# Can grandpa understand the Bell's Theorem?

1. Apr 9, 2011

### miosim

Can a grandpa understand the Bell's Theorem?

Einstein said that you don't truly understand something unless you can explain it to your Grandma. I think that this should apply also to a grandpa.
I am a grandpa who is struggling to understand the Bell's Theorem. I read a number of popular books and articles, tryed Wikipedia, followed discussions on this forum, and even tried to read the original Bells’ paper, but I still cannot grasp the logic and the experimental proof of this theorem. The popular explanation of the experiment in terms of red and blue balls may be a good illustration but still doesn’t make sense to me as an explanation.
Of cause my inability to understand math is a biggest problem, but the controversial concepts of quantum mechanics don’t give me such a problem regardless that they are also based on math.
In spite of my shallow background in math (say high school level) I believe that this shouldn’t prohibit me to understand the physical concept assosiated with this theorem.
Actually, I view math as a formalized logic and logic works only within well defined area of knowledge. Therefore I am careful with the logical and mathematical deductions applied to subatomic events that are obviously not fully understood yet.
So I am asking for a help in understanding the Bell's theorem and its experimental proof in terms of physical concepts (of cause if we truly understand them). For the start I have a specific questions: How come the formalism of quantum theory leads to the Sine correlation while EPR formalism leads to Linear correlation (see Fig.below)?

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2. Apr 9, 2011

### Jonathan Scott

I'll have a go.

On the macroscopic scale, quantum mechanics goes over to classical mechanics. This means that for example if a set of particles with a known direction of spin or polarization are observed in some other direction, the average value of the result should match the classical component of the original measurement in the same direction.

The average value of some measurement is given by sum of the probability of each result (the relative rate at which it occurs) times its value, so for example if the possible results are +1 and -1 and the average value is 0.7 then this means that the probability of +1 is 0.85 (or 85%) and the probability of -1 is 0.15 (or 15%), giving an average value of 0.85*(+1)+0.15*(-1) = 0.7 as required. If we use "+1" to mean the same and "-1" to mean different, then this average value is also known as the correlation.

The unexpected bit is that this same correlation also applies when a matched pair of particles are separated and observed in two different directions. The QM result for the correlation is the same as if one of them was in a pure state and the other is measured at some angle to that pure state. This seems quite sensible mathematically, but leads to a weird effect.

Consider the example of observing the spin of a matched pair of spin-1/2 particles (with total spin 0) travelling in opposite directions, using observing devices which can be rotated about the axis along the line of travel and give results spin "up" or "down". A classical spin measurement would give a value proportional to the cosine of the angle between the measurement directions.

If the two devices are aligned in matching orientations (in opposite directions to allow for the initial state), then QM says that 100% of the results should match. If either of the two devices is turned at 90 degrees to the original orientation, then QM says that the average correlation should be zero, so 50% of the results should match and 50% should be different. If either device is turned to 45 degrees from the original orientation, then the classical projection of one direction on the other is cos 45 degrees, which is about 0.7 (70%), so to get this correlation we need 85% of results to be the same and 15% to be different.

This gives the following:

1. Devices aligned: 100% of results same
2. Turn one device by 45 degrees: 15% of results are now different
3. Turn other device by 45 degrees other way: 15% of results are now different
4. Turn both devices by 45 degrees in opposite directions (so they are now at 90%): 50% of results are now different

If either device is turned by 45 degrees on its own, this could result in 15% of measurements at that device being different from what they would have been at the original orientation, so if both are turned then the maximum effect this could have according to classical physics is to make 30% of the results different from the other end, but according to QM, which is confirmed by experiment, we actually get 50% different results in this case. This apparent impossibility is a simple example of Bell's theorem.

For this to be possible, then it appears that we must assume that somehow changing the direction of measurement at one end also affects the result at the other end, either via some instant communication, or via some "memory" of previous results causing the other end to modify the next measurement. Experiments have confirmed that the QM results still hold when the distance between the devices is large enough to rule out light-speed communication between the devices, and they have used fast switching between multiple observing devices to effectively interleave two separate experiments, ensuring that any "memory" effect would be unable to give the correct average result.

