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(Non)locality and (in)determinism

Imagine two experimentalists, Alice and Bob, situated in their laboratories. Alice chooses and pushes one of two buttons, A0 and A1, on her apparatus. Bob observes on his apparatus one of two indicating lamps, b0 and b1, lighting. The four combinations are logically possible: (A0,b0), (A0,b1), (A1,b0) and (A1,b1).

It may happen that only two combinations, (A0,b0) and (A1,b1), occur in the experiment. Then one concludes that A has influence on B.

It may happen that all the four combinations occur with some (conditional) probabilities P(b0|A0), P(b1|A0) = 1 - P(b0|A0), P(b0|A1) and P(b1|A1) = 1 - P(b0|A1). If P(b0|A0) differs from P(b0|A1), one concludes that A has influence on B.

Here is a more complicated scenario: Alice pushes one of two buttons, A0 and A1; also Bob pushes one of two buttons, B0 and B1. Alice observes one of two outcomes, a0 and a1; also Bob observes one of two outcomes, b0 and b1. Logically, 16 combinations are possible:

\textstyle (AX, BY, ax, by)

where each of X,Y,x,y is 0 or 1. Imagine that only 8 combinations occur, with the following (conditional) probabilities:

P( {ax,by}{|}{AX,BY} ) = \begin{cases} \frac{1}{2}, & \mbox{if } x \oplus y = XY \\ 0, & \mbox{otherwise} \end{cases}

That is, the two outcomes are perfectly anticorrelated (either (a0,b1) or (a1,b0), equiprobably) when (A1,B1) is chosen. In the three other cases ((A0,B0), (A0,B1), (A1,B0)), the two outcomes are perfectly correlated (either (a0,b0) or (a1,b1), equiprobably).

Does it imply that some influence exists (A on B, or B on A), or not?

The question is important, since the answer depends on our fundamental assumptions about nature.

On one hand, Alice cannot send a message to Bob, using her buttons A0, A1 and his indicators a0, a1. (Nor Bob to Alice.) In this sense the answer is negative (influence need not exist).

On the other hand, no one is able to design apparata that behave as required, without using a kind of influence (A on B, or B on A). In this sense the answer is affirmative (some influence must exist).

Thorough logical analysis reveals that the affirmative answer follows from the assumptions of local realism and counterfactual definiteness.