Okay, hello everyone.

Let me just remind you about the Bell's Inequality or rather the Klausel-Holthorne-Schimonie

inequality.

So the setup is very simple.

They have some source and the source from time to time emits a pair of particles.

And these particles are measured in devices that work like for example Stern-Gerlach apparatuses

where the outcome can be either one of two possible outcomes, namely for example spin-up

or spin-down.

And the idea is to imagine that somehow the properties of these particles are there even

before the measurement, even before I inquire one or the other property.

And the idea is that these detectors which are placed here, they have settings like the

axis of the Stern-Gerlach magnet and these settings can be chosen to have different values.

So for example the axis can be little a or little a prime and here again the axis can

be little b or little b prime in some other direction.

And correspondingly if you imagine that the properties are there in advance then you can

say I will have an outcome capital A that would correspond to the outcome if I were

to measure along the axis little a and it just depends on this random invariable lambda

that I cannot control that may be different for each one of the experiment.

Could also depend on some randomness inside the detectors presumably but what A does not

depend on the outcome, measurement outcome capital A does not depend on is the setting

at b.

And so then there would also be an outcome that I would obtain if I were to measure along

a prime and correspondingly here I would talk of b and b prime.

And all these outcomes are just plus or minus one or if you also want to have the possibility

that you do not detect anything then you would add zero to those outcomes.

And so you have these four random variables and then you just derive a very simple inequality

for these four random variables.

And this inequality is valid whenever we have four random variables that can only take the

values plus minus one or zero and the inequality I will write down once more is like this the

average of a b minus average or plus average of a b prime plus the average of a prime b

minus the average of a prime b prime has to be less than or equal to two.

Okay and so this is the Klaus and Moulton-Simoni inequality which slightly generalizes Bayes

original proof because in the original proof he just assumed that for parallel detector

settings you would get perfect anti-correlations as would ideally be the case but what do you

do if you have an experiment where you just get 98 percent anti-correlations then you

cannot use the proof anymore.

So this is much more general.

But of course it follows the same idea.

So it's also called a Bell inequality.

And now let me remind you once more what are the special assumptions that go into this

because at the face of it this seems to hold always because you always can talk about these

measurement results being plus or minus one or zero and then this is completely general

so there was no magic in the derivation.

However remember that if we write a times b we really mean the result at detector a

for a setting little a times the result at detector b for a setting little b and correspondingly

if we talk of a prime then that is result a again but then the result at detector b

for another setting b prime.

And now this is what we used in the proof even though we didn't write it in the complicated

form.

So in particular we employed the fact that if I choose those settings a and b or I choose