by QM, and that strictly contains the set of models allowed by LR. [By models allowed by QM I mean QM models allowing any quantum measurements on any state, pure or mixed, on any bipartite system, of any pair of dimensions. The Tsirelson bound is not restricted to particular measurements, states, dimensions. These models: NS, QM, LR are just subsets of R^16. They are all closed, convex, compact; they are all relatively open inside the affine subspace of 16-tuplets satisfying the no-signalling equalities and the normalisation equalities. Adding the nonnegativity constraints gives us the NS set: it is a convex polytope. The LR set is also a convex polytope. That's Fine's result on the necessary and sufficient character of the 8 one-sided CHSH inequalities.]

[Justo]

Justo's 4 *functions* do not "originate" from the data. Those functions are mathematical functions which are considered within a certain mathematical physical model, which is hoped to be able to reproduce certain expectation values (in fact: theoretical correlations) which are also derived in another mathematical physical model. Quantum mechanics has formulas for correlations too. They do not "originate" from any streams of data.

Not literally, the four functions are supposed to represent actual data. If your mathematical model, be it LHV or QM, does predict the results of real experiments, what's the point of performing experiments to falsify your theory? That's the whole point of the paper, highlighting the irrelevance of the CFD hypothesis.
The other point explained in the paper is that even if you assume that CFD is meaningful, it is nonetheless irrelevant for the problem investigated by Bell.
How come? Well it turns out that the BI can be derived from different hypotheses, for instance
Joint probabilities ------> BI (1)
Non-commuting observables ----> BI (2)
CFD and locality --------> BI (3)
Violation of the inequality forces the rejection of any and all of the antecedents. None of them invalidate the others. For instance, we can correctly conclude that a joint probability describing the result of the experiment does not exist. It is equally correct to conclude that the observables involved in the experiment (when you use QM to make the prediction) do not commute.
So it is very funny when highly educated people with PHDs pompously declare the Bell inequality is meaningless because it only proves the non-existence of a joint probability.
One very widespread nonsense among professional and sometimes high-profile physicists is the claim the Bell theorem only proves that realism is false and quantum mechanics is local. Sometimes realism means CFD, others hidden variables.
All the previous just reveal that none of them understand that Bell later derived his inequality from only two assumptions:
Local Causality and Measurement Independence ------> BI (4)
This means that by rejecting CFD in (3) you cannot retain locality because theorem (4) is still valid. Those who understand the problem and opt for retaining Local Causality attack Measurement Independence.
So, even not accepting the inconsistency of the CFD hypothesis, its irrelevance to the problem studied by Bell is unavoidable.

Joy previously correctly observed the inconsistency of the CFD hypothesis. I used to agree with him on this point, if the BI depended on CFD it would be meanless.
Now Minkwe found what could be a theoretical loophole in Bell's derivation, i.e., theorem (4). He found reasons to doubt that Bell's derivation describes the results of real experiments. He observed what seems to be a doubtful assumption that De Baere already observed in 1984 and probably the Bell establishment never paid any attention or those who know simply know and don't bother to explain because have more important things to do.
Well, I don't have more important things to do, so I bother to explain it in section 4.2 of our paper.

[gill1109]

Justo: “represent” means something completely different from “originate from”. And “represent” is not the same as “are equal to”. Inside a mathematical model, various expressions *represent* things in the real world. Of course the idea behind deriving BI’s starting from the usual formulation of LHV, is that the function A(a, lambda) *represents* “what the measurement outcome would be, if the setting were “a” and the hidden variables (which need not be localized anywhere at all) happened to be “lambda”. Consequently, whatever the settings on Alice’s and Bob’s side happen to actually be, A(a, lambda) and A(a’, lambda) *represent* the outcomes which would be observed by Alice if her setting was a, or if it was a’, … when the hidden variables happen to equal “lambda”. So if you think of lambda as being “real”, then we have CFD and locality. If you think of LR just as a mathematical model which could be hoped to explain observed correlations, that’s also fine. Bell’s conclusion was that such models could not explain quantum predictions. And now we know that such models cannot explain laboratory reality. Bell noticed that Bohmian theory was a hidden variables theory that perfectly *completed* quantum theory. But it was non-local. He wondered if that could be fixed. His answer: no. At least, not without super-determinism or retro-causality.

[Justo]

Justo: “represent” means something completely different from “originate from”. And “represent” is not the same as “are equal to”. Inside a mathematical model, various expressions *represent* things in the real world. Of course the idea behind deriving BI’s starting from the usual formulation of LHV, is that the function A(a, lambda) *represents* “what the measurement outcome would be, if the setting were “a” and the hidden variables (which need not be localized anywhere at all) happened to be “lambda”. Consequently, whatever the settings on Alice’s and Bob’s side happen to actually be, A(a, lambda) and A(a’, lambda) *represent* the outcomes which would be observed by Alice if her setting was a, or if it was a’, … when the hidden variables happen to equal “lambda”. So if you think of lambda as being “real”, then we have CFD and locality. If you think of LR just as a mathematical model which could be hoped to explain observed correlations, that’s also fine. Bell’s conclusion was that such models could not explain quantum predictions. And now we know that such models cannot explain laboratory reality. Bell noticed that Bohmian theory was a hidden variables theory that perfectly *completed* quantum theory. But it was non-local. He wondered if that could be fixed. His answer: no. At least, not without super-determinism or retro-causality.

Discussing what I meant by "represent" or what you did by "originate from" makes us sound like two politicians trying to prove their honesty.
Perhaps you missed the main point. Let us consent that CFD is a very logical and consistent assumption. It is, nonetheless, irrelevant for the physical problem concerning the Bell theorem. Why? Because Bell derived the inequality from only two assumptions 1)Local causality 2) Measurement (or statistical) independence. Therefore, CFD is irrelevant for the same reason that the existence of a joint probability is.

[Heinera]

I'm sorry that you guys have completely lost me on this CFD discussion. For me CFD has two meanings, depending on context:

The mathematical context: The functions A(a, lambda) and B(b, lambda) are mathematically well defined on their respective domains. In other words, something that should be completely uncontroversial.

The philosophical or ontological context: I have no idea. Probably some foggy stuff.

But I do know one thing: It is trivially true that there can be no CFD in experimental data. The combination of the terms simply has no meaning. So just by involving CFD with experiments should be a red flag for yourself indicating that you probably don't know what you are talking about.

[Justo]

I'm sorry that you guys have completely lost me on this CFD discussion. For me CFD has two meanings, depending on context:

The mathematical context: The functions A(a, lambda) and B(b, lambda) are mathematically well defined on their respective domains. In other words, something that should be completely uncontroversial.

The philosophical or ontological context: I have no idea. Probably some foggy stuff.

But I do know one thing: It is trivially true that there can be no CFD in experimental data. The combination of the terms simply has no meaning. So just by involving CFD with experiments should be a red flag for yourself indicating that you probably don't know what you are talking about.

I mostly agree with what you say. Owed to its very nature a counterfactual statement is not (at least directly and surely in Bell's case) falsifiable. However, even if assume that CFD is consistent with experiments (as Gill does), I have argued that it is irrelevant for the BI.

[Heinera]

I mostly agree with what you say. Owed to its very nature a counterfactual statement is not (at least directly and surely in Bell's case) falsifiable. However, even if assume that CFD is consistent with experiments (as Gill does), I have argued that it is irrelevant for the BI.

OK, but with my mathematical definition of CFD there can be no BI if A(a, lambda) and B(b, lambda) are not mathematically well defined. In fact there would not even be a paper by Bell in 1964, because that was what he assumed. I agree with you that the proof of the inequality does not need to further rely on any CFD than that. (And I'm pretty sure Richard agrees with us here.)

[Justo]

I mostly agree with what you say. Owed to its very nature a counterfactual statement is not (at least directly and surely in Bell's case) falsifiable. However, even if assume that CFD is consistent with experiments (as Gill does), I have argued that it is irrelevant for the BI.

OK, but with my mathematical definition of CFD there can be no BI if A(a, lambda) and B(b, lambda) are not mathematically well defined. In fact there would not even be a paper by Bell in 1964, because that was what he assumed. I agree with you that the proof of the inequality does not need to further rely on any CFD than that. (And I'm pretty sure Richard agrees with us here.)

CFD does mean that those functions are well defined. What CFD means, when used to derive the BI, is that you can make counterfactual predicción using those functions and then perform experiments to falsify those predictions.
Basically is that, of course, there are personal interpretations that some people use, for instance, that there exist elements of physical reality and staff like that mostly irrelevant concerning Bell theorem.

[Justo]

I should have said "CFD does not mean" instesad of "CFD does mean"