# 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?

• Sorry, could you try to clarify your fourth question? I don't quite understand what you're asking there... – Danu May 24 '14 at 17:11
• I guess this is the root question in a way. Conservations says when ever to particles interact the sum of their quantum values must equal that of the progenitors. Ie pair production makes a pair of particles whose charge, spin, momentum etc all are equal to the particle (photon) which created it. There for if you know the states of the photon and the states of one of the particles, then you know the state of the other particle. I guess this is realism, and if you answer the other questions, you'll answer this question. – jeffpkamp May 24 '14 at 17:32
• Perhaps an explanation via sets comparing them would help? youtube.com/watch?v=qd-tKr0LJTM – user6972 May 27 '14 at 17:36
• I like this article by David Mermin, "Quantum mysteries for anybody", web.pdx.edu/~pmoeck/pdf/Mermin%20short.pdf, and for a simple proof, I don't think you can do better than this arxiv.org/abs/1212.5214 – innisfree Jan 27 '15 at 15:31

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.

• I already understood most of what you described. The part that I don't understand is how the tests for bells inequalities work and what the results mean. The idea behind entanglement is that measurement of one particle alters its partner in some detectable way when measured at 45 degree angle ways, whereas L-R says measuring one doesn't affect the other because the values are already set. How do Bell tests prove the QM scenario? Are +45 angles more or less correlated than expected? Has anyone done a blind experiment where an eavesdropper could secretly randomly sample and show an effect? – jeffpkamp May 27 '14 at 14:00
• The essence of what Bell experiments really do is make measurements on a large number of entangled particle pairs and observe the correlation over the entire set. If it is higher than the Bell inequality then that contradicts LR. There is not observable effect of one particle on another when measuring an entangled particle. It is just in the statistics. The real meat is in Bell's proof that LR implies Bell's inequality. This means that experiments that contradict Bell's inequality also contradict LR. The correlation predicted by QM is higher than allowed by LR via Bell's inequality. – Daniel Mahler May 27 '14 at 17:03
• You assumed that hidden variables should be integer, while the hidden variable can take real numbers as well. In the case of Bell's inequality , local realism and quantum mechanics does not show any difference if we consider hidden variables as real numeric variables. – Alberto Sep 13 '16 at 20:31

My understanding is that the measurement at 45° matches the measurements at 0° and 90° more than it should (assuming local hidden variables) given how often 0° and 90° match.

Think of two detectors that move between 0°, 45°, and 90°, so that you get the 90° measurement when one is at 0° and the other at 90° and the 45° measurement when one is at 45° and the other at either 90° or 0°. When measuring 45° and one of the other two angles, you get a match 85% of the time. So 90° matches 45° 85% of the time, and 0° matches 45° 85% of the time – how often must 90° and 0° match? At least 70% of the time – 0°, 45°, and 90° would all match 70% of the time, and for the other 30%, half the time 45° would match with 0° and half the time it would match with 90°. 45° would match either angle 85% of the time - 70% when all three angles match, plus 15% when 45° matches one but not the other.

But when 90° and 0° are measured, they only match 50% of the time. What’s the most that 45° can equally match the other two? 50% of the time all three match, then the other 50% of the time when 90° and 0° do not match, 45° can only match one or the other. If it matches one half the time and the other the other half of the time, the highest percentage you can get is 75%. 50% for when all three match, then 25% of the time matching 90° and not 0° and 25% of the time matching 0° and not 90°.

1. What we actually see - 45° measurements matching 85% of the time, 90° measurements matching 50% of the time. This suggest that the angle of measurement of one particle has a correlation with the angle results of the measurement on the other particle.
2. Two separate things. If just looking at the results of 45° measurements, it says that 90° measurements must match at least 70% of the time (70% of the time when 0°, 45°, 90° all match plus 15% each for 0° and 90° when 45° matches one and not the other) . However, if looking at 0° and 90°, then it says that 45° can’t match the other two more than 75% of the time (50% when all three match plus 25% for each angle when 45° matches them and not the other).
3. Quantum predictions say that there can be a correlation between the angle of measurement of one particle and the result of the measurement on the other - even when there isn’t enough time between the final setting of the angle of measurement for one particle and the measuring of the other for light to travel between the two locations.
4. The correlation between the particles are connected to actions done to one of the particles.
5. I’d only argue with the wording of “completely random” for the 90° angle

(1) Quantum mechanics predicts 25% or more will correlate. (2) Bell says hidden variables should correlate 33% or more of the time. (3) The difference between the two is Bell's inequality. (4) They are different things unless I misunderstand your question. Two objects are entangled if you can measure or observe one of them and instantly know something about the other one. Conservation could mean some physical quantity in an isolated system is constant. (5) Your description of local realism seems complicated. My understanding is that a particle cannot get its instructions from a distant source that would take faster than light communication. Instead the particle most likely carried the instruction from its beginning.

• What is the differences between "particle most likely carried the instruction from its beginning." and "particle most likely had its state from its beginning." – Alberto Aug 21 '16 at 17:24
• @Alberto I said a "particle most likely carried the instructions from the beginning" but I did not say "most likely had it's state from its beginning". That would be close to the same thing. I said a "particle cannot get its instructions from a distant source" because that would take faster than light communication. That was my point. – Bill Alsept Aug 21 '16 at 18:07

To understand Bell's theorem it is not at all necessary to know anything about quantum mechanics. Essentially, it is sufficient if you believe that quantum theory predicts that it is violated even if the two measurements are space-like separated so that getting information what is measured at the other place is forbidden even by relativity.

http://ilja-schmelzer.de/realism/game.php gives a simple explanation how Bell's theorem works.

Bell proves first that, once both measure the same direction they get 100% correlation, but there cannot be information what has been measured on the other side, all measurement results should be predefined. Then he chooses three angles 0, 120 and 240 degrees. Assume now both measure different angles. Then we know two of the three values, all predefined, all + or -. Once out of three values + or - there is at least one pair equal, the probability of getting equal results should be at least 1/3.

Quantum theory predicts only 1/4 of getting equal results.

The straightforward solution, the one which is realized in the existing hidden variable theories like the de Broglie-Bohm interpretation, is that one of the hidden variables is a hidden preferred frame, and that the hidden variables can send information faster than light. But a hidden preferred frame, even if contradicted by nothing, is anathema in modern physics, and people prefer to reject realism, causality, logic and everything else, and fall into complete mysticism, only to avoid a preferred frame.

• The problem is achieving perfect correlation. David Mermin brings up an interesting point in his book “Boojums All The Way Through”. Chapter 12 gives an excellent description of the experiment and part four shows how incredibly difficult it is to achieve perfect correlations. Usually the data comes only from runs in which both detectors actually flash or flash close enough together to assume their correlated. In the end this cherry picking changes the outcome. It’s possible that Un-skewed results could agree with quantum mechanics predictions. – Bill Alsept Nov 13 '17 at 17:32
• To avoid this purely practical problem, there are the CHSH inequalities, which do not have this necessity. Slightly more complicated to understand the trick mathematically, so Bell's theorem in the original form remains easier to understand, but for practical tests one does not use it. – Schmelzer Nov 14 '17 at 18:34

## protected by Qmechanic♦Jul 8 '14 at 18:01

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