Bell's theorem and experiments for dummies

[PRo]

I know that this place may not be appropriate for such discussions, but here goes.
Is there a possibility to dumb down those alleged proofs of ontological randomness to a very low level (me included) of understanding mathematics and physics?
Please help me to correct any errors and improve the following attempt to assess Bell's theorem and experiments.

Assumptions
(A1) A source emits a pair of particles with some opposite pieces of information.
(A2) The pieces of information from (A1) remain constant over time.
(A3) A detector can measure the pieces of information from (A1) via three methods [X, Y and Z], but only one method at a time can be used. Each method measures a specific part of the information.
(A4) There are only two values possible for the measurments from (A3). [+ and -]

Theoretical Results
A source emits pairs of particles as per the above assumptions.
One particle is sent to a detector (Alice) and the other to another detector (Bob).
There are maximum 8 combinations of particles and 9 combinations of measurements:

Code: Select all

    Particles for          Results [Alice|Bob]          Opposite
    Alice and Bob   1   2   3   4   5   6   7   8   9   results
    X Y Z   X Y Z  X|X X|Y X|Z Y|X Y|Y Y|Z Z|X Z|Y Z|Z  probability
A   + + +   - - -   O   O   O   O   O   O   O   O   O   1
B   + + -   - - +   O   O   I   O   O   I   I   I   O   5/9
C   + - +   - + -   O   I   O   I   O   I   O   I   O   5/9
D   + - -   - + +   O   I   I   I   O   O   I   O   O   5/9
E   - + +   + - -   O   I   I   I   O   O   I   O   O   5/9
F   - + -   + - +   O   I   O   I   O   I   O   I   O   5/9
G   - - +   + + -   O   O   I   O   O   I   I   I   O   5/9
H   - - -   + + +   O   O   O   O   O   O   O   O   O   1

From the above we get the following equivalent of Bell's inequality (?):
Probability of getting opposite results at the detectors > 5/9

Actual Results
We consider the case of two entangled electrons, for which their spin is measured at three angles: X=0°, Y=120°, Z=240°.
Probability of getting opposite results at the detectors = 1/2 (?)

Conclusions
This is where I'm stuck, because we also have the following inequality (if we make the theoretical experiment for an equal amount of measurement combinations):
The ratio between the number of opposite results and total number of measurements > 3/9 = 1/3

For me all this seems to be like having a dice with two faces "O" and the other four "I", for which the probability is 1/3 to get "O". Then we make the actual throws and somehow reliably get "O" 1/4 of the time, which is logically possible... ?

[Mikko]

The assumption that all cases A .. H are equally probable is not justified. By symmetry A and H should have the same probability. Same way B .. G should have the same probability. But there is no symmetry between A and B, so these probabilities need not be the same. And even unsymmetric probabilities might be possible.

When the result inferred from the assumptions differ from the real result, at least one of the assumptions is too strong, perhaps totally wrong. Which one and how wrong are still an open questions. One possibility is that the measurement of one particle immediately affects the other particle, in spite of the distance between them. Perhaps some experiment could refute this possibility (called "non-locality") but the experiment discussed here doesn't. Anyway, if you consider such non-locality too incredible, you may consider relaxing some other assumption. Perhaps you can imagine other experiments to guide in the selection of the assumptions.

[PRo]

The assumption that all cases A .. H are equally probable is not justified. By symmetry A and H should have the same probability. Same way B .. G should have the same probability. But there is no symmetry between A and B, so these probabilities need not be the same. And even unsymmetric probabilities might be possible.

Doesn't using the > sign eliminate this issue?

When the result inferred from the assumptions differ from the real result, at least one of the assumptions is too strong, perhaps totally wrong. Which one and how wrong are still an open questions. One possibility is that the measurement of one particle immediately affects the other particle, in spite of the distance between them. Perhaps some experiment could refute this possibility (called "non-locality") but the experiment discussed here doesn't. Anyway, if you consider such non-locality too incredible, you may consider relaxing some other assumption. Perhaps you can imagine other experiments to guide in the selection of the assumptions.

Putting aside the spooky action at a distance, one of the things that I'm saying is that the actual results are still logically possible within the framework of assumptions A1 to A4, if my analogy with the dice is correct. But this brings into question how the particles are emitted, how random the process is?

I can't edit my OP, there the last inequality should've been (forgot the equal sign):
The ratio between the number of opposite results and total number of measurements > 3/9 = 1/3
Doesn't this show that the actual results (if for this case is indeed 1/2) don't contradict the theoretical expectations?

I have to say that I found this forum via Joy Christian's activity on the web about the subject. If I understand correctly his objection is that the codomain of the measuring function (?) is wrong. But beyond this I don't have the ability to follow his reasoning. Does this mean that assumptions (A2) or/and (A4) must be reformulated and then we won't have any contradiction between the theory and experiments?

