Hidden variables and Bell's inequality

[morrobay]

[Mentor's note: Moved from a thread about field theories as this is just about the basic meaning of the theorem for ordinary entangled particles]]

The essence of the argument for me is that for a hidden variable theory to reproduce the predictions of QM it's going to have to produce a correlation function that has a functional dependence on the relative angle of the detector settings. So in an appropriate frame if Alice changes her mind at the last moment about her measurement setting there can be no way this 'information' is transmitted to Bob's location before Bob's measurement - certainly not with a locally causal field. The role of the hidden variables is to make explicit the reasons for an observed correlation. So although the correlation happens because of some prior connection we can't apply the same reasoning to the last minute change of setting, which for want of a better word occurs pretty much in the 'present'. It's that potential change that must, somehow, be accommodated within our hidden variable description. How does that happen within a causal locally realistic description? I'm not seeing any possible physical explanation of that (within the context of a hidden variable theory).

Suppose that there are photon spin outcomes that are pre existing from entanglement or from local hidden variables.
For every detector angle Alice selects there is a result that is correlated with every detector angle Bob selects.
With 1⁰ units that would be 360 x 360 mutual outcomes.
* In order to have the probabilities that produce correlations that match quantum mechanical predictions;
opposite results cos² (β - α ) or same sin² (β - α ) you allow for the probability that Alice gets a given result on her particle (for any detector angle)
to vary with the direction that Bob chooses to measure on.
This could be a local hidden variable model for relative angles between detectors
*In part from post #12 comparison between quantum entanglement and a classical version.

 

[gill1109]

In order to have the probabilities that produce correlations that match quantum mechanical predictions;
opposite results cos² (β - α ) or same sin² (β - α ) you allow for the probability that Alice gets a given result on her particle (for any detector angle)
to vary with the direction that Bob chooses to measure on.
This could be a local hidden variable model for relative angles between detectors.

In a good experiment, Alice and Bob choose their measurement angles freely. Moreover, the time elapsed between choice of angle and registration of measurement outcome (on each side of the experiment) is so short (relative to the distance between the two measurement locations), that there is no way that Alice's angle could be known at Bob's location before Bob's measurement outcome is fixed.

 

[Truecrimson]

You can't refute the claim "Everything is determined ahead of time". But you can certainly refute a superdeterministic theory that makes definite predictions. If your superdeterministic theory predicts that "Bob and Alice will always choose the same detector setting", and they DON'T choose the same detector setting, then that particular superdeterministic model is refuted.

I totally agree with you. To clarify, "Everything is determined ahead of time" is what I call superdeterministic. Morrobay's hidden variable theory seems to fall into that type.

For every angle Alice sets at detector A there is a pre existing / hidden variable correlated outcome relative to any angle at detector B.

 

[morrobay]

I totally agree with you. To clarify, "Everything is determined ahead of time" is what I call superdeterministic. Morrobay's hidden variable theory seems to fall into that type.

Where do you draw the line between a (super)deterministic hidden variable theory where probability outcomes vary
with β - α and match QM predictions
And experiments that show maximum inequality violations, S_qm = 2√2 When a = -45⁰ , a' = 0⁰ b = 22.5⁰ , b' =22.5⁰
The outcome probabilities for above relative θ also vary in a determined, "ahead of time" and pre existing way ?

 

[Truecrimson]

I don't know what you are talking about. Experiment results are experiment results. They are not theory. There is a theory (QM) that predicts those results without positing pre-existing values of the observables.

 

[morrobay]

I don't know what you are talking about. Experiment results are experiment results. They are not theory. There is a theory (QM) that predicts those results without positing pre-existing values of the observables.

Let me restate the question (originally from post # 45) with emphasis on local hidden variables
rather than pre existing values that are associated with superdeterminism.
This local model allows for probability outcomes at A & B to vary with β - α based on physical mechanisms (interactions with particle , hidden variables and detector) and in agreement with QM predictions : S = 2√2
Then how can this LHV model be distinguished from QM theory when they both produce the same results ?
Ie: Could the LHV model produce the same curve that QM predicts
With the former the result of physical mechanisms that are not completely understood.