Various studies have been made of the assumptions behind such models and the ways in which the laws of physics could be modified in an attempt to remove such weirdness, but Bell's theorem proves that it is not possible to find a classical "explanation" for how QM works except by violating at least one major fundamental principle of physics, for example by using faster-than-light communication.

3. Apr 9, 2011

### miosim

Jonathan Scott,

Thanks a lot. Almost everything you are saying makes sense to me, but I have to do more homework to have a better understanding for this situation. For now I have just one question regarding your explanation of Bell’s theorem:

When you refer to classical physics that predicts 30% correlation (vs. QM that predicts 50% correlation) what classical physics do you mean? Is it classical Malus' law? I am trying to understand the direct connection between different mechanisms (classical vs. QM) and different predicted results.

Thanks,

Mark

4. Apr 9, 2011

### JesseM

Here's an analogy I wrote up a while ago, see if it helps:

Suppose we have a machine that generates pairs of scratch lotto cards, each of which has three boxes that, when scratched, can reveal either a cherry or a lemon. We give one card to Alice and one to Bob, and each scratches only one of the three boxes. When we repeat this many times, we find that whenever they both pick the same box to scratch, they always get the same result--if Bob scratches box A and finds a cherry, and Alice scratches box A on her card, she's guaranteed to find a cherry too.

Classically, we might explain this by supposing that there is definitely either a cherry or a lemon in each box, even though we don't reveal it until we scratch it, and that the machine prints pairs of cards in such a way that the "hidden" fruit in a given box of one card always matches the hidden fruit in the same box of the other card. If we represent cherries as + and lemons as -, so that a B+ card would represent one where box B's hidden fruit is a cherry, then the classical assumption is that each card's +'s and -'s are the same as the other--if the first card was created with hidden fruits A+,B+,C-, then the other card must also have been created with the hidden fruits A+,B+,C-.

The problem is that if this were true, it would force you to the conclusion that on those trials where Alice and Bob picked different boxes to scratch, they should find the same fruit on at least 1/3 of the trials. For example, if we imagine Bob and Alice's cards each have the hidden fruits A+,B-,C+, then we can look at each possible way that Alice and Bob can randomly choose different boxes to scratch, and what the results would be:

Bob picks A, Alice picks B: opposite results (Bob gets a cherry, Alice gets a lemon)

Bob picks A, Alice picks C: same results (Bob gets a cherry, Alice gets a cherry)

Bob picks B, Alice picks A: opposite results (Bob gets a lemon, Alice gets a cherry)

Bob picks B, Alice picks C: opposite results (Bob gets a lemon, Alice gets a cherry)

Bob picks C, Alice picks A: same results (Bob gets a cherry, Alice gets a cherry)

Bob picks C, Alice picks picks B: opposite results (Bob gets a cherry, Alice gets a lemon)

In this case, you can see that in 1/3 of trials where they pick different boxes, they should get the same results. You'd get the same answer if you assumed any other preexisting state where there are two fruits of one type and one of the other, like A+,B+,C- or A+,B-,C-. On the other hand, if you assume a state where each card has the same fruit behind all three boxes, so either they're both getting A+,B+,C+ or they're both getting A-,B-,C-, then of course even if Alice and Bob pick different boxes to scratch they're guaranteed to get the same fruits with probability 1. So if you imagine that when multiple pairs of cards are generated by the machine, some fraction of pairs are created in inhomogoneous preexisting states like A+,B-,C- while other pairs are created in homogoneous preexisting states like A+,B+,C+, then the probability of getting the same fruits when you scratch different boxes should be somewhere between 1/3 and 1. 1/3 is the lower bound, though--even if 100% of all the pairs were created in inhomogoneous preexisting states, it wouldn't make sense for you to get the same answers in less than 1/3 of trials where you scratch different boxes, provided you assume that each card has such a preexisting state with "hidden fruits" in each box.

