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A correlation function is a statistical correlation between random variables at two different points in space, time, or other parameter space, usually as a function of the variable distance between these points. In QFT, field autocorrelation functions are propagators, so use the "propagator" tag, instead.

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Is there a Bell inequality for $2\times 2$ and $1\times1\times1\times1$ configuration?

Bell inequality is violated in entangle experiments is a known phenomenon. The standard experiment is conducted with two parties and is explained in https://en.wikipedia.org/wiki/Bell%27s_theorem#Bell …
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