Bell's theorem and local realism

[DrChinese]

Well, the playbook idea that you sketched could be implemented by one of t'Hooft's machines, couldn't it?

Unless his machine is a "Bell Playbook Reader" like a Kindle, I doubt it. Because anything less can probably be falsified as it will rely on some element which is not hidden. That makes it susceptible to experimental falsification. Which I would expect to be "easy" to do, in the theoretical sense.

 

[ueit]

No, it doesn't. The coupled Maxwell-Lorentz equations are local. What that implies is that if you want to compute \vec{E}(\vec{r},t), it is sufficient to know the values of \vec{E}, \vec{B} and the positions of charged particles in the region of spacetime consisting of all points \vec{r'}, t' such that

• 0 < t - t' < \delta t,
• |\vec{r'}-\vec{r}| < c \delta t.

You don't need to know anything about points more distant than c \delta t. The evolution of the electric field only depends on facts about local conditions, not facts about the whole universe.

Sure, but you forget that the local values of electric and magnetic field are related to their far-away sources. If you know the field in your location, true, you don't need to know about the field sources (distant charges). But this does not imply in any way that the local field is independent of its distant sources. It's your choice to either measure the local field directly or compute it from position/momenta of nearby and distant charges. You don't need both, it will be redundant.

That's not true. The trajectory of a charge depends on local values of the fields. The evolution of the fields depends only on NEARBY charges. Distant charges are irrelevant (if you know the local values of fields in the recent past).

But, as I've pointed above, "the local values of fields in the recent past" is a function of position/momenta of all field sources (in the past). So, the trajectory of a charge is not independent from the position/momenta of all the other charges.

From the fact that the theory is deterministic it also follows that the future position/momenta of all charges is a function of position/momenta of all charges in the past.

We can therefore conclude that the trajectory of a charge is also not independent from the position/momenta of all the other charges in the future.

That's just not true. You're glossing the distinction between a local theory and a nonlocal theory.

I think I have clearly pointed out where your reasoning fails. It becomes obvious if you think in terms of gravity. GR is local, therefore you can predict Earth's trajectory if you know the space curvature in its vicinity. You do not need to know anything about the Sun, Moon or any other object. But this doesn't imply that Earth's trajectory is independent of the Sun. And the reason is that the local curvature itself DOES depend on the Sun.

 

[ueit]

But that isn't good enough. The relevant facts about the detectors is not their positions at the time of splitting. What's relevant for the predictions of QM are the positions of the detectors at the time of detection. The detectors could very well change positions while the particles are in-flight (after they have split).

As i have argued above, once you establish that the spin of the particles depends on the past position/momenta of all the other charges it follows that it also depends on their future position/momenta due to determinism.

It is true that if you knew the positions and velocities of every particle in the universe, then you could in principle predict the positions the detectors would be in at the time of detection. But that's a LOT more complicated than allowing the actual orientation of the spin magnetic moment to depend on the local EM field at the moment of splitting. As a matter of fact, the local EM field would be pretty much irrelevant. (If the detectors are electrically neutral, then they have a negligible effect on a distant EM field.) What it would take for a local deterministic model to reproduce the predictions of quantum mechanics is a supercomputer that can simulate the rest of the universe. And it would have to come up with the result of its computation essentially instantly.

This approach seems completely implausible to me.

First of all, do you agree that the freedom assumption fails for classical EM?

 

[ueit]

Most definitely not, and we know that from Bell! Certainly there are many more things that determine and ultimately affect the outcomes of experiments in a local classical (deterministic) universe. The orientation of angle settings, for example. The orientation of a particle is not determined in any way by randomizing devices which select settings those outside of a light cone, as was done in the experiment of Weihs et al (1998).

The point is that superdeterminism is NOT anything like any Laplacian demon operating in a clockwork universe. In such a universe, we would not get a value from experiment that matches QM. In superdeterminism, there is a mechanism in place that PREVENTS the selected sample from matching the true universe of counterfactual values. And I say that NO superdeterministic theory can ever reproduce all of the results of QM.

