For a subset of 2 dimensional unit parameter vectors, Bell's correlation formula with local hidden variables reproduces the quantum correlation. This is unexpected considering a general no-go LHV claim derived from the same function.
Bell's theorem witnesses that the predictions of quantum theory cannot be reproduced by theories of local hidden variables in which observers can choose their measurements independently of the source. Working out an idea of Branciard, Rosset, Gisin and Pironio, we consider scenarios which feature several sources, but no choice of measurement for the observers. Every Bell scenario can be mapped into such a correlation scenario , and Bell's theorem then discards those local hidden variable theories in which the sources are independent. However, most correlation scenarios do not arise from Bell scenarios, and we describe examples of (quantum) non-locality in some of these scenarios, while posing many open problems along the way. Some of our scenarios have been considered before by mathematicians in the context of causal inference.
Bell's Theorem witnesses that the predictions of quantum theory cannot be reproduced by theories of local hidden variables in which observers can choose their measurements independently of the source. Working out an idea of Branciard, Rosset, Gisin and Pironio, we consider scenarios which feature several sources, but no choice of measurement for the observers. Every Bell scenario can be mapped into such a \emph{correlation scenario}, and Bell's Theorem then discards those local hidden variable theories in which the sources are independent. However, most correlation scenarios do not arise from Bell scenarios, and we describe examples of (quantum) nonlocality in some of these scenarios, while posing many open problems along the way. Some of our scenarios have been considered before by mathematicians in the context of causal inference.
The strongest attack against quantum mechanics came in 1935 in the form of a paper by Einstein, Podolsky and Rosen. It was argued that the theory of quantum mechanics could not be called a complete theory of Nature, for every element of reality is not represented in the formalism as such. The authors then put forth a proposition: we must search for a theory where, upon knowing everything about the system, including possible hidden variables, one could make precise predictions concerning elements of reality. This project was ultimatly doomed in 1964 with the work of Bell Bell, who showed that the most general local hidden variable theory could not reproduce correlations that arise in quantum mechanics. There exist mainly three forms of no-go theorems for local hidden variable theories. Although almost every physicist knows the consequences of these no-go theorems, not every physicist is aware of the distinctions between the three or even their exact definitions. Thus we will discuss here the three principal forms of no-go theorems for local hidden variable theories of Nature. We will define Bell theorems, Bell theorems without inequalities and pseudo-telepathy. A discussion of the similarities and differences will follow. 1
It is proved that Bell's inequality can be traced back to a classical probabilistic formula that is invalid clearly, while the spin correlation formula in quantum mechanics to a formula that has confirmed both by facts and quantum mechanics. Two misunderstandings about Bell's theorem are pointed out: Firstly, the hidden variable theory Bell's used is a special one, but it is regarded as a general form of such theories. Secondly, there are two outcomes obtainable from Bell's hypotheses: one is compatible with quantum mechanics and the other leads to Bell's inequality. Unfortunately, the former is regarded as a property of local hidden variable theory, while the latter as self-evident. G. Lochak has revealed the first misunderstanding, and thereby he pointed out that Bell's inequality has nothing to do with locality, but he did not find the second one, so that he still analyzed this problem starting from hidden variable theory, which is actually irrelative to Bell's inequality. (The Journal of American Science. 2005;1(2):42-50).
A local hidden variable model exploiting the detection loophole to reproduce exactly the quantum correlation of the singlet state is presented. The model is shown to be compatible with both the CHSH and the CH Bell inequalities. Moreover, it bears the same rotational symmetry as spins. The reason why the model can reproduce the quantum correlation without violating the Bell theorem is that in the model the efficiency of the detectors depends on the local hidden variable. On average the detector efficiency is limited to 75%.