But now suppose Alice and Bob look at all the trials where they picked different boxes, and found that they only got the same fruits 1/4 of the time! That would be the violation of Bell's inequality, and something equivalent actually can happen when you measure the spin of entangled photons along one of three different possible axes. So in this example, it seems we can't resolve the mystery by just assuming the machine creates two cards with definite "hidden fruits" behind each box, such that the two cards always have the same fruits in a given box.

You can modify this example to show some different Bell inequalities, see post #8 of this thread for one example.

5. Apr 9, 2011

### Jonathan Scott

Sorry about any confusion in my terminology - this is a classical inequality rather than a prediction.

If you assume the classical position that changing the measurement direction of one device does not change the results at the other one, then changing both devices cannot result in more difference in the results than the sum of the differences caused by changing the two devices independently, which is 15% + 15% = 30%. For a classical experiment, we would normally expect less than that, in that we would expect that when we change both devices, some of the changes would switch results at both ends, in which case they would still match and the total differences would therefore be less than 30%. This means that if you construct a classical model which explains the first three experiments, it is impossible for it to explain the fourth one.

There is no unique "classical" model involved here. Various different classical models can be constructed to match QM predictions in any specific case, but it is not possible for any single such model to match QM in all cases, for example in all four of the cases previously listed. That is the essence of Bell's theorem.

6. Apr 9, 2011

### miosim

Jonathan Scott and JesseM,

I think that I start to understand the overall strategy of this theorem. It seems to me that it expects a lower probability to hit a target by the classical deterministic particles having deterministic “bullet like trajectory” but expects the higher probability to hit a target by QM “particles” having some sort of self-adjusting mechanism. Please correct me if I am wrong.

7. Apr 9, 2011

### JesseM

The notion of local realism is very general, it doesn't require determinism or particles having bullet-like trajectories, the particle could be performing all types of crazy swerving, as long as there were no FTL influences on its swerves it would still fail to violate the Bell inequalities in these experiments.

8. Apr 9, 2011

### rogerl

Re: Can a grandpa understand the Bell's Theorem?

Have you tried this site?

http://quantumtantra.com/bell2.html

It's the easiest to understand illustration of Bell's Theorem in the web. Your grandpa can understand even just looking at the illustration.

9. Apr 9, 2011

### miosim

Re: Can a grandpa understand the Bell's Theorem?

Rogerl,

From the site you suggested:

“Zero angle = 100% Match.
Right angle = 0% Match.
Angle between Zero and Right angle = Cosine Squared (Angle) Match.”

I assume that this formula is for the QM model. Where is this formula coming from?

What is the formula for the deterministic (non-QM) model? Where is this formula coming from?

10. Apr 9, 2011

### miosim

Looks like I am off track again and should start from very beginning and move slowly step by step.

JesseM, could you please explain me again how we calculate a specific % correlation for the real physical model (local realism) and for QM model, say at 22.5 degree between polarizers.

I would like to understand the real physical objects this mathematical analysis is based on and not just a math itself that could be misleading. For example the mass of the atomic nuclear isn’t a sum of the masses of its protons and neutrons. It is why I feel uneasy to extrapolate probability analysis performed on the scratching cards to the quantum entities.

Thank you,

Mark

11. Apr 9, 2011

### rogerl

Re: Can a grandpa understand the Bell's Theorem?

It's just a formula for polarizer. Anyway. I think someone has mentioned the web site is correct. I first read it in Nick Herbert book Quantum Reality. Try it too.

12. Apr 9, 2011

### JesseM

Knowing how QM derives the cos^2 relationship is irrelevant to understanding Bell's theorem, if you are really curious you'll have to pick up a textbook on QM to learn about multiparticle wavefunctions, how they evolve over time, and how they are used to make probabilistic predictions about measurement outcomes. But Bell's theorem is just about what's true under local realism, it's only at the end once you've derived an inequality which must hold in local realism that you check the inequality against QM and see that QM predicts this same inequality is violated, demonstrating that QM is incompatible with local realism.