Can I ask you to point out exactly where my argument fails? Which of the points I've made (1-10) are false in your opinion? I know your opinion is different, but you have to justify it with arguments. If you want to use Bell again you have to point out why my argument against the freedom assumption fails.

 

[stevendaryl]

Sure, but you forget that the local values of electric and magnetic field are related to their far-away sources. If you know the field in your location, true, you don't need to know about the field sources (distant charges). But this does not imply in any way that the local field is independent of its distant sources. It's your choice to either measure the local field directly or compute it from position/momenta of nearby and distant charges. You don't need both, it will be redundant.

That's true. Everything can affect everything else. But the point of a local theory is that everything that's relevant about distant particles and fields is already captured in the values of local fields and the positions/momenta of local particles. So the evolution equations don't need to take into account anything other than local conditions.

This is in contrast to a nonlocal theory, where the evolution equations must potentially take into account everything. Superdeterminism can turn a nonlocal theory into a local theory, but at the cost of requiring, essentially, a local representation of distant facts. The normal electromagnetic fields do not have anywhere near enough information to reproduce nonlocal EPR correlations.

But, as I've pointed above, "the local values of fields in the recent past" is a function of position/momenta of all field sources (in the past).

I think that's a very bad way of thinking about it. The evolution equations for the E&M field do not require any knowledge about distant particles and fields. Everything of relevance about distant variables is already included in the values of local variables.

The reason I say that it is a bad way of thinking about it is that glosses over the very important distinction between local and nonlocal theories.

I think I have clearly pointed out where your reasoning fails. It becomes obvious if you think in terms of gravity. GR is local, therefore you can predict Earth's trajectory if you know the space curvature in its vicinity. You do not need to know anything about the Sun, Moon or any other object. But this doesn't imply that Earth's trajectory is independent of the Sun. And the reason is that the local curvature itself DOES depend on the Sun.

You keep wanting to make things abstract, but I don't see how the abstraction gives any insight. Yes, everything depends on everything else, but in the case of gravity, the dependencies are very constrained. In the case of an EPR-type experiment, the dependencies are completely unconstrained. In such an experiment, Alice chooses a detector orientation, \vec{a} and Bob chooses a detector orientation, \vec{b} and the particle detected by Alice must choose to go right or left (in a Stern-Gerlach set-up), and similarly for the particle detected by Bob. In order for a deterministic model to generate the correct statistics (those predicted by QM), it seems that each particle's decision must depend on BOTH \vec{a} and \vec{b}. So the question is: how does Bob's particle know the value of \vec{a}, and how does Alice's particle know the value of \vec{b}?

Your answer seems to be: Alice's state was known ahead of time, and her choice of \vec{a} was (by assumption) deterministic, so \vec{a} is actually computable from this knowledge. But it's not just knowledge about Alice. Since Alice can make her decision based on who gets a hit in the baseball game, the computation would have to involve the states of the baseball players, as well. And since a fan at the baseball game might throw a paper airplane to distract the batter, you would have to know the state of the fans, as well. The computation is completely unconstrained, in that it might require knowledge of the whole rest of the universe.

You give the analogy of the position of the Earth in the future. Well, there is always the possibility that the Moon will explode and fragments will knock the Earth off its course. Then our prediction would be wrong. Positions of planets are only predictable under the assumption that nothing too weird is going to happen. If we tried to take into account weird stuff, then the future position of the Earth would not be predictable, in any practical sense. It would be computationally impossible.

The same thing would apply to an EPR type experiment. There might be a way to guess the most likely setting Alice will choose, based on incomplete knowledge of Alice. But to be certain of Alice's choice would be computationally impossible (given finite resources to do the computation). So if EPR correlations were explained by superdeterminism, it would either require infinite precision and infinite processing power in the little cellular automata, or else the correlations could be destroyed by Alice using a sufficiently difficult-to-predict algorithm for choosing her setting.

