This video is going to be about the proof of Bell's theorem, but I made another video
explaining the significance of Bell's theorem and I think it would really helpful to watch
that first -- So if you haven't watched that video then you can click this link to be taken
to it.
So hopefully Nobody Who Hasn't Watched The Other Video is here.
Remember that what we want to prove today is Bell's theorem that entangled particles
communicate to each other faster than light.
Before I start on the details of the proof, I'll explain the main idea.
Last video I told you guys about something called Spin, and some weird experimental facts
about it.
I won't show you how, but it is possible to explain Spin in the frame work of Quantum
Mechanics.
However, say we didn't believe anything could go faster than light.
Then, because of EPR, quantum mechanics couldn't be true.
Therefore, we'd have to look for an alternative theory where nothing ever acted faster than
light.
We'll call all the theories in this category 'local theories'.
What this proof is going to show is that, no matter how cleverly we try, we could never
come up with a local theory that can explain spin.
We'll do that by a proof by contradiction.
In other worlds we start by believing it is possible and see what happens.
There's one particular experiment with spin that we'll try explain.
Let's say I have some source of electrons with entangled spins.
This means that if A and B are a pair, if I choose to measure them both in the same
axis, one will be up and the other down.
Now, A and B are sent off to two machines very far from each other, which we will call
Alice and Bob.
Alice and Bob can each measure the spin of their electron in three different axes.
These are the 3 axes and we choose them so that they are 120 degrees apart from each
other.
When A arrives at Alice, Alice randomly chooses an axes to measure A's spin in, and Bob does
the same thing with B.
But they do it independently from each other, so Alice could measure spin around 2 and Bob
around 3 for example.
If the machines measure their electron to be spin up then they flash red, or if it's
spin down they flash white.
Someone notes if they're the same colour or not.
This experiment is repeated many times.
The question is, what fraction of the time will their colours be different?
The answer is actually half, and if you're willing to take my word for that then you
can skip this part and move to the most interesting part of the proof.
If not, then stick around and we'll figure out why that's the case, using the rules of
spin from last time.
Ok, I'm assuming you've decided to stay.
Let's say Alice measured around axis 1, and happened to get spin down.
Let's see how likely Bob is to measure spin down as well.
The reason I can just pick a particular case like this is because all the other cases will
work out in the same way, to the same result.
Well, one 1/3 of the time, Bob will choose axis 1 as well, so since A and B are entangled,
Bob's particle will definitely have spin up in that direction so the colours don't match.
What about if Bob measured around axis 2?
Because A is down in the first axis, B must be spin up in the first axis, so we just have
to figure out, how much of this spin is pointing in the 2nd direction.
Actually as you can see, this vector is partly pointing in the negative direction of axis
2.
What does this mean?
It means that if you measure the spin in axis 2, it is either going to be up or down, but
more likely down.
So in this case A and B are more likely than not going to have the same sp in and the colours
are more likely to be the same.
What about axis 3?
This is a really similar story to 2.
As you can see, this vector is pointing in the negative direction of axis 3, so most
of the time B will be down if it is measured in axis 3, so again they A and B will most
likely have the same colour.
let's review those 3 cases again: If Bob is on axis 1, Alice and Bob definitely
won't have the same colour.
If Bob is on axis 2 or 3 though, more often than not Alice and Bob will have the same
colour.
It works out so that overall there's an exactly 50% chance that Alice and Bob have the same
colour and 50% chance they don't.
Ok, so now, we're going to see if a local theory could explain this result.
Firstly let's think about what happens when Alice and Bob happen to pick the same axis.
A and B have to be opposite spins.
We know that if nothing goes faster than light, the only way this is can happen is if the
particles decided there spins while they were in contact.
We don't know how they decide who gets which spin but the awesome thing about this argument
is that it doesn't matter.
All that matters is that each particle A and B have decided what there spins are in each
of the 3 directions before hand.
There are 8 possibilities:
This one is that A is spin up in axis 1 and 3 but spin down in axis 2.
Then of course B is the opposite.
It would take a while if we had to go through every single possibility, but luckily for
me we don't.
We'll just look at the case were A is spin up in axis 1 and 3 and spin down in 2 and
the case where A is spin up in every axis.
Why can we get away with this?
Because we can get the other cases either by swapping which particle we call A and B,
which doesn't matter, or swapping the names of the axes, which also doesn't matter.
Convince yourself that that's true.
Ok, so let's consider the case that A is all spin up.
If so what's the chance that Alice and Bob will have the same colour?
No matter which axes get picked A is always up, and B is down, so they're always different
spins whenever they adopt this plan.
What about the other plan?
This is A's spins in each axis and underneath is B's.
Alice and Bob randomly pick an axes, and there are a few possibilities for what happens:
Alice could pick 1 then Bob could pick 1 as well, so they are different colours
Or he could pick 2 which means that they'd have the same colour etc, and we just go through
and tally up all 9 possibilities and find that 5 out of the nine times they are different
colours.
So that means that whether or not A and B picked a plan like this or like the previous
type, they will always be more likely to have the same colour.
But as we already know, this isn't the right result
So you see, local theories just don't have the right tools to explain spin.
I think that's because spin is a crazy phenomena that is nothing like the classical properties
of matter that we're used to.
On the hand any local theory is quite classical because it assumes particles can't effect
each other instantly over distances which really limits how weird it can be and in this
case, its just too classical to explain something like spin.
So that's Bell's theorem.
I promised to cover a couple of topics last time so I'll work on that, but don't expect
them too soon.
On that optimistic note, see you next time.