As for local realism, it never predicts any "specific % correlation", it just shows that the correlations cannot violate certain inequalities. Look at my lotto card analogy--the basic logic is that if Alice and Bob always get the same result when they choose to scratch the same box (i.e. the same fruit is uncovered on both their cards), yet their choice of which box to scratch is random so the source manufacturing the cards doesn't "know in advance" what they'll pick, in that case there's only one way the source can guarantee they'll always get the same result on any trial where they scratch the same box, and that's to have hidden properties associated with the cards ('hidden fruits') that predetermine what fruit they will show for all three boxes (with the source always making sure to give them identical predetermined responses for all three boxes, so no matter which boxes Alice and Bob pick, if they choose the same ones they'll get the same response). If the source didn't manufacture the cards with such hidden properties that predetermined their responses, if the two cards just "made up on the spot" what fruit would appear when a given box was scratched, then the two cards would have no way to coordinate their responses in a way that would ensure they give the same response whenever Alice and Bob choose to scratch the same box (since Alice and Bob make their choices at a separation which makes it impossible for Alice's choice to have a causal influence on Bob's card at the moment he scratches it, assuming no FTL signals). Do you follow this logic? If so, then it's just a matter of showing that if they always have the same predetermined responses to all three boxes, then on any trial where Alice and Bob choose different boxes to scratch, the probability they will see the same fruit must be greater than 1/3.

The situation with the particles is no different--instead of Alice and Bob having three possible boxes to choose from on each trial, they have three possible orientations of their polarizers which they have agreed to choose between. And instead of picking a box and getting either a cherry or a lemon, when they pick a polarizer orientation they either get the result that the photon goes through the polarizer and is detected by a detector behind it, or it's reflected by the polarizer and is detected by a differently-placed detector. And for a certain type of experimental setup, they will also find that whenever they choose the same setting, they both get the same result (for a different setup they will always get opposite results, but the logic is the same either way). So, in a local realist universe the only way to explain this is to say that the source sent out both photons with "hidden" properties that predetermined what they would do if they encountered a polarizer at each of the three possible orientations, with the source always creating pairs with identical predetermined results for all three orientations. And again this leads to the conclusion that if Alice and Bob pick different orientations the probability of getting the same result must be at least 1/3, when in fact it can be lower than this in QM.

13. Apr 9, 2011

### Upisoft

It seems your problem is to understand the weirdness of quantum events. I'll try to explain it with an example from a very old book I've read when I was a child. I'm 41 now.

Suppose you have red and green balls and a machine that lets only the red balls to go through. You have a source that generates balls with 50% of being red or green. So you put your machine and as expected you get about 50% of them go through.

So you pick up another machine that does the same thing and put it to check if your first machine is working correctly. As expected all of the balls get through.

Now you see that your balls, like billiard balls, also have a number on them it could be 1 or 2. You have machine that lets balls labeled with 1 to go through. You put your machine at the source and you find out that about 50% of them pass through. Well, we have machine that generates random balls red or green with label 1 or 2, both with 50% chance.

So the next step is to find if there is a correlation between red color and the label 1. If there is no correlation we will expect about 25% of the balls to be red and having label 1 at the same time. So we put our machine that filters out greens and then we put our machine that filters out 2s. We get 25%, so nothing that strange. Our generator makes totally random balls without any correlation between color and label.

Now it is getting strange. For some reason someone in the lab doubts our result and gives us another machine that lets only red balls to pass through. We put it as 3rd and we expect to get all balls passing through it, as they are surely red. After all the first machine is letting only red balls to pass through.

Well, surprise. Only half of the balls pass through the 3rd machine.

So classically you expect that 25% will pass through these 3 machines. But QM balls behave differently and only 12.5% pass through.

What happens is that the measurement of the "label" destroys the information about the "color", so after we have measured the label with the second machine we have balls with random color.

14. Apr 9, 2011

### DrChinese

miosim, one thing to consider is this: the only factor in determining correlations in Bell tests is the angle between. That in and of itself is very revealing. Now imagine that the values were predetermined. You will see that there must exist the same relationship between ANY pairs of possible angles. Now finally, try to imagine those relationships must hold simultaneously. After a while, you will see that only a linear type relationship supports that. Obviously, that is not the case and it is also not what QM predicts, as shown on the graph you presented.