Essentially, the only way that the EPR correlations could always hold is if every particle has a complete description of the whole rest of the universe, and the processing power to simulate the future evolution of the universe.

But there is no reason to argue about it: If you really believe that a superdeterministic theory can reproduce the predictions of quantum mechanics, then try to create one. It's basically the quantum Randi challenge:

We have two teams: The Red Team and the Blue Team. The Red Team gets to pick algorithms for Alice and Bob to decide their settings. The Blue Team gets to pick an algorithm for the electron and positron to decide whether they go left or right. Can the Blue Team reproduce the statistics predicted by QM?

In a superdeterministic model, the Blue Team would be able to know the algorithms chosen by the Red Team. But what is the Blue Team going to do with this knowledge? It could try simulating the running of the Red Team algorithms, in order to predict what the settings will be. That would work, but it would require the Blue Team to have potentially unlimited processing power.

 

[stevendaryl]

First of all, do you agree that the freedom assumption fails for classical EM?

No, I don't. You cannot determine the positions and momenta of distant particles knowing only local fields.

 

[stevendaryl]

Can I ask you to point out exactly where my argument fails? Which of the points I've made (1-10) are false in your opinion? I know your opinion is different, but you have to justify it with arguments. If you want to use Bell again you have to point out why my argument against the freedom assumption fails.

I don't think that there is any doubt that everything in the universe is correlated with everything else. That's not the question. The question is whether that correlation is strong enough that the locations of distant particles can be computed using only local knowledge.

 

[atyy]

In Bell's theorem there's a "free will" assumption, which means we assume that measurements settings at spacelike separation can be set independently (in the probabilistic sense). Is "superdeterminism" different from saying that the free will assumption fails?

 

[stevendaryl]

Unless his machine is a "Bell Playbook Reader" like a Kindle, I doubt it. Because anything less can probably be falsified as it will rely on some element which is not hidden. That makes it susceptible to experimental falsification. Which I would expect to be "easy" to do, in the theoretical sense.

I'm saying that a "Bell Playbook Reader" could very well be implemented as a cellular automaton, couldn't it?

If Alice's and Bob's settings could be known ahead of time, then it would not be difficult to reproduce the QM predictions for EPR using local hidden variables. If the universe is deterministic, then whatever mechanism chooses the hidden variables could, in principle, predict Alice's and Bob's settings from past information. I think that's a ridiculous model, but I don't think it's logically impossible.

Actually, now that I think about it, this kind of deterministic model for EPR reminds me a little bit of the Bohm model. In the latter case, nonlocal interactions are introduced to make the statistics work out, but it's assumed that there are no observable nonlocal interactions. In the case of the playbook reader, potentially unlimited computing power is introduced to make the statistics work out, but it's assumed that this computing power is unavailable for any other purpose. (If Alice tapped into that kind of computing power to make a pseudo-random choice, then that would defeat the ability to predict Alice's setting. A computer can't, in real time, predict the behavior of an equally powerful computer.)

 

[DrChinese]

Can I ask you to point out exactly where my argument fails? Which of the points I've made (1-10) are false in your opinion? I know your opinion is different, but you have to justify it with arguments. If you want to use Bell again you have to point out why my argument against the freedom assumption fails.

Well sure, and I don't think this path is new. Bell is well accepted after all.

According to you, and assuming there is some classical action (local and realistic), there are correlations between events that are separated. Even in classical determinism, I say there are NOT mechanisms which relate observables used in Bell tests.

Now I get the idea that a planet follows a distinct orbit around a star, and that is pre-determined even though the planet and the star are spacelike separated. Thus a prediction can be made with certainty on the path of the planet and of the star even if a decision is made as to how to observe each at the last possible time. The results seem observer independent. But they are not actually non-local. And if there is any interaction between the observer and that being observed which is material to the outcome, then that part of your argument explicitly fails. I forget which number that is.

