Bell's Theorem with Easy Math - Stuck

[Badvok]

If A(a,λ)=a+Q+R and B(b,λ)=b+Q+S, then λ includes factors that are local to both measurement devices, yet A(a,λ) is unaffected by anything that happens at device b and B(b,λ) is unaffected by anything that happens at device a.

Sorry, I guess this is maths that I don't understand, how can A(a,λ)=a+Q+R not include S when S is part of λ.

 

[Nugatory]

Sorry, I guess this is maths that I don't understand, how can A(a,λ)=a+Q+R not include S when S is part of λ.

Continuing with the trivial examples (and using trivial examples because I'm pretty sure that you're just getting hung up on Bell's notation here):

A(a,λ)=a+Q+R+(S-S) includes S but the value of A still doesn't depend on S.

More generally, λ is a set of conditions, and nothing requires that you use every member of that set in the definition of every function of that set. If a theory says that A(a,λ) uses the B-local conditions (except in the trivial self-cancelling sort of way that I just did), then that theory is non-local. Bell's theorem is a statement about the behavior of theories that are not non-local in this sense,

 

[DrChinese]

Sorry, I guess this is maths that I don't understand, how can A(a,λ)=a+Q+R not include S when S is part of λ.

It could, IF you wanted to switch to a NON-LOCAL version of hidden variables.

But otherwise, the shared variables do not include information about the measuring devices. The measuring devices can include any number of variables though, as long as a doesn't depend on b and vice versa.

 

[Badvok]

Continuing with the trivial examples (and using trivial examples because I'm pretty sure that you're just getting hung up on Bell's notation here):

A(a,λ)=a+Q+R+(S-S) includes S but the value of A still doesn't depend on S.

More generally, λ is a set of conditions, and nothing requires that you use every member of that set in the definition of every function of that set. If a theory says that A(a,λ) uses the B-local conditions (except in the trivial self-cancelling sort of way that I just did), then that theory is non-local. Bell's theorem is a statement about the behavior of theories that are not non-local in this sense,

Thanks, but Bell then goes onto express an expectation value as the integral with respect to λ of the product of A, B, and the probability distribution of λ. Again, I'm unsure how that can work when there are different λs.

 

[DrChinese]

Thanks, but Bell then goes onto express an expectation value as the integral with respect to λ of the product of A, B, and the probability distribution of λ. Again, I'm unsure how that can work when there are different λs.

Shared set λ (since λ are those local variables present when entanglement begins); while sets a and b are not shared. So there are 3 total sets of variables. The only restriction is that a is not shared with b, and vice versa.

 

[DrChinese]

And again, I would suggest trying to provide a specific example to work through so you can see the difficulties with your ideas. For example, suppose there is some formula, the answer to which is +/- or 1/0 or similar. Make the components of that formula such that we can get an answer with different inputs. Try to fix it so that the result is a perfect correlation when a and b are the same on one parameter (which we will associate with angle setting).

For example: suppose we get 0 if the result of our function is even, 1 if the result is odd. Our function is simply a sum of the inputs (this is not supposed to be a serious example in any physical sense.

The EntangledSourceHV1 (shared) is 6.
The AliceHV1 (not shared) is 9.
The BobHV1 (not shared) is 13.
The AliceMeasementAngle is 2.
The BobMeasementAngle is 2.

Alice's result = A(EntangledSourceHV1, AliceHV1, AliceMeasementAngle) = A(6+9+2)=1 (since sum is odd)
Bob's result = A(EntangledSourceHV1, BobHV1, BobMeasementAngle ) = A(6+13+2)=1 (since sum is odd)

So this works out for the perfect correlation at angle=2, so that is good. And you can add as many HVs as you like using this idea.

Now, try varying the measurement hidden variables with each side. You will see that as long as they change in tandem, everything is fine - but not otherwise. But if they change in tandem, then they are not observer independent, are they?

 

[Badvok]

OK, it doesn't matter what factors affect the measurements so long as they are the same for both and not linked to the observer setting. If they were slightly different, e.g. magnetic field strength, then that would simply affect how close to perfect correlation the experiment could get but it would still be able to achieve better than classical physics would predict.

Still a bit confused about why A(a,λ) and B(b,λ) need to be restricted to ±1 though. Is this just to make the maths easier or is this a fundamental part of the proof itself?

I see the following image (or variations of it) on a lot pages that discuss Bell's Inequalities.
http://upload.wikimedia.org/wikipedia/commons/7/77/StraightLines.svg
This is used to illustrate the difference between the QM prediction (and experimental results) and a 'local realist' prediction. The straight lines of the LR prediction obviously arise simply from constraining A and B to ±1 but I do wonder what the graph would look like without this constraint.

 

[Ilja]

The restriction follows from the particular experiment considered here. Once what is observed are only two possible results, up and down, this particular experiment cannot be explained by theories which allow for three, four or more possible results.

There may be other experiments, with other possible results and, therefore, other mathematical proofs and other resulting inequalities. But this is not quite relevant. If our world is local (better Einstein-causal) and realistic, this particular experiment needs an explanation in terms of such an Einstein-causal realistic theory. Once this is impossible for the particular experiment, Einstein-causal realism is dead.

 

[Badvok]

Thanks again for all your help with this.

If anyone is interested I've knocked up a little JavaScript model to play with the various assumptions so you can see the effects they have (though of course nothing allows you to get closer to the quantum predictions/real test results). You can even tweak the rate at which entangled photons are generated so you can get closer to a realistic simulation. It is on my home server here. There's no advertising or anything nasty there, just a very simple page with some script. Feel free to take it, and reuse it or change it if you wish. If I get the time I may later expand it to include some pretty graphical animations.