# Bell's theorem

#### DrChinese

Gold Member
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.
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

Staff Emeritus
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

Staff Emeritus
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

Staff Emeritus
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

Mentor
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

Mentor
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

Staff Emeritus
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

Staff Emeritus
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:
$A_i, A'_i, B_i, B'_i$
we compute 4 correlations:

1. $\langle A, B \rangle$ = average of $A_i B_i$
2. $\langle A, B' \rangle$ = average of $A_i B'_i$
3. $\langle A', B \rangle$ = average of $A'_i B_i$
4. $\langle A', B' \rangle$ = average of $A'_i B'_i$

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 $A'_i$. She can't measure both. Similarly, Bob must decide whether to measure $B_i$ or $B'_i$. 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.

#### atto

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 $A'_i$. She can't measure both. Similarly, Bob must decide whether to measure $B_i$ or $B'_i$. He can't measure both.
The experiments must provide these lists, and this is just the raw data, measured during experiment on both sides.

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.
In that case you lose completely the context of these Bell-type inequalities.

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.
To such, let say: a free-version of the problem, applies quite different inequality,
and it has higher limit, because up to 4, thus QM still breaks nothing!

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.
QM shows nothing special in this area. We know very well the formal mathematical truths are universal, unbreakable, indestructible.
The experimental tests/verification of the mathematical theorems are completely pointless.

#### Nugatory

Mentor
The experimental tests/verification of the mathematical theorems are completely pointless.
The experimental tests are not verifying the correctness of the mathematical theorem - we know that it's correct (unless there's an error in the proof and no one has found one in the past century, so that's not a serious possibility).

The theorem is stated in the form (I've already posted a link to Bell's original paper, and for this discussion we probably need to focus on that) "If A then B", and therefore "If not B then not A". That's the theorem, and no one is arguing about it.

The purposes of the experiments is to discover whether B is false; if it is then the mathematical logic of the theorem "if not B then not A" tells us that A is false.

#### atto

The theorem is stated in the form (I've already posted a link to Bell's original paper, and for this discussion we probably need to focus on that) "If A then B", and therefore "If not B then not A". That's the theorem, and no one is arguing about it.

The purposes of the experiments is to discover whether B is false; if it is then the mathematical logic of the theorem "if not B then not A" tells us that A is false.
I don't know what represent the A, B.

The EPR-tests were designed to verify some inequalities, never the whole reality, nor the basics of math.

#### DrChinese

Gold Member
The point I am trying to make is about the failure of the freedom assumption, not about locality. ...
The "failure of freedom assumption" means that Alice and Bob's choice of measurement settings are not free. In that view, those too is a function of the parameters you claim are somehow tied up in the other parameters you are mentioning. But that cannot be! There is no known influence of those parameters on the human brain!! (Except of course in superdeterminism.) And if you care to postulate some connection, it can be ("easily") falsified.

Just to remind everyone what is at stake here, let's use my usual example of Type II entangled photons with possible angle settings 0, 120 and 240 degrees. The QM prediction for Alice and Bob to match is 25% when their settings are different. The local realistic prediction is not less than 33%. So for an example where Alice is checking at 0 degrees and Bob is checking at 120 degrees for a run, we might expect something like this (and in this case Alice and Bob are told to make their setting choices according to DrChinese):

0 120 240 Alice&Bob Match / Total Matches / Total Permutations
+ - - 0 1 3
- + - 0 1 3
- - + 1 1 3
+ - + 0 1 3
+ - + 0 1 3
+ - - 0 1 3
+ + - 0 1 3
- + + 1 1 3
Total 2 8 24
(sorry these don't line up quite right)

Note that it is certainly possible to have Alice and Bob see 25% match rate (2 of 8 runs). But regardless of how you pick ‘em, the total match rate cannot be less than 33% (8 of 24 permutations, and note the 16 of the permutations are counterfactual). So whenever we say a local realistic theory is occurring, we have something like the above. And that means that there is something privileged about Alice and Bob’s choice of settings. That is because the 0/120 degree combination of settings has a 25% match rate (matching QM), while the 0&240 combo has a 37.5% match rate (3 of 8) and the 120&240 combo also has a 37.5% match rate (3 of 8). In any local hidden variable theory purporting to mimic QM via loopholes or failed implied assumptions, the true universe (including counterfactuals) cannot match the observed sample.