Regardless, in a deterministic world, such correlations are extremely limited. They certainly don't lead to predictions for Bell tests that match experiment. That such is true is seen by asking: why don't classical dice correlate more than by chance (on the average, 1 of 6 times a pair will match) ? You are saying that there is a stochastic connection, and yet that is picked out of the blue. There is no hypothetical connection between the observer's choice of measurement that restricts him or her or otherwise guides the results. You need *Superdeterminism* for that!

 

[stevendaryl]

In Bell's theorem there's a "free will" assumption, which means we assume that measurements settings at spacelike separation can be set independently (in the probabilistic sense). Is "superdeterminism" different from saying that the free will assumption fails?

That's what superdeterminism means, in practice.

However, I don't think that the assumption of "free will" is necessary. The real assumption is that the algorithms (whatever they are) for choosing the settings are too complicated to be predictable by any single mechanism in whatever is supposed to choose the hidden variable values. The settings might be deterministic, but they can depend on absolutely anything (as I said, they could depend on events from a baseball game, or anything else). So the only way that the settings could be guaranteed to be predictable is if the mechanism that chooses the hidden variable had access to a complete simulation of the universe (or at least of everything in the region surrounding the experiment).

 

[atyy]

That's what superdeterminism means, in practice.

However, I don't think that the assumption of "free will" is necessary. The real assumption is that the algorithms (whatever they are) for choosing the settings are too complicated to be predictable by any single mechanism in whatever is supposed to choose the hidden variable values. The settings might be deterministic, but they can depend on absolutely anything (as I said, they could depend on events from a baseball game, or anything else). So the only way that the settings could be guaranteed to be predictable is if the mechanism that chooses the hidden variable had access to a complete simulation of the universe (or at least of everything in the region surrounding the experiment).

The assumption of "free will" is necessary. For example, the assumption of independence could fail with fine tuning of initial conditions, since the apparently independent apparatuses were at the same location at the big bang.

 

[stevendaryl]

The assumption of "free will" is necessary. For example, the assumption of independence could fail with fine tuning of initial conditions, since the apparently independent apparatuses were at the same location at the big bang.

I perfectly well understand that fine tuning of initial conditions can theoretically explain everything. However, it's a very unsatisfactory satisfaction: As Dr. Chinese said, you have to carefully choose, at the beginning of time, the precise values for all positions and momenta just to make Bell's inequalities work out. Such fine tuning is certainly a logically possible explanation, but variants of such fine tuning could explain absolutely everything. All the experiments ever purported to demonstrated relativity or QM could very well be just freak malfunctions of equipment that just happen to malfunction in just the right way to seem to agree with the theoretical predictions. Invoking fine-tuning to explain EPR correlations is not much (if any) more satisfying than that.

 

[stevendaryl]

I perfectly well understand that fine tuning of initial conditions can theoretically explain everything. However, it's a very unsatisfactory satisfaction: As Dr. Chinese said, you have to carefully choose, at the beginning of time, the precise values for all positions and momenta just to make Bell's inequalities work out. Such fine tuning is certainly a logically possible explanation, but variants of such fine tuning could explain absolutely everything. All the experiments ever purported to demonstrated relativity or QM could very well be just freak malfunctions of equipment that just happen to malfunction in just the right way to seem to agree with the theoretical predictions. Invoking fine-tuning to explain EPR correlations is not much (if any) more satisfying than that.

So, I don't think that invoking "free will" is a very good way to look at things. We don't really have any idea what "free will" means. Its only role in arguments about Bell is that it is something that is not predictable. To me, rejection of superdeterminism is not really about free will. It's about the rejection of a class of theories that are logically possible, but are useless because they explain too much. Variations could be used to explain absolutely anything at all.

On the other hand, if there were a superdeterministic theory that explained HOW the fine-tuning came about, I would find that more satisfying.

