quantum mechanics - Bell's theorem for dummies, how does it work?

I've been reading up on theoretical physics for a few years now and I feel like I am starting to get an understanding of particle physics, at least as much as you can from Wikipedia pages. One thing I have tried to understand but fails to make sense to me is Bell's Theorem. I understand that the premise of the ERP paper was that the "wave form collapse" couldn't work because it would require the two particles which made up the convoluted wave form to communicate instantly, violating the information speed limit. Instead they suggested that it was due to hidden variables (ie the values are already set, whether they have been measured or not).

My question is can any one explain to me how Bell's experiment works and how it disproves this in terms that don't require an in-depth understanding of the math behind quantum mechanics?

My current understanding of the experiment is you have two people who are reading a quantum value of entangle quantum particles (for my understanding lets say the spin state of a positron-electron pair produced by a pair production event). For each particle pair the two readers measure the spin at a randomly chosen angle.

Here is where I need clarification: if I understand it correctly, local realism hypothesis states that when measured on the same axis the spin states should always be opposite (.5 + -.5 =0, ie conservation) when measure on opposite axis the spin states should always be the same ( .5 - .5 = 0 ) and when measured 90 degrees apart, the values are totally random. This part I get. I believe these results are predicted to be the same by both local realism and quantum mechanics. The inequalities between the two hypotheses rise when the particles are measured on axes which are between 0-90 degrees off axis from each other, correct?

What I would like to have explained is the following:

1. 
   

   What are the predictions made by quantum mechanics?

   
   


2. 
   

   What are the predictions made by local realism?

   
   


3. 
   

   How do they differ?

   
   
   


4. 
   

   How is entanglement different from conservation?

   
   


5. 
   

   Any corrections in regard to my explanation above?

   
   

Answer

Bell's theorem shows that standard QM is inconsistent with local realism. Local realism is a very general principle that was not originally thought to make any testable physical predictions. A major part of Bell's achievement was showing that Bell's inequality is implied by local realism, while standard QM predictions violate it. Experiments like Aspect's have since shown that Bell's inequalities are violated in reality, refuting local realism, in a way that is consistent with standard QM.

I think your issue is with the definition of local realism:

when measured on the same axis the spin states should always be opposite (.5 + -.5 = 0, ie conservation) when measured on the opposite axis the spin states should always be the same (.5 - .5 = 0) and when measured 90 degrees apart, the values are totally random.

This is just what standard QM predicts for entangled particles.

Local realism states that what happens at any point can only be directly affected by the state in its immediate neighbourhood, any long range effects must be mediated by particles or field disturbances travelling at (sub)luminal velocities, and that all behaviour is deterministic.

If entangled particles are far enough apart that one can perform measurements on both of them in a way that ensures the measurement events are separated by a space-like interval then local realism would require the particles to carry enough hidden variables to predetermine the outcome of each possible measurement, since any effect from one measurements would not have time to propagate to the other measurement to enforce the correlated observations.

Local realism and Bell's inequalities are not violated when only measurements separated by integer multiples of 90 degrees like in your description are considered. The discrepancy between QM and local realism only appears when oblique angles are considered, reaching a maximum when the angle between the measurements is 45 degrees (plus some multiple of 90 degrees), when the correlation between the measurements becomes $\sqrt{2}$ greater than allowed by Bell's inequality and therefore by local realism.

Spin conservation is really a separate issue. It just says that if the total spin of an isolated system was $x$ at some point in the past then it will always be $x$ and vice versa. Entanglemnt provides a way of satisfying conservation laws without assigning definite values of the conserved quantities to the individual components.

Bell's theorem is really about local realism and not really about QM. Experimental results could in principle violate Bell's inequality but not agree with QM predictions either. This would still rule out local realism and all theories satisfying it. The fact that QM does predict correlations higher than allowed by Bell's inequality and experimental results do agree with those predictions is kind of incidental.