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Elementary Exposition of a Bell Inequality Violation

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

Contents of this page:

I.
Abstract

II.
Physical Observations

III.
Essence of Weirdness

IV.
Conclusion

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...