M1L7f.txt

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I've used the word "Bell" or "Bell state" so often,
I think it's time to talk about what Mr. Bell is famous for,
namely the Bell's inequality.
I said already in the introduction
for the quantumness of light and entanglement
that it is the EPR paradox and the Bell's inequalities which
a lot of people, including myself, think is the deeper
essence of quantum physics.
It really shows that quantum physics is not just
classical physics with wave character.
It goes way beyond it.
So I want to demonstrate that with Bell's inequality.
And the formulation, which is very simple
and I want to present it here, is an inequality, which
is-- I mean, there are many Bell's inequalities now.
You have different states, different detectors.
And you can derive Bell's inequalities, which are then
violated by quantum physics.
And what I want to present here is the CHSH inequality.
That's Clauser, Horne, Shimony, and-- yes, thank you.
So the situation is the following.
We have something which decays, something
which emits two photons, maybe an atom in an excited state.
And it does a click-clock, a two-photon cascade.
One photon goes to Bob.
One goes to Alice.
Or I always try to sort of stress similarities
between light and atoms.
We discussed the experiment where
you take a mercury molecule.
You dissociate it, and then Bob and Alice each get an atom.
And you can say these photons has a polarization,
or the atom has a spin.
But Bob and Alice can now use different Stern-Gerlach filters
in x and y at obscure angle circular bases.
I mean, you name it.
But what happens is, because it's a spin up,
spin down, horizontal-vertical polarization,
it's a two-level system which is emitted.
And after Stern-Gerlach filter, you have only two combinations.
We call it plus and minus.
So in its most general form, what we assume
is that Bob does measurements in a basis.
If you think about spatial orientation
of a Stern-Gerlach filter called S,
where the outcome is plus/minus 1, T, where
the outcome is plus/minus 1.
And Alice has her own choices.

And we want to assume, now, that everything
is classical probability, that if you just
write down this expression-- don't ask me
where it comes from.
This is probably something which people wrote up
after they found something interesting
and tried to prove it or simplify it.
You just write down QS, RS, RT minus QT.
You can then rewrite it by factoring out S and T.
And now the next step is, because Q and R are there
in each measurement 1 or minus 1, either this is 0
or that is 0.

No, sorry-- one is 0, one is 2.
And therefore, this kind of funny combination
of letters for every measurement is either plus or minus 2.
OK.
Now we do many measurements.
And the probability for a certain outcome,
that the variable Q, there's a certain probability
that the outcome is q.
So you sort of use this pretty much probabilistic thing
that the system has a certain probability to be.
And this is, of course, the assumption.
The probability is that the particle
comes with a certain probability in the state q, r, s, t.
Of course, you should scream q, r, s, t commute.
But this is classical now-- don't
compute with each other quantum mechanically.
But we come to that in a moment.
So then you simply put, by multiplying
each event with its probability, you put now brackets around it.
These are expectation values.
And what you have is an inequality,
that this expression, which is the correlation
between certain measurements, between the quantity QS,
RS, and so on, is smaller or equal than 2.
I mean, it's very, very basic, and that's why I wrote it down.
I don't want to spend a lot of time on it.