 

[zonde]

This local model allows for probability outcomes at A & B to vary with β - α based on physical mechanisms (interactions with particle , hidden variables and detector) and in agreement with QM predictions : S = 2√2

It can't do that. If LHV is not exploiting loophole it can't violate Bell inequality.

Edit:

Suppose that there are photon spin outcomes that are pre existing from entanglement or from local hidden variables.
For every detector angle Alice selects there is a result that is correlated with every detector angle Bob selects.

The whole point is that Alice's particle does not know Bob's detector angle so it can't select result based on that information.

 

[morrobay]

Wrong. There is nothing in the hidden variables which makes any assumption somehow "inspired" by point particles. The hidden variables may be anything, they may even live in 26 dimensional spaces or somewhere else, the other ingredients are the decisions of the experimenters what to measure and the results of the measurements, above are, obviously, macroscopic items, thus, in no way assume anything about microscopic theory.

This is what I cannot understand : " The hidden variables may be anything " So if hidden variable is such a flexible term ,
then why cannot hidden variables account for inequality violations ? A local hidden variable theory that matches QM predictions.

 

[Nugatory]

This is what I cannot understand : " The hidden variables may be anything " So if hidden variable is such a flexible term ,
then why cannot hidden variables account for inequality violations ? A local hidden variable theory that matches QM predictions.

Have you tried working through the mathematical logic in Bell's original proof of his theorem? That's really the best answer to your question. The next answer would be "No matter what they are, as long as they are local they cannot explain the correlations".

Basically, Bell shows that no matter what the hidden variables are, you need to use the hidden variables at both detectors (that is, non-local) to correctly calculate the correlations.

 

[DrChinese]

This is what I cannot understand : " The hidden variables may be anything " So if hidden variable is such a flexible term ,
then why cannot hidden variables account for inequality violations ? A local hidden variable theory that matches QM predictions.

This is the crux of Bell's Theorem: there are no such theories. That this is true, you can determine for yourself with some simple trial and error with photons (Type I produces matches at the same angles, mismatches at theta=90 degrees). Pick any 3 angles (not the same or different by multiples of 45 degrees). You cannot hand pick a dataset which will yield the proper QM average predictions - cos^2(theta) - for all 3 angles. If you can't even hand pick them, obviously there is no theory which will work either.

Example angles:
0, 30, 60
0, 22.5, 67.5
0, 10, 20

 

[entropy1]

Isn't the sheer fact that Alice can in principle change her measurement base at any time she wishes before measuring, a certain clue to that her measurement result, if correlated with Bob's, is communicated FTL (or non-local) to Bob? All terminology depending on the interpretation of QM of course.

 

[DrChinese]

Isn't the sheer fact that Alice can in principle change her measurement base at any time she wishes before measuring, a certain clue to that her measurement result, if correlated with Bob's, is communicated FTL (or non-local) to Bob? All terminology depending on the interpretation of QM of course.

No, you cannot say what is communicated or where. Or even whether it is from Alice to Bob or Bob to Alice. We would all be speculating at that point.

 

[eloheim]

This is what I cannot understand : " The hidden variables may be anything " So if hidden variable is such a flexible term ,
then why cannot hidden variables account for inequality violations ? A local hidden variable theory that matches QM predictions.

Here's the point: Imagine your little algorithm for Bob's result. It says, "if Bob measures at angle x, consult Alice's angle (y) for that same measurement, to get Bob's result (z)." Simple enough.

Now imagine the particles have been sent in opposite directions for a very long period of time, so they're very far apart. Bob sets his angle and makes his measurement. But to calculate his result, you need to know what angle Alice is picking for her (simultaneous) measurement halfway around the globe.

Bob gets his result as SOON as he makes his measurement (obviously), but how could the little algorithm give him that instant result when it needs to know what angle Alice is picking thousands of miles away? Of course, it can't.

Yes, it can give you a list of what the results WOULD be, given that Alice chooses whichever angle such-and-such for her particle measurement, but without actually knowing what Alice is up to at that moment, you can't calculate a specific, actual result when Bob needs it. Which is as soon he makes the measurement.

I hope this helps, perhaps? :-)