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Physics Monkey

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My question is this: do these questions usually still make sense if we replace entanglement by classical correlation? For example, I can send Alice and Bob each a random bit but with both bits perfectly correlated. At the level of the typical entanglement/superluminal signaling question, are these any different e.g. since Alice has the same (completely mixed) state in both case? I realize this is a bit soft, so your opinions and impressions are very welcome. Basically what I want to know is if people are really asking about entanglement or merely any kind of correlation.

And let me clear, I am certainly very aware that entanglement is not the same as classical correlation, and of Bell's theorem, and so on. In my mind, this question is really about the very basic issues and about pedagogy.