 

[atyy]

I perfectly well understand that fine tuning of initial conditions can theoretically explain everything. However, it's a very unsatisfactory satisfaction: As Dr. Chinese said, you have to carefully choose, at the beginning of time, the precise values for all positions and momenta just to make Bell's inequalities work out. Such fine tuning is certainly a logically possible explanation, but variants of such fine tuning could explain absolutely everything. All the experiments ever purported to demonstrated relativity or QM could very well be just freak malfunctions of equipment that just happen to malfunction in just the right way to seem to agree with the theoretical predictions. Invoking fine-tuning to explain EPR correlations is not much (if any) more satisfying than that.

Well if it's a loophole it's a loophole. Much more important than aesthetic satisfactoriness is we cannot devise a superdeterministic theory that can be used by us to describe our universe, since we don't have access to such fine grained data to determine the fine tuned initial condition.

Edit: Unless the dynamics and fine tuned initial condition were both compact enough, and were fine tuned to be placed in 't Hooft's head too.

 

[DrChinese]

Congrats to stevendaryl on hitting 2000 posts!

 

[atto]

When drawing conclusions from this most important and profound theorem, I wonder if somebody has interpreted its proof of the falseness of local realism as implicitly referring to elementary particles as realistic objects.

This is a contradiction.
If Bell proved formally - mathematically some limit exists, then there is no possibility to go over these limit.

Simply: there are only two mutually exclusive possibilities:
1. Bell proof is wrong - a mistake
2. QM is a wrong model, because predicts impossible correlations, what proved Bell.

The logic makes no compromises!

 

[stevendaryl]

This is a contradiction.
If Bell proved formally - mathematically some limit exists, then there is no possibility to go over these limit.

Simply: there are only two mutually exclusive possibilities:
1. Bell proof is wrong - a mistake
2. QM is a wrong model, because predicts impossible correlations, what proved Bell.

The logic makes no compromises!

No, those are not the only possibilities. Bell showed that the predictions of QM cannot be reproduced by a certain type of theory--a local hidden-variables model. QM is not such a theory. So there is no contradiction between Bell's theorem and QM.

 

[stevendaryl]

Congrats to stevendaryl on hitting 2000 posts!

Thanks. Wow. That sounds like a lot, but I see it's nothing compared to your record.

 

[atto]

No, those are not the only possibilities. Bell showed that the predictions of QM cannot be reproduced by a certain type of theory--a local hidden-variables model. QM is not such a theory. So there is no contradiction between Bell's theorem and QM.

In mathematics there is no alternative worlds, especially: one with the parameters, and other without.
Everything that has been proven there is certain, indisputable, there is no alternative.

For example: |a+b| <= |a| + |b|, for any real number a, b.

This has been proven, it's true - unconditionally.
You never find the: a,b which breaks this inequality, because these don't exist.
If you now create a theory which anyway breaks this inequality, then the theory will be false.

 

[rubi]

In mathematics there is no alternative worlds, especially: one with the parameters, and other without.
Everything that has been proven there is certain, indisputable, there is no alternative.

For example: |a+b| <= |a| + |b|, for any real number a, b.

This has been proven, it's true - unconditionally.

Bell has also proven an inequality, but not unconditionally. Instead, it requires some assumptions and if a theory (like QM) violates these assumptions, then it needn't satisfy the inequality. There is no mathematical contradiction.

 

[stevendaryl]

In mathematics there is no alternative worlds, especially: one with the parameters, and other without.
Everything that has been proven there is certain, indisputable, there is no alternative.

For example: |a+b| <= |a| + |b|, for any real number a, b.

This has been proven, it's true - unconditionally.
You never find the: a,b which breaks this inequality, because these don't exist.
If you now create a theory which anyway breaks this inequality, then the theory will be false.

Bell proved an implication, a statement of the form:

"If X is true, then Y is true."

He did not prove "Y is true."

If X is false, then Y might be false.

A lot of mathematical theorems (I would say the vast majority of them) have the form of an implication: If x, y and z are integers, and each is greater than 0, then x^3 + y^3 \neq z^3. The theorem isn't true without the condition. For example, x=0, y=0, z=0 is a counter example.

