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@Ankit It's gotta break somewhere. You can see the crack begins on the surface, then expands perpendicular to the surface. Tug on opposite sides of a piece of silly putty and it'll break in much the same way - the tear will be perpendicular to the surface. So the question is why doesn't the tear go diagonal or curve?
For any given run of the experiment, "either correlated or anticorrelated" is the same thing as "uncorrelated". Think about it: if Bob has two binary numbers, they will always be either correlated or anticorrelated (i.e. they are either the same or different). Alice doesn't get to choose which of those two cases happens. So there is no way Alice can use this to send Bob a message. Therefore it is non-signalling.
I usually describe hidden variables (particle A and particle B decide how they will respond to various measurements, etc.) then say that quantum goes beyond that. Then they say "can you give a concrete example?" and I explain CHSH. This line of conversation usually does not turn out in a good way. So I too am interested in a better way to explain this.
@Nathaniel sure, but again there is a subtlety as classical probabilistic "universes" can still interfere constructively, just not destructively. But since you bring up interference I will mention my own paper arXiv:1305.2186.
The parallel universes explanation (or the exponential size of state space) is unsatisfactory, in my opinion, since this would also apply to classical probability distributions (which can be viewed as vectors). The difference is that probability distributions admit a hidden variables interpretation. But the distinction between probability distributions and quantum states is subtle. I like the quote that opens Prong 2 of Aaronson's Are Quantum States Exponentially Long Vectors?.
Chapter 9 of arXiv:1206.0785 "The Quantum Frontier" lists some of the claimed sources of quantum speedup (not necessarily the ones you list), and explains why these explanations are unsatisfactory.
That's why I asked there originally. But this has to do with quantum information, a topic which spans the math, cstheory, and physics sites. Although many QI folk subscribe to all three sites, physics seems to have a larger population. The question itself is math, but QI people have the intuition on trace preserving maps and Kraus operators.
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