Local Hidden Variable Model That Equals QM Predictions?

[morrobay]

With E a,b = ( dλ ( a,a' b,b' λ) Suppose λ depends independently/locally on detector setting choices at A and B.
For example suppose there are 360 detector settings at A and 360 at B , corresponding to 360 particle/detector interactions/outcomes at A = ± 1 and also at B = ± 1. Then as θ = β-α there are 360² possible thetas.
If these thetas when applied to: P++ = P-- = 1/2 (sin θ/2)² and to P-+ = P+- = 1/2/(cos θ/2)²
are in agreement with QM predictions: Bell inequality violations
( that of course include the 360 cases when α = β , sin 0 = 0 , cos 0 = 1)
Then could a computer simulation for the 360² θ's verify if this local,deterministic hidden variable model fits the facts ?

 

[PeterDonis]

Bell's theorem shows that it is mathematically impossible for a model of the type you describe to match the QM predictions for all possible combinations of angles for A and B. Such a model can match QM predictions for some combinations, but not all combinations. So no such model can fit the facts.

 

[entropy1]

You seem to assume knowledge of θ. I am not sure if that is local.

 

[Zafa Pi]

With E a,b = ( dλ ( a,a' b,b' λ) Suppose λ depends independently/locally on detector setting choices at A and B.
For example suppose there are 360 detector settings at A and 360 at B , corresponding to 360 particle/detector interactions/outcomes at A = ± 1 and also at B = ± 1. Then as θ = β-α there are 360² possible thetas.
If these thetas when applied to: P++ = P-- = 1/2 (sin θ/2)² and to P-+ = P+- = 1/2/(cos θ/2)²
are in agreement with QM predictions: Bell inequality violations
( that of course include the 360 cases when α = β , sin 0 = 0 , cos 0 = 1)
Then could a computer simulation for the 360² θ's verify if this local,deterministic hidden variable model fits the facts ?

I believe that both @PeterDonis and @entropy1 are correct, but the latter is more directly germane to your proposal. Alice & Bob are far apart (not local) when setting their angles and neither knows what the other's setting is. So how does one know which of your 360² to select? If A & B are close and can communicate then all bets are off, i.e. they can make any correlations they want.

 

[morrobay]

I believe that both @PeterDonis and @entropy1 are correct, but the latter is more directly germane to your proposal. Alice & Bob are far apart (not local) when setting their angles and neither knows what the other's setting is. So how does one know which of your 360² to select? If A & B are close and can communicate then all bets are off, i.e. they can make any correlations they want.

All the 360² possible thetas are applied for computing all the possible outcomes of the model to see if it equals QM predictions.
As well as recorded experimental outcomes. E (a.b) = ∫ dλ C (a a' b b' λ)
Model outcomes only depend on settings at space like separated A and B and λ (particle properties)
For photons all the thetas are applied to E (a,b,) = cos² θ - sin² θ
S = E ( ab ) - E(ab') + E (a'b) + E (a'b')
S_hv ≤2 S_QM = 2√2
For spin 1/2 particles all the thetas are applied to the cos² and sin² formulas to produce model curve for comparisons to this graph.
Or could be applied to any inequality of this sort: N(a+b-) + N(b+c-) ≥ N(a-c+)
So knowledge about theta (β-α) is not applicable for the model testing QM predictions. Only a computer simulation for all thetas..

 

[DrChinese]

All the 360² possible thetas are applied for computing all the possible outcomes of the model to see if it equals QM predictions...

The thing is, as PeterDonis mentioned, this is exactly what has been looked at many times before. Bell's Theorem precludes this from working. You only need to check it for 3 pairs of angles to see the problem:

0/120 degrees
120/240 degrees
0/240 degrees

Try hand inserting actual values for these 3 angles 0/120/240 for a series of trials (it is easier if you work with the correlated case rather than the anti-correlated case). These cannot have pairwise values that match (or mismatch depending on setup) less than 1/3 (unless you know in advance which pair you are going to choose). QM predicts 1/4. This more or less corresponds to the graph you provided for the 30 degrees and 60 degrees cases.

When you hand insert values - that's so you can cherry pick to try and make it work out - you realize quickly that you can only make things work out if you cheat. I.e.you know which pair of angles you are selecting in advance. And if you cheat like that, you can make any formula work out. Even the QM prediction.

 

[Zafa Pi]

@morrobay, are you proposing a way to violate a Bell inequality without using entangled particles?

 

[morrobay]

@morrobay, are you proposing a way to violate a Bell inequality without using entangled particles?

