Elementary Exposition of a Bell Inequality Violation

Written September 26, 1995, by Matthew Cook. May be freely redistributed.

Contents of this page:

I. Abstract

"Bell's Theorem" is a much-touted theorem showing that "hidden variables" alternatives to quantum mechanics cannot be correct.

The name "hidden variables" is somewhat misleading. It sounds sneaky, but all it means is that there is useful information present in the universe that quantum mechanics fails to use. There is nothing particularly strange about this idea.

Here I give a demonstration of Bell's Theorem, using only the most elementary ideas. What it shows is that "ordinary" (what quantum mechanics would call "hidden variables") theories are inconsistent with certain observed experimental results.

If humans were smarter, theorems like this would be inherently obvious, since they are, after all, true. They wouldn't have the name "Bell" attached to them, and people would immediately see that the experimental results in part II below require a rethinking of things. But this is unfortunately not the case.

Nothing needs to be known about quantum mechanics. Quantum mechanics is in fact totally irrelevant. You don't even need to know about the cosine function. Here we just look at a certain (rather general) type of theory and show that it can't explain a certain physical observation. All that's required is a basic understanding of probability.

Since it is just this sort of rather general type of theory that has been classically and common-sensically been assumed to be able to explain physical observations, demonstrations such as the one below in part III show us that some rethinking is needed about how we model the universe.

II. Physical Observations

It is physically possible to set up a special kind of "emitter" that emits a pair of photons in opposite directions. I don't know much about how or why it works, but it does. In fact, I think it's pretty irrelevant how it works.

Anyway, the photons emitted are related as regards polarization. You have probably played at some point with polarized sheets of plastic. They're pretty nifty. Anyway, if we set up the following experiment, then we can play with rotating the polarized sheets and seeing what happens. When doing so, we will talk about the "angle" of a sheet, meaning its direction of polarization.

               |\                                    |\
               | \                                   | \
         [     |  |    photon             photon     |  |     ]
detector [     |  |   <~~~~~~~  emitter  ~~~~~~~>    |  |     ] detector
         [     |  |                                  |  |     ]
                \ |                                   \ |
                 \|                                    \|
            polarized sheet                       polarized sheet
Here are the results of our playing: (with experimental error factored out)

1. If we remove both sheets, our detectors tell us that the photons are emitted in pairs (i.e. perfect correlation: left detection <= => right detection, where "=>" means "implies").

2. If we put in one sheet, the detectors tell us that the sheet absorbs the photon on that side half the time, and lets it pass through half the time. The angle of the sheet doesn't affect this.

3. If both sheets are at the same angle, we get perfect correlation on the two sides. (i.e. left detection <= => right detection)

4. If the sheets differ by 30 degrees, than a detection on one side means there is a 3/4 chance of seeing a detection on the other side.

5. If the sheets differ by 45 degrees, than a detection on one side means there is a 1/2 chance of seeing a detection on the other side. (No correlation.)

6. If the sheets differ by 60 degrees, than a detection on one side means there is a 1/4 chance of seeing a detection on the other side.

7. If the sheets differ by 90 degrees, then we get perfect anti-correlation. (i.e. detection is either at one side or the other but never both)

8. In general (this generalizes 3 through 6 above), if the sheets differ by an angle t, and we get a detection on (say) the left side, then there is a chance of Cos[t]^2 that we also get a detection on the right side. (And of course vice-versa.)

Of these observations, only 1, 2, 3, 4, and 6 will be used below.

III. Essence of Weirdness

Now that we have the data, let's try to make a theory that explains it. We will fail, but that's what's so interesting. First, here are some principles of any "reasonable" theory:

1. The emission is not affected by presence or angle of sheet or detector.

2. A photon is not affected by the sheet or detector on the opposite side.

3. A photon is not affected by the photon on the opposite side hitting the sheet or getting detected.

4. Whether the photon passes through the sheet or not depends somehow (maybe even probabilistically) on the state of the photon and sheet.

These (1-4) are all aspects of saying that effects are local. In fact, they're getting kind of boring.

There just isn't any real meat to our theory yet. Obviously we'll have to analyze our data slightly in order to figure out what the meat should be.

So let's make a Venn Diagram showing what's happening if the right hand sheet is at 0 degrees, and the left hand sheet is at +30 degrees.

To draw our diagram, we note the following: The left hand photon passes through half the time, as does the right one. Furthermore, if one of them passes through, then there is a 3/4 chance that the other one does too. This gives us our picture:

|                             |         |     Key:
|  L@           R@            |  R@     |
|                             |  L~     |       L = left photon
+         +         +         +         +       R = right photon
|                             |         |
|                             |         |       @ = "splat"
|                             |         |           (gets absorbed by sheet)
+---------+---------+---------+---------+       ~ = "whew"
|         |                             |           (passes through sheet)
|  L@     |                             |
|  R~     |            R~        L~     |       e.g. L@ = left photon gets
+         +         +         +         +                 absorbed by sheet
|         |                             |
|         |                             |       + = reference point, no meaning
|         |                             |
Hmm, not much to go on yet. Let's add in the results of when the left sheet is at -30 degrees instead of +30 degrees.

