The GHZ-State in conflict with local realism Consider three, with respect to their polarisation, entangled particles in the following state:
$|\psi\rangle = \frac{1}{\sqrt2}(|H\rangle_1|H\rangle_2|H\rangle_3 + |V\rangle_1|V\rangle_2|V\rangle_3)$
In the book I read (The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation - Anton Zeilinger) it is stated that from the point of local realism, if particle 1 is measured to have horizontal polarisation, the other two particles must have opposite circular polarisation and if particle one is measured to have vertical polarisation the other two particles are ident in circular polarisation.
I can only obtain this statement, if I change basis of two particles to the circular polarisation state basis. This basically means, I use the superposition principle and therefore a quantum mechanical description. Is this really legitimate? Is there any way of seeing this in a classical image?
 A: Since  $|H\rangle=  \frac{1}{\sqrt2}(|R\rangle + |L\rangle)$
and $|V\rangle=  \frac{1}{i\sqrt2}(|R\rangle - |L\rangle)$ the state
$|\psi\rangle = \frac{1}{\sqrt2}(|H\rangle_1|H\rangle_2|H\rangle_3 + |V\rangle_1|V\rangle_2|V\rangle_3)$
and the state $\frac{1}{\sqrt2}(|H\rangle_1
\frac{1}{\sqrt2}(|R\rangle_2 + |L\rangle_2) \frac{1}{\sqrt2}(|R\rangle_3 + |L\rangle_3)$
 + $|V\rangle_1
\frac{1}{i\sqrt2}(|R\rangle_2 - |L\rangle_2)
\frac{1}{i\sqrt2}(|R\rangle_3 - |L\rangle_3))$
are exactly the same state, just like writing a vector in two different coordinate systems is the same vector.
Some people use the word superposition to make it sound like something deep is going on. The reality is much much simpler, systems are vectors in a vector space (a particular kind of a vector space called a Hilbert space that has an inner product that gives it nice properties). And you are free to use any basis you want for a vector space.
So we can expand out $\frac{1}{\sqrt2}(|H\rangle_1
\frac{1}{\sqrt2}(|R\rangle_2 + |L\rangle_2) \frac{1}{\sqrt2}(|R\rangle_3 + |L\rangle_3)$
 +$ |V\rangle_1
\frac{1}{i\sqrt2}(|R\rangle_2 - |L\rangle_2)
\frac{1}{i\sqrt2}(|R\rangle_3 - |L\rangle_3))$ to get $\frac{1}{2\sqrt2}(|H\rangle_1
(|R\rangle_2 + |L\rangle_2)(|R\rangle_3 + |L\rangle_3)$
 -$ |V\rangle_1
(|R\rangle_2 - |L\rangle_2)
(|R\rangle_3 - |L\rangle_3)),$ which equals
$\frac{1}{2\sqrt2}(|HRR\rangle+ |HRL\rangle+ |HLR\rangle+ |HLL\rangle$ $-|VRR\rangle+ |VRL\rangle+ |VLR\rangle- |VLL\rangle)$
Which does not match what you said, each possibility (RR, RL, LR, LL) is equally possible if you project onto the first particle being $|H\rangle$ or onto the first particle being $|V\rangle.$
And your whole statement seemed a little weird in that it makes a giant distinction between horizontal and vertical, as if the universe cares whether you tilt your head 90 degrees.
Let's investigate what happens if you are in the state $|\psi\rangle = \frac{1}{\sqrt2}(|H\rangle_1|H\rangle_2|H\rangle_3 + |V\rangle_1|V\rangle_2|V\rangle_3)$
  and you project onto the first particle being $|R\rangle$ or $|L\rangle.$
Like all situations with vectors, computing a projection can be easier if you choose the basis aligned with the projection. So let's write that same vector $|\psi\rangle = \frac{1}{\sqrt2}(|H\rangle_1|H\rangle_2|H\rangle_3 + |V\rangle_1|V\rangle_2|V\rangle_3)$
 In a basis with all $R$ and $L$ so we can jump straight to $\frac{1}{2\sqrt2}(|HRR\rangle+ |HRL\rangle+ |HLR\rangle+ |HLL\rangle$ $-|VRR\rangle+ |VRL\rangle+ |VLR\rangle- |VLL\rangle)$ since that does two out of three of them. Then we just need to replace the $H$ for particle one with some $R$ and $L.$
So $\frac{1}{2\sqrt2}(|HRR\rangle+ |HRL\rangle+ |HLR\rangle+ |HLL\rangle)$ becomes
$\frac{1}{4}(|RRR\rangle+ |RRL\rangle+ |RLR\rangle+ |RLL\rangle$+$
|LRR\rangle+ |LRL\rangle+ |LLR\rangle+ |LLL\rangle).$
And similarly $\frac{1}{2\sqrt2}(-|VRR\rangle+ |VRL\rangle+ |VLR\rangle- |VLL\rangle)$ becomes $\frac{1}{4i}(-|RRR\rangle+ |RRL\rangle+ |RLR\rangle- |RLL\rangle$+$
|LRR\rangle- |LRL\rangle- |LLR\rangle+ |LLL\rangle).$
And once again every possibility happens with equal frequency.