Now suppose Alice and Bob left their settings alone for long enough to have 1,000,000 runs instead of just 8. The Alice&Bob pair has 250,000 matches (same 25%) and this is the local realistic summary (give or take a few) when we extrapolate:

0&120: 250,000 of 1,000,000, or 25% (this is the Alice&Bob setting)
0&240: 375,000 of 1,000,000, or 37.5% (this is a counterfactual setting)
120&240: 375,000 of 1,000,000, or 37.5% (this is a counterfactual setting)

Clearly, there is something “preferred” about the Alice & Bob setting pair, else the results would be consistent! If you are getting 25% there, you are getting something much different on the counterfactual ones. Note that we have agreed that we get the same result when Alice and Bob make independent decisions (ignoring the instructions from DrChinese) and they make their decisions outside each others’ light cones. Weihs et al (1998).

So I appreciate that you are saying it is possible to have a violation of the freedom assumption if classical dependencies exist. But perhaps you can explain how, out of the 1000000 runs, the results match the QM prediction AND yet are wildly different from the local realistic average, using any classical idea at all. Because you are essentially saying that the results are observer dependent (Alice&Bob results are different from the 0&240 and 120&240 combos) while simulanteoulsy saying that the choice of Alice&Bob’s settings is correlated to DrChinese’s instructions above, transmitted through PhysicsForums.com via this post.

Wait, perhaps I have special powers! That would explain a lot. Or perhaps you can acknowledge that superdeterminism, that mystical theory which is yet to be unveiled, is nothing at all like determinism. And if we follow the requirements of superdeterminism to their logical conclusion, it will be seen that a local realistic rendering must bear elements that are unscientific by almost any standard.

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#### atto

You just try to analyze the consequences of the impossible lists of outcomes/measurements, which break the inequality.
But these lists don't exist, fortunately, there is nothing to analyze.

Although, on the other hand, you can analyze this scenario.
We assume that Alice has a knowledge of the Bob's chooses;
for example she has a magic crystal ball, through which she sees images instantly from a distance, and so on.

That fantastic 'possibility' has been even implemented in many movies, for example: Star Trek, Stargate, etc. :)

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#### Nugatory

Mentor
I don't know what represent the A, B.

The EPR-tests were designed to verify some inequalities, never the whole reality, nor the basics of math.
Have you read the paper yet? (I'll repeat the link so people who new to this thread won't have to dig back through it to find it: http://www.drchinese.com/David/Bell_Compact.pdf)

When I say that the theorem is of the form If A then B, A represents the "vital assumption" stated after equation 1 in the paper, and formalized in the integral in equation 2; and B represents the inequality.

#### atto

Have you read the paper yet? (I'll repeat the link so people who new to this thread won't have to dig back through it to find it: http://www.drchinese.com/David/Bell_Compact.pdf)

When I say that the theorem is of the form If A then B, A represents the "vital assumption" stated after equation 1 in the paper, and formalized in the integral in equation 2; and B represents the inequality.
OK. This is a formal proof of the inequality.
So, it's true - it can't be violated in any way.

#### rubi

OK. This is a formal proof of the inequality.
So, it's true - it can't be violated in any way.
It can be violated by a theory in which $P(\vec a,\vec b)$ isn't of the form as given by equation (2). This is the case for QM.

#### Nugatory

Mentor
OK. This is a formal proof of the inequality.
So, it's true - it can't be violated in any way.
No, it is a formal proof that if a certain precondition holds, then the inequality cannot be violated in any way.

It's like the Pythagorean Theorem, which says that if a triangle is a right triangle then the sum of the squares of the lengths of the two sides will equal the square of the length of the hypotenuse - it can be and is violated by any triangle that is not a right triangle.

The experiments that measure whether Bell's inequality is violated are analogous to measuring the sides of a given triangle to see if the sum of the squares of the lengths of the two shorter sides is equal the square of the length of the long side. If it's not, then the Pythagorean theorem tells us that that triangle is not a right triangle.

#### stevendaryl

Staff Emeritus
The experiments must provide these lists, and this is just the raw data, measured during experiment on both sides.
...
QM shows nothing special in this area. We know very well the formal mathematical truths are universal, unbreakable, indestructible.
The experimental tests/verification of the mathematical theorems are completely pointless.
I think you are confused about this topic. You seem to be expressing a view that is at odds with what everyone else has said about Bell's inequality. As I said recently in a different thread, the fact that something is an establishment view doesn't make it right, but it makes Physics Forums the wrong place for you to be arguing about it.

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