[Mikko]

The assumption that all cases A .. H are equally probable is not justified. By symmetry A and H should have the same probability. Same way B .. G should have the same probability. But there is no symmetry between A and B, so these probabilities need not be the same. And even unsymmetric probabilities might be possible.

Doesn't using the > sign eliminate this issue?

Yes, if you assume that all probabilities are non-negative.

Putting aside the spooky action at a distance, one of the things that I'm saying is that the actual results are still logically possible within the framework of assumptions A1 to A4, if my analogy with the dice is correct. But this brings into question how the particles are emitted, how random the process is?

As I said, you might consider different probabilities for A .. H. You could also consider varying probabilities. But as long as probabilities of measurement diretions (whether X or Y or Z) are always equal the result holds. So the source randomness doesn't matter as long as the choice of measurement and particle properties are independent. The probability that the results are different is still at least 5/9 (asuming that no probabilities can be negative), which is greater than the observed 1/2. So either some other assumption is false or the choice of measurement and particle properties are not independent.

[PRo]

I don't know what are negative probabilities. Maybe the inequality about the probability of getting opposite results being > 5/9 isn't equivalent to Bell's and indeed wrong. If that's the case, can you help me fix it?

But if indeed the actual results give 1/2, then is it correct to say that it doesn't contradict the initial assumptions?
For e.g. we can have pair B of particles for measurements 1, 3, 5, 6, 7, 8, 9 and pair C of particles for the other measurements 2 and 4. That would give us a ratio of 1/3 for opposite results.
So, to conclusively contradict the assumptions shouldn't the actual results give a ratio smaller than 1/3?

Could it be that Bell's inequality should be derived as follows (?):

Code: Select all

    Particles for          Results [Alice|Bob]
    Alice and Bob   1   2   3   4   5   6   7   8   9
    X Y Z   X Y Z  X|X X|Y X|Z Y|X Y|Y Y|Z Z|X Z|Y Z|Z
A   + + +   - - -   O   O   O   O   O   O   O   O   O
B   + + -   - - +   O   O   I   O   O   I   I   I   O
C   + - +   - + -   O   I   O   I   O   I   O   I   O
D   + - -   - + +   O   I   I   I   O   O   I   O   O
E   - + +   + - -   O   I   I   I   O   O   I   O   O
F   - + -   + - +   O   I   O   I   O   I   O   I   O
G   - - +   + + -   O   O   I   O   O   I   I   I   O
H   - - -   + + +   O   O   O   O   O   O   O   O   O
Opposite results    1  4/8 4/8 4/8  1  4/8 4/8 4/8  1
probability

Then we would have an inequality that isn't violated by the actual results (if those are indeed 1/2):

Probability of getting opposite results at the detectors > 4/8 = 1/2 (?)

[FrediFizzx]

Could it be that Bell's inequality should be derived as follows (?):

Seems like you are caught up in the mess that Bell's original inequality can't be applied successfully to a real EPR type experiment. That is why the CHSH inequality was created. However, it has problems also which you can find out more about by exploring this forum.

[FrediFizzx]

I have to say that I found this forum via Joy Christian's activity on the web about the subject. If I understand correctly his objection is that the codomain of the measuring function (?) is wrong. But beyond this I don't have the ability to follow his reasoning.

Yes, that I believe is correct. Bell's argument restricts the +/1 outcomes to the real line where they should actually be points on a parallelized 3-sphere. Joy's model for the EPR scenario is actually quite easy to understand and his analytical results have been validated numerically by the computer program GAViewer. I will be happy to explain more if you are interested.

[Mikko]

I don't know what are negative probabilities.

It simply means that you put a negative number in a place of a probability in some formula. Usually considered nonsense, but in a situation that does not make sense anyway somebody might consider.

Maybe the inequality about the probability of getting opposite results being > 5/9 isn't equivalent to Bell's and indeed wrong.

There is nothing wrong in the result. It is related to Bell's inequality, but that is not really important. You can consider it as a distinct result in the same situation. But it has the same implications as Bell's inequality: at least one of your assumptions is wrong.

But if indeed the actual results give 1/2, then is it correct to say that it doesn't contradict the initial assumptions?

Your assumptions mean that the probability is at least 5/9. Experimentally correct result is 1/2. Both can't be right.

For e.g. we can have pair B of particles for measurements 1, 3, 5, 6, 7, 8, 9 and pair C of particles for the other measurements 2 and 4. That would give us a ratio of 1/3 for opposite results.

That alone contradicts the assumptions. The production of the pair of particles and the choice of the measurements are made at the same time in three different places so not having the same measurements for each type of pair is a violation of locality.