So I can't get your results. Maybe I made a mistake. Or, since you mentioned something about realism it is possible that by realism your text meant "doing things wrong."
It is unfortunate that this happens often enough that I have to entertain the possibility.  There are perfectly fine realist ways to do quantum mechanics correctly. All you need to be a realist and do things correctly is to have a thing that determines what happens and does so in a proper way that agrees with experiments.
Sometimes people think this can't be done because they imagine ways to be a realist that they want to pretend is general enough and then show that those don't work. Instead of actually looking at what people that make correctly functioning realist pictures do.  For instance, if you have a spin degree of freedom it is entirely possible to get correct answers by having a spatial wavefunction that has values that aren't just complex numbers but are spin valued and then have it evolve according to the evolution equations.
People that make realist models that also want to do it correctly actually do this.
People that want to pretend you can't be a realist pretend people don't do that. They pretend like a realist's goal must be to assign an up/down left/right clockwise/counterclockwise etcetera choice for each possible direction and type of spin measurement. But the evolution equations are quite clear about describing an evolving wave that has a larger spin degree of freedom, so a realist that wants to do it correctly models that (or models something equivalent if they'd rather).
Why would I want to assign a $\pm 1$ to a bunch of different directions and types of measurements when that technique doesn't model what we actually observe? Why would anyone assume that people do except to show they shouldn't? It's good to tell people not to do that. But instead of being a realistic about assigning $\pm 1$ to a bunch of different directions and types of measurements you can be a realist about a wave in configuration space that has spin values that evolves according to evolution equations that give experimentally confirmed results.
For instance you can realistically model a polarization measurement by looking at a differential beam deflection that alters the spin as the beams separate and does it in a wave where the amount of beam going into each separation is proportional to the amount of that type of spin that was in the incoming spin. Why? Because that's what we actually do and observe. But then which part of the beam ends up in which of the separated beams will depend on exactly what equipment you set up and how you ran the experiment. But that is what really happens, it isn't failing to be a realist to be realistic.
But someone that doesn't want to be a realist and is just trying to scare you into not being one either might (incorrectly) claim that the result shouldn't depend on the details of how you set up your measurement. But it does matter. And someone telling us we shouldn't do things we have to do in order to do it correctly is someone that is setting people up for failure.
There is a whole research area where you make restrictions on what you are allowed to do, then show that it doesn't work. You can learn things that way, but the more time you spend working with wrong systems the more your intuition is bent towards thinking about things in incorrect ways. And everyone will keep making up their own new ways to do things wrong because wrong things don't build on older wrong things.
If you learn to do things correctly, your intuition can start to help you. And you can build on it in the sense that at the worst you get multiple ways to do things to be able to check your work. And it might get used again since people like to do things correctly. So it is empowering.
I would not suggest learning about alleged realism from someone that doesn't use realist models to do correct physics.  If the realistic model is nonlocal, so be it. If the realist model has the results depend on exactly how you did your measurement, so be it. But if it gives wrong results, don't even call it realist, call it wrong. And you don't have to use realist models explicitly, but of course if someone watches you do the math they could always make a model that has that math be a realistic model of how it behaves.
Don't get mad at them if they do so (and aren't making you do it too). Don't freak out if it is nonlocal or depends on the context the subsystem is placed in on the context of how it interacts.  After all, if they aren't forcing you, why care what they do, and why should they listen to you about what to do if the things you tell them would make them get wrong answers and their methods get right answers.
And the real problem with all the acrimony is that two methods that make the same predictions aren't really very different. It simply isn't worth fighting over. One might be easier to remember, another easy to visualize, one easier to learn first, another easier to compute with, and another might be faster to compute with or easier to take limiting cases or more stable computationally. There are so many reasons why one method might be better in a particular situation, so different things that give the same answers never have to be enemies.
Just don't read too much into the fact that different things can make he same predictions. If they make the same predictions they are potentially equally good. The only reason to favor one in a particular situation would be if it helps in that particular more than the cost of switching and learning.
If you asked the soft question about how to decide what to learn or teach, then clearly you want to weigh the efficiency of teaching, access to prior literature, ability to communicate results, ease of use, whether it creates barriers or misunderstandings, etc.
And that's why it can be good to look at a correct way of doing Physics as merely telling you how to make your predictions, but if it starts to go meta and claim that its way is the only way to make correct predictions then that has veered away from telling you what predictions to make and is telling you how you must make them.  Must make is different than can make.
So when you study quantum mechanics, pay attention to theories that tell you how you can make predictions. Focus on how to make them correctly and how to use theories that make correct predictions. If your theory has a limited domain of validity, know the limits.  But if you veer off into someone telling you that their way is the only way then watch out. Sure, they might be warning you about the danger of careless mix and match, like taking part of one system and part of another, for instance you wouldn't want to use two coordinate systems and just throw coefficients from both systems around without regard for when they are coming from which.
But if they instead are really trying to tell you about other methods, stop. Ask yourself if it matters. Does it affect how you use this system, are they warning you about a domain of validity? Or are you starting to learn a new method, in which case separate the two methods and learn the new method properly and from a good source. Not every source is a good source for every method.
And learning about realism should be done from a source that knows how to be realist and still agree with experimentally confirmed results. Otherwise what is the point? You probably aren't trying to learn how to do things wrong.