Not at all. This is a train of thought continued from the Entangled Particles thread, page 1. See my post #32. Item 2. Correlations encoded during preparation of entanglement.

 

[Zafa Pi]

Not at all. This is a train of thought continued from the Entangled Particles thread, page 1. See my post #32. Item 2. Correlations encoded during preparation of entanglement.

Are you trying to give an explanation for the correlations that infect entangled particles? Why do you need a 360 by 360 table as @DrChinese pointed out a 3 by 3 should suffice? If you want a classical explanation, how about dBB or ER = EPR?

I don't know why I'm saying all this since I really don't understand what you're trying to do.

 

[entropy1]

Not at all. This is a train of thought continued from the Entangled Particles thread, page 1. See my post #32. Item 2. Correlations encoded during preparation of entanglement.

Yes and the probability for opposite spins : P-+ = P+- = 1/2 ( cos θ/2)² That are dependent on both detector settings , θ = β - α
What are the alternative explanations for the correlations between spacelike separated entangled particles that do not include a superluminal signal ?
In this paper http://www.mathpages.com/home/kmath731/kmath731.htm and elsewhere are the terms:
1. " QM non separability"
2. Correlations " encoded during preparation of entanglement .
Then how are 1 and 2 in accord with P++ = P-- = 1/2 (sin θ/2)² that imply a dependence on detector settings A and B for the distant correlations that violate Bell's inequality , but again do not include superluminal signals ?

What is the matter about Item 2, @morrobay ?

 

[Zafa Pi]

What is the matter about Item 2, @morrobay ?

Hidden variables = item 2, for what its worth.

 

[morrobay]

Hidden variables = item 2, for what its worth.

If you scroll down in the paper to page 5/13 (on printout) to :
A plot of the simple linear correlation profile and the qm profile is shown below.
PLOT
These profiles agree only for measurements differing by 0, π/2 and π. For all other cases the simple pre-programmed linear model fails to match the qm predictions,
( sin(θ/2)² and cos(θ/2)² formulas. ) That is understood.
Now the following is exactly what my question in this topic is:
" This raises the interesting question of whether any pre programmed response profile can reproduce the predictions of qm (and the experimental results)
Suppose each particle is programmed with a more complicated profile of responses as a function of the measurement angle"
This would incorporate the 360² thetas in my proposal.
The paper then continues to show why this clearly ruled out.
I notice that you Zafa Pi have a math background. Perhaps you could elaborate on why this model is ruled out. thank you.
http://www.mathpages.com/home/kmath521/kmath521.htm

 

[Jilang]

My understanding is that all probabilities must add to one and none can be negative. If you allow the negative ones however...

 

[DrChinese]

Suppose each particle is programmed with a more complicated profile of responses as a function of the measurement angle"
This would incorporate the 360² thetas in my proposal.
The paper then continues to show why this clearly ruled out. ...

Perhaps you could elaborate on why this model is ruled out.

If you can see it is ruled out (much as Bell first showed us), what more is there?

As to the 360^2: the 3 angles I provided (0, 120, 240 degrees) should ALSO be enough to demonstrate why your idea doesn't work. Just write out a sequence of those - say 10. You will see that no matter what values you provide, the average will be at least 1/3. The quantum mechanical prediction is 1/4. It works the same way on most any 3 angles, but the important point is that experiment rules out your idea (unless there is superluminal signalling).

And my 3 angles are way easier to calculate than yours - by a factor of more than 10,000,000. Just sayin'...

 

[Zafa Pi]

If you scroll down in the paper to page 5/13 (on printout) to :
A plot of the simple linear correlation profile and the qm profile is shown below.
PLOT
These profiles agree only for measurements differing by 0, π/2 and π. For all other cases the simple pre-programmed linear model fails to match the qm predictions,
( sin(θ/2)² and cos(θ/2)² formulas. ) That is understood.
Now the following is exactly what my question in this topic is:
" This raises the interesting question of whether any pre programmed response profile can reproduce the predictions of qm (and the experimental results)
Suppose each particle is programmed with a more complicated profile of responses as a function of the measurement angle"
This would incorporate the 360² thetas in my proposal.
The paper then continues to show why this clearly ruled out.
I notice that you Zafa Pi have a math background. Perhaps you could elaborate on why this model is ruled out. thank you.
http://www.mathpages.com/home/kmath521/kmath521.htm

Neither A or B knows what angle the other measured, and they are far apart. How do your preprogramed entities know what values to deliver from your table? Are you proposing they somehow know what A and B did and thus immediately give the appropriate responses to match those of the entangled photons?