So, to conclusively contradict the assumptions shouldn't the actual results give a ratio smaller than 1/3?

There is no "should" about the actual result. But the assumptions require at least 1/2.

Could it be that Bell's inequality should be derived as follows (?):

It doesn't matter how do you derive Bell's inequality or whether you derive it at all. You don't need it. Your own inequalities describe the situation the way you can understand so they are in a sense better than Bell's.

Code: Select all

    Particles for          Results [Alice|Bob]
    Alice and Bob   1   2   3   4   5   6   7   8   9
    X Y Z   X Y Z  X|X X|Y X|Z Y|X Y|Y Y|Z Z|X Z|Y Z|Z
A   + + +   - - -   O   O   O   O   O   O   O   O   O
B   + + -   - - +   O   O   I   O   O   I   I   I   O
C   + - +   - + -   O   I   O   I   O   I   O   I   O
D   + - -   - + +   O   I   I   I   O   O   I   O   O
E   - + +   + - -   O   I   I   I   O   O   I   O   O
F   - + -   + - +   O   I   O   I   O   I   O   I   O
G   - - +   + + -   O   O   I   O   O   I   I   I   O
H   - - -   + + +   O   O   O   O   O   O   O   O   O
Opposite results    1  4/8 4/8 4/8  1  4/8 4/8 4/8  1
probability

Then we would have an inequality that isn't violated by the actual results (if those are indeed 1/2):

Probability of getting opposite results at the detectors > 4/8 = 1/2 (?)

The actual result is 1 for those measurements where the same chosen directions happen to be the same. For the remaining cases the actual probability is 1/4. The chosen directions are recorded with the observed results, so there each column can be analyzed separately.

[PRo]

For e.g. we can have pair B of particles for measurements 1, 3, 5, 6, 7, 8, 9 and pair C of particles for the other measurements 2 and 4. That would give us a ratio of 1/3 for opposite results.

That alone contradicts the assumptions. The production of the pair of particles and the choice of the measurements are made at the same time in three different places so not having the same measurements for each type of pair is a violation of locality.

I'm not sure what you mean by: not having the same measurements for each type of pair.

[Mikko]

I'm not sure what you mean by: not having the same measurements for each type of pair.

In the assumption A3 only one of three possible measurements are performed. You have not assumed anything about which one, but you will get right there a spooky action at distance unless you assume something about it. In a typical experiment the particle pair is created first and the particles separated before one of the three measurements are chosen. Therefore the properties of the particles cannot depend on the choice that later will be made. But the choice of the measurement is made before the particle arrives to the measurement site, so the choice of measurement cannot depend on the particles. You proposed different measurements for different particle pairs, but there is no way to do that without violating locality. (The assumption that the properties of the particle do not affect the choice of the measurement is often called "free will", the idea being that the experimenters can choose whichever measurement they want.)

[PRo]

What I'm proposing is that an hypothetical experiment is performed with all conditions fulfilled and we get the 1/3 ratio. So I ask, can this be possible? And it's possible if the source emits such and such pairs of particles for such and such combination of measurements, all this without violating anything. The same way I can find any ratio above 1/3, including 1/2.
I see no restriction that such scenarios can't happen even for the 1/2 ratio which seems to be experimentally validated.

[PRo]

Is there also the restriction that for the entire duration of the experiment the source emits pairs of particles with the same properties? If yes, then indeed the scenarios I highlighted can't happen. Until now I thought that the source emits the particles randomly, in the sense that their properties can randomly differ from one emission to another.

[PRo]

David Mermin describes this experiment but I don't find (maybe I missed it?) in it the restriction that the source is set up to emit only one type of particle pair for a complete set of measurements. Without such a restriction I don't see what prevents the scenarios I highlighted (which can give any ratio above or equal to 1/3) from happening. So how exactly is the source set up to emit the particle pairs?

[Mikko]

What I'm proposing is that an hypothetical experiment is performed with all conditions fulfilled and we get the 1/3 ratio. So I ask, can this be possible? And it's possible if the source emits such and such pairs of particles for such and such combination of measurements, all this without violating anything. The same way I can find any ratio above 1/3, including 1/2.
I see no restriction that such scenarios can't happen even for the 1/2 ratio which seems to be experimentally validated.

The source does not now what measurements the experimenters will choose to make. (Knowing it would be even spookier than action at distance if the measurement is not chosen yet.) Therefore it cannot but draw from the same probability distribution in each case.

[Mikko]

Is there also the restriction that for the entire duration of the experiment the source emits pairs of particles with the same properties

Only if the properties are the same in every experiment (which would be against the assumption A1). The actual result is that for each particle both measurement results are observed with equal probability.