 

[atto]

Bell has also proven an inequality, but not unconditionally. Instead, it requires some assumptions and if a theory (like QM) violates these assumptions, then it needn't satisfy the inequality. There is no mathematical contradiction.

Unfortunately, the Bell's inequality is an example of this type certainty - mathematical tautology.

Write the inequality, and try to break it;
you'll find that this is impossible - absolutely.

 

[atto]

Bell proved an implication, a statement of the form:

"If X is true, then Y is true."

He did not prove "Y is true."

If X is false, then Y might be false.

A lot of mathematical theorems (I would say the vast majority of them) have the form of an implication: If x, y and z are integers, and each is greater than 0, then x^3 + y^3 \neq z^3. The theorem isn't true without the condition. For example, x=0, y=0, z=0 is a counter example.

If it was as you say, there would be no discussion of non-classical correlations, entanglement of particles, and so on.

 

[Nugatory]

Write the inequality, and try to break it;
you'll find that this is impossible - absolutely.

You will find that it is impossible to break the inequality if you assume that the result of the measurement at a detector can be written as a function of the state of the particle that hits that detector and the state of the detector. That's what Bell's theorem says.

However, the inequality can be broken if you assume that the result of a measurement at a detector is a function of the state of the particle that hits that detector, the state of the detector, and the angle between that detector and the other detector.

 

[atto]

In the mathematical theorems there are no particles, so there is no question of any particle parameters.

You just have three (or four) series of numbers with values ​​from the set {1, -1}.

You calculate these correlations by well-known formulas, and check the inequality.
Your mission is to show the series, which break the inequality. That's all.

 

[Nugatory]

In the mathematical theorems there are no particles, so there is no question of any particle parameters.

You will find a copy of Bell's paper presenting his theorem here: http://www.drchinese.com/David/Bell_Compact.pdf

The statement of the theorem starts with the words "Consider a pair of spin one-half particles..." and proceeds from there.

 

[stevendaryl]

If it was as you say, there would be no discussion of non-classical correlations, entanglement of particles, and so on.

Why do you say that? Bell proved that for every theory of a certain type, (local realistic), his inequality holds. QM violates his inequality. Therefore, QM is NOT a theory of that type. There is no contradiction. Neither Bell nor QM is wrong.

The interesting question is what does it MEAN to have a theory that is not a local realistic theory. That's what all the discussions are about.

Bell's theorem does not prove QM is wrong, and QM does not prove Bell's theorem is wrong.

 

[stevendaryl]

In the mathematical theorems there are no particles, so there is no question of any particle parameters.

You just have three (or four) series of numbers with values ​​from the set {1, -1}.

You calculate these correlations by well-known formulas, and check the inequality.
Your mission is to show the series, which break the inequality. That's all.

Let's go through what Bell proved. Part of it is a mathematical theorem. It's just a fact, and I don't think there is any dispute about it. The second part is the application of this fact to physics. That is NOT pure mathematics. You can't apply mathematics to physics without making assumptions, and if you prove that something is impossible, you really have only proved that, under those assumptions, it is impossible.

You can make the mathematical part of Bell's theorem into a claim, as you say, about 4 series of numbers, each of which is +1 or -1.

Given 4 lists of numbers, each being +1 or -1:
A_i, A&#039;_i, B_i, B&#039;_i
we compute 4 correlations:

1. \langle A, B \rangle = average of A_i B_i
2. \langle A, B&#039; \rangle = average of A_i B&#039;_i
3. \langle A&#039;, B \rangle = average of A&#039;_i B_i
4. \langle A&#039;, B&#039; \rangle = average of A&#039;_i B&#039;_i

Then we can prove that a certain inequality must relate these 4 numbers. There is no dispute about that. It's a mathematical theorem. QM certainly does NOT prove this theorem wrong.