15. Apr 9, 2011

### miosim

It seems to me that there is no shortcut in understanding the Bell’s theorem, or at least not for me. I need to spend more time studying Bell’s theorem, but I may better try to understand its physics prior following Bell’s mathematical proof. I don’t remember seeing any justification that statistical analysis used in Bell’s theorem is adequate for the reality it models and this bothers me.

I appreciate for everybody help.

Mark

16. Apr 10, 2011

### Jonathan Scott

There is no unique real physical model. The problem is that if you construct any such model that matches SOME of the QM results, it is impossible for it to match ALL of the QM results.

For the spin case, when we measure spin classically at an angle $\theta$ to the original known spin, we expect the result component to be $\cos \theta$ times the original spin. In QM, we can only get "up" or "down" results, so the probability of the results must be given by $(1 + \cos \theta)/2$ and $(1 - \cos \theta)/2$ respectively, in order that the probabilities add up to 1 and the average value is $\cos \theta$. By well-known trigonometric identities, these probabilities are equal to $\cos^2 (\theta/2)$ and $\sin^2 (\theta/2)$. Similar results apply for the photon polarization case, except that the angles are halved.

If you consider the set of possible results from observing pairs of particles, then if you make one change which causes 15% of the results to be different from the other end, and then you make another change which independently would also have caused 15% of the results to be different, then the total effect of that cannot affect more than 30% of the results. This means that if you have a classical model which correctly predicts the first three cases, it is impossible for it to correctly predict the 50% different case.

Note however that it IS possible to produce a classical model which gives the correct correlations when instead of turning the both devices by 45 degrees we turn just one of them by 90 degrees. This means we have to be very careful about hidden assumptions of rotational symmetry.

17. Apr 10, 2011

### miosim

Jonathan Scott,

Thank you for these helpful tips. I really need to study more methodically the underlying QM interactions (and associated math) between quantum particles and polarizer prior to study Bell’s theorem. It would take some time, but I belief that it is worth it.

18. Apr 10, 2011

### JesseM

Again, Bell's theorem is about deriving what would be true in a local realist theory, you really don't need to know anything about QM. What's more, studying QM will only give you some abstract mathematical derivation of the cos^2 rule, it's notoriously hard to get any sort of concrete physical picture out of QM so if you're hoping that your studies will give you any sort of physical insight into where the cos^2 rule comes from you're probably wasting your time.

Did you look at my post #12 on why local realism implies there must be predetermined results for each detector setting? If there was anything there you had trouble understanding, just ask.

19. Apr 10, 2011

### miosim

I tried to understand your post #12, but I can’t follow this explanation without fully understanding link between “local realism” and Bell’s inequities.

Now, the main question for me is why the “local realism” probability function is linear (see Figure in post #1)? As I understand the “local realism” system is still a quantum mechanical system and therefore should be described by the wave function and lead to the same cos^2 correlation rule.

The “local realism”, as I understand it, shouldn’t change the math of wave function but only offers different interpretation.

20. Apr 10, 2011

### JesseM

Well, the whole explanation is trying to show you a series of steps that follow from local realism, ultimately leading to a particular Bell inequality (namely the one that says that when the experimenters Alice and Bob scratch different boxes or choose different detector settings, the probability that they will get identical results should be greater than 1/3). So it would help if you would explain the first statement in post #12 that you don't understand or you don't see how it follows from the assumption of local realism plus the previous statements.
You're dealing with a different Bell inequality than the one I was discussing, the one I'm discussing is simpler and doesn't involve any specific probability function, I think it would help to try to understand this simpler version (which is still a valid Bell inequality) first. You don't mention where you got the graph in your opening post so I don't know the exact derivation, but I think based on this post by DrChinese the answer is probably that there is no requirement that the LR probability function be linear, it's just that this is the probability function that's the closest possible to the QM predictions while still being possible under local realism...look at the graph he posts in his comment, where the LR prediction is a straight line in blue, after which he says: