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For a project, I'm planning to study Bell's inequality, which as far as I can gather is taken to rule out hidden variable theories of QM. I'm looking for recommendations of decent sources which derive the inequality, so I can get my head around the assumptions made and exactly where the inequality is applicable. I should add that I'm not looking for a "general outline" kind of source, I need something a bit more rigorous!

glS
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Lammey
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    Holevo's book "Probabilistic and statistical aspects of quantum theory" has a very formal and clean definition of the hidden variable concept, and of course of how it fails. Lubos Motl blog (the Reference Frame) has a lot of nice posts about it I suggest you to search there, too.

    As a final suggestion, you could be interested in the Mermin GHZ experiment too (look for the original paper). I find it easier to understand and remember than the Bell experiment.

     – giulio bullsaver Oct 18 '14 at 17:13
  • Possible duplicates: https://physics.stackexchange.com/q/129140/2451 , https://physics.stackexchange.com/q/14377/2451 and links therein. – Qmechanic Oct 18 '14 at 17:18
  • There is a fairly trivial "handwaving" explanation for why hidden variables don't work: the measurement process exchanges energy with the quantum system that depends on the unknown state of the classical measuring system, hence it can not be folded into the state of the quantum mechanical system. The easiest physics experiment to observe this on phenomenologically is, IMHO, the Stern-Gerlach experiment. Formally you do this with a density matrix or like in the answer by J Thomas. – FlatterMann Apr 15 '23 at 19:53
  • This paper discusses the assumptions made in deriving and interpreting Bell's theorem, see especially Section 7 https://arxiv.org/abs/quant-ph/9906007 – alanf Aug 16 '23 at 09:38
  • As far as textbooks go, the last chapter of Griffiths has a pretty clean discussion of Bell’s inequalities. – Jahan Claes Dec 24 '23 at 00:42

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Here is a simple and rigorous description. It says nothing about how and why QM sometimes gives results that are incompatible with hidden variables, but it makes it completely clear what it is that hidden variables can't do.

Understanding Bell's Theorem

The theorem is about explaining the statistics observed by two experimenters, Alice and Bob, that are making measurements on some physical system in a space-like separated way. The details of their experiment are not important for the theorem. What is important is that each experimenter has two possible settings, named 0 and 1, and for each setting the measurement has two possible outcomes, again named 0 and 1.

[....] Having their settings and outcomes defined like this, our experimenters measure some conditional probabilities $p(ab|xy)$, where $a,b$ are Alice and Bob’s outcomes, and $x,y$ are their settings. Now they want to explain these correlations. How did they come about?

Well, they obtained them by measuring some physical system $\lambda$ (that can be a quantum state, or something more exotic like a Bohmian corpuscle) that they did not have complete control over, so it is reasonable to write the probabilities as arising from an averaging over different values of $\lambda$. So they decompose the probabilities as

$p(ab|xy)=\sum p(\lambda|xy) p(ab|xy\lambda)$

The first assumption that we use in the proof is that the physical system $\lambda$ is not correlated with the settings $x$ and $y$, that is $p(\lambda|xy)=p(\lambda)$. I think this assumption is necessary to even do science, because if it were not possible to probe a physical system independently of its state, we couldn’t hope to be able to learn what its actual state is. It would be like trying to find a correlation between smoking and cancer when your sample of patients is chosen by a tobacco company.

And so on. Each probability step is simple and logical.

J Thomas
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