QM, and in particular, the EPR experiment, does not provide us with such a set of 4 lists of numbers. That's because in a twin-pair experiment, the experimenters (call them Alice and Bob) must make a choice: For each run i of the experiment, Alice must decide whether to measure A_i, or to measure A&#039;_i. She can't measure both. Similarly, Bob must decide whether to measure B_i or B&#039;_i. He can't measure both.

So an EPR experiment, you don't get 4 lists of numbers, each of which is either -1 or +1. You get 4 values, 2 of which are +1 or -1, and 2 of which are ?, meaning unmeasured.

If you assume that there really are 4 numbers for each i, and that those 4 numbers are either +1 or -1, but we just don't know what two of them are, that leads to a contradiction. But QM doesn't say that there are 4 numbers associated with each run. It only says there are two numbers, the numbers actually measured by Alice and Bob. To assume that there are 4 numbers goes beyond QM to some "hidden variables theory" that is supposed to explain QM. Bell proved that there is no such hidden variables theory. There is no way to replace the ? by +1 or -1 everywhere so that the statistics for unmeasured values obey the predictions of QM.

So QM, together with Bell's theorem, shows us that quantum measurements are not simply a matter of measuring a variable that had a pre-existing value whether you measured it or not. What is it, if not that? Well, that's the big question.

 

[ueit]

That's true. Everything can affect everything else. But the point of a local theory is that everything that's relevant about distant particles and fields is already captured in the values of local fields and the positions/momenta of local particles. So the evolution equations don't need to take into account anything other than local conditions.

This is in contrast to a nonlocal theory, where the evolution equations must potentially take into account everything.

The point I am trying to make is about the failure of the freedom assumption, not about locality. The theory is local, OK. Once you know the local field (which would be rather difficult, as it would require infinite resolution and accuracy) you can ignore distant sources, OK. So what?

You want to describe the local field at the locations of Alice, Bob and Source (source of entangled particles) as a brute fact (electric and magnetic field vectors in each point). This is your choice. It is impossible to posses such an information but this is your problem.

Now, my choice is different. I want to calculate the local fields at Alice, Bob and Source as a function of the field sources. For a limited number of sources this is in principle computable. Let's say, for simplicity, that Alice, Source and Bob are placed on the Z axis of some reference frame and they are not moving relative to each other. In this conditions we can express the fields in the following way:

E,B (Alice) = f(q1, q2,...qn, x1, x2,...xn,y1, y2,...yn, z1, z2,...zn, mx1, mx2,...,mxn, my1, my2,... myn, mz1, mz2,...mzn)

E,B (Source) = f(q1, q2,...qn, x1, x2,...xn,y1, y2,...yn, z1+AS, z2+AS,...zn+AS, mx1, mx2,...,mxn, my1, my2,... myn, mz1, mz2,...mzn)

E,B (Bob) = f(q1, q2,...qn, x1, x2,...xn,y1, y2,...yn, z1+SB, z2+SB,...zn+SB, mx1, mx2,...,mxn, my1, my2,... myn, mz1, mz2,...mzn)

where:

n = number of charges in the universe
q = electric charge
xi, yi, zi = position of charge i
mxi, myi, mzi = momentum of the charge
AS = Alice-Source distance
SB = Bob-source distance

Now, looking at the equations above, can you maintain that the local fields at Alice, Source and Bob are independent parameters? (to be clear, I mean independent in a strict mathematical way, I know that there is no non-local instantaneous conection between them)

If you replace the Alice Z coordinate in the Alice's field equation you get the fields at Source, or Bob. The three local fields are as dependent as you can get.

At this point we can ignore the distant sources. Our experiment begins and the evolution of Alice, Bob and Source only depends on the local fields. Now, this is the place where your reasoning fails. Their evolution is still not independent because the dependency was already there in the initial values of their local fields. As the time passes, those correlations are maintained (We are simply doing the same mathematical transformation on the three correlated fields). In the absence of some indeterministic process those correlations will remain forever.

I will give you an answer to all the points you have raised, but now I have to depart from the computer, sorry.