Just for the heck of it, let's do this in such a way so that the left photon passing through at +30 degrees has the least correlation possible with the left photon passing through at -30 degrees. (Since we can't actually perform both these measurements simultaneously on the left photon, we don't have direct experimental data on what this correlation is.)

| R(0)@   |  R(0)@            | R(0)@   |     Key:
| L(+30)@ |  L(+30)@          | L(+30)~ |
| L(-30)~ |  L(-30)@          | L(-30)@ |       L = left photon
+         +         +         +         +       R = right photon
|         |                   |         |
|         |                   |         |       (n) = sheet at angle n
|         |                   |         |
+---------+---------+---------+---------+       @ = "splat"
| R(0)~   |  R(0)~            | R(0)~   |       ~ = "whew"
| L(+30)@ |  L(+30)~          | L(+30)~ |
| L(-30)~ |  L(-30)~          | L(-30)@ |       e.g. L(+30)@ = left photon
+         +         +         +         +                      gets absorbed
|         |                   |         |                      by sheet at
|         |                   |         |                      +30 degrees
|         |                   |         |
We minimized the correlation of L(+30) and L(-30) by doing the following:

i. In the upper part of the diagram (where the right photon goes splat), we made the quarter for L(-30)~ totally disjoint from the quarter for L(+30)~.

ii. In the lower part of the diagram (where the right photon passes through), we made the quarter for L(-30)@ totally disjoint from the quarter for L(+30)@.

Now we're getting somewhere. We have shown that the minimum correlation between L(+30) and L(-30) is given by the diagram above. In particular, the region where L(+30)~ and L(-30)~ is as small as possible in the above diagram. Don't just take my word for it -- take a minute to convince yourself of this.

Well, now we can fruitfully use observation #3 from part II: We know that the region where L(+30)~ must be exactly the same as the region where R(+30)~. So in fact we know that if L(-30)~, then the chance of R(+30)~ is at *least* 1/2.

But wait a minute -- we also know this chance from experimental observation. Specifically, observation #6 in part II tells us that this chance is 1/4. Wait a minute! Something's wrong! We have a contradiction here, because 1/4 is not at least 1/2.

Note that our Venn Diagram treatment is correct. If one allows for probabilistic happenings at the photon-sheet interaction, it is somewhat trickier to show that the Venn Diagram treatment is correct, but it still is. (For example, extend an "ordinary" Venn Diagram by making each point (representing a possible state of things just after the photon pair has been emitted) have a vertical extent coming 1 cm out of the paper, which gets divided according to the probabilities of various things happening. Then we can make our "probability-aware" Venn Diagram by looking at the volumes of the regions involved.)


Here's the same argument all over again with no Venn Diagram, just in case you weren't convinced the first time:

Principle III-4 tells us that the photon state plus the sheet state gives us the chance of the photon passing through sheet. The sheet's state is given by its angle, so the photon's state must give a function f:[0, 360) -> [0, 1] which gives the probability of passing through a sheet at a given angle. This function can of course vary from emission to emission.

In fact, we can tell what this probability is for a given angle by looking at what the other photon does when presented with a sheet at the same angle. Since the photons always do the same thing when presented with sheets of the same angle, the action of the other photon tells us for sure what this photon would do. If the other photon passes through a sheet at angle x then f(x)=1, and if the other photon is absorbed by the sheet then f(x)=0. So in fact the photon-sheet interaction cannot be probabilistic.

So a photon-pair emission endows both photons with a function f indicating whether or not they will pass through sheets at various angles. This function must be the same for both photons, since they always react in the same way to sheets at the same angle.

Now we're getting somewhere with our theory!

Suppose we look at 8 million emissions. Then according to observation II-2, about 4 million of them will have f(0)=1, and observation II-6 tells us that of those about 1 million will also have f(60)=1, so a total of about 1 million cases have f(0)=f(60)=1.

Now, of the original 8 million emissions, observation II-2 also tells us that about 4 million of them will have f(30)=1. Of those 4 million, according to observation II-4, about 3 million will have f(0)=1, and about 3 million will have f(60)=1. We don't know exactly how much overlap there is between the 3 million cases where f(0)=f(30)=1 and the 3 million cases where f(30)=f(60)=1, but since there are only 4 million total where f(30)=1, we can see that there must be at least about 2 million emissions where f(0)=f(30)=f(60)=1.

But how can there be at least 2 million emissions where f(0)=f(30)=f(60)=1 if there are only 1 million emissions where f(0)=f(60)=1? We have reached a contradiction!

Where did we go wrong? We must have gone wrong in our principles. That is to say, there must be something going on that is non-local in nature. Hmph. How unpleasant.

IV. Conclusion

So in trying to come up with a theory, we instead came up with a contradiction. The weird thing about it is that the contradiction didn't seem to ensue from any theory -- it came up before we had any theory, as if it were inherent in the measurements. But nature is not contradictory, so there must instead be a flaw in our logic or assumptions. This is the essence of Bell's Theorem: Any theory making the assumptions 1-4 (part III) can't be correct.

Quantum Mechanics says that our mistake was to assume locality as we did, by assuming distant events are not related. Quantum Mechanics says that such distant events *can* be related -- not in a cause-and-effect way, but rather just in a correlated way, since all parts of a given experiment are described by one big wave function, which can contain correlations between distant parts.

But we still have the strange situation that if we assume that various events are occurring in our experiment, and that that they only have local effects, then we reach a contradiction. And yet the non-local effect, if any, is not detectable in any way, which makes it hard to argue that it exists at all. For example, it is clear that it doesn't matter which photon you consider to be making the decision "first", so it is not a standard cause-and-effect relationship. It is a relationship of a kind that we unfortunately don't have a name for, much less an intuition for.

One is left asking, "So does one measurement affect the other or not???" Tune in next time for the answer to this vexing question...