 

[cube137]

just a simple inquiry.. I know there are non-local hidden variables.. but are there non-deterministic hidden variables too or are all hidden variables deterministic?

if there are nondeterministic-non local hidden variables.. how does this differ to Copenhagen then?

 

[Zafa Pi]

@morrobay, You can find the clarification in Bell's original paper.

I'm having computer problems with this site, so I'm quitting for a while.

 

[morrobay]

Neither A or B knows what angle the other measured, and they are far apart. How do your preprogramed entities know what values to deliver from your table? Are you proposing they somehow know what A and B did and thus immediately give the appropriate responses to match those of the entangled photons?

1⁰ through 360⁰ measurement outcomes are recorded at both A and B. Then all 360² thetas
can be produced for comparing experimental and calculated results.. The question now is not on experimental setup
but rather a section in this paper ; http://www.mathpages.com/home/kmath521/kmath521.htm
( Scroll down to the first plot shown of simple linear correlation profile and QM profile.Then in the paragraph below the plot start here:
" Suppose each particle is programmed with a more complicated profile of responses as a function of the measurement angle"
Can you elaborate on the math shown that rules out such a particle being in agreement with QM
predictions in relation to the Bell inequality ?

 

[Nugatory]

Can you elaborate on the math shown that rules out such a particle being in agreement with QM
predictions in relation to the Bell inequality ?

What sort of elaboration are you looking for? If you could tell us which part is not clear and needs further explanation, we may be able to help.

 

[morrobay]

1⁰ through 360⁰ measurement outcomes are recorded at both A and B. Then all 360² thetas
can be produced for comparing experimental and calculated results..

Before referencing the math in question I want to restate/update. The original idea was that there could be pre programmed and fixed responses for entangled particles all 360⁰ settings such that outcomes at any combination of angles at spacelike A and B could agree with QM predictions and therefore violate the inequality. Ie the long distance non classical correlations are encoded during entanglement preparation. So there are 360² possible thetas.
Now there is a sidetrack question: If as said above,360 measurement outcomes are recorded at A and 360 at B . Not at parallel settings but random so that each A and B have outcomes from stream of identically prepared entangled particles for 360 settings. . Is it valid to combine them all to make up the 360² thetas. β-α
In other words is it valid for this particular model to combine an outcome at setting at A , 80⁰ from one pair. And then from another pair an outcome at B , at 333⁰ . To say this way: in this model could the results from two different pairs, A at 80⁰ and B at 333⁰ be equal to outcomes for one pair measured at A, 80⁰ and B 333⁰ ? If this is invalid then all 360² measurements could be made.

Now the math in question: (4) The integral that = - cos(θ).
If this equals the QM prediction for the correlation from the said pre pro grammed particle then why is it ruled out in the following paragraph:
"This is because the increase in correlation is proportional to the increase in θ arising from the transition at α = π - θ

 

[Nugatory]

The original idea was that there could be pre programmed and fixed responses for entangled particles all 360⁰ settings such that outcomes at any combination of angles at spacelike A and B could agree with QM predictions and therefore violate the inequality. Ie the long distance non classical correlations are encoded during entanglement preparation.

And that is not possible. Bells theorem basically says that if the results are encoded during preparation, then the correlations must obey the inequality. And as @DrChinese has repeatedly pointed out, you should try finding an encoding that leads to a violation of the inequality for just three angles ($0$,$2\pi/3$, $4\pi/3$); you won't be able to.

Now there is a sidetrack question: If as said above,360 measurement outcomes are recorded at A and 360 at B . Not at parallel settings but random so that each A and B have outcomes from stream of identically prepared entangled particles for 360 settings. . Is it valid to combine them all to make up the 360² thetas. β-α
In other words is it valid for this particular model to combine an outcome at setting at A , 80⁰ from one pair. And then from another pair an outcome at B , at 333⁰ . To say this way: in this model could the results from two different pairs, A at 80⁰ and B at 333⁰ be equal to outcomes for one pair measured at A, 80⁰ and B 333⁰ ? If this is invalid then all 360² measurements could be made.

You can do that, but the resulting table will not have the property that when the two angles are the same, the results of the two measurement are never the same. Thus, it fails to match observation even before we consider any inequalities.

 

[entropy1]

@morrobay Perhaps you mean to form a table for A and B out of actual experimental data. You have to keep in mind then that the prefabricated data for A and B is local. The orientation of the detectors may be changed while the pair of particles is already produced (and the HV with them). In that case, if A=0°, then we have a set of 360 outcomes for possible angles for B. But if we, say, choose 10° for A, we have the same 360 possible outcomes for B! A and B are separated, and will not agree to QM outcomes.