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torekp comments on The Philosophical Implications of Quantum Information Theory - Less Wrong Discussion
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I disagree. Following Pearl, I define "A causes B" to mean something like: (DO:A) raises the probability of B.
Bob's choice in the evening to make strong measurements along the beta-axis, raises the probability of Alice's noon measurements along the beta-axis measurements having been the ones that showed the best correlation. It doesn't raise the probability of any individual measurement being up or down, but that's OK. Even on a many worlds interpretation, where perhaps every digital up/down pattern happens at some "world" and the overall multi-world distribution is invariant, "probability" refers to what happens in our "world", so again that's OK.
Correlation can only be observed after the fact, in the evening, not at noon. So isn't this just a case of Bob affecting Bob+Alice's immediate future, where they go over the results? Why do I say Bob's choice affected Alice's results? Because correlation is a two-way street, and in this case there isn't much traffic in the forward direction. Alice's measurements only weakly affect Bob's results.
Yes, but there's still some terminological sleight-of-hand going on here. It is only fair to say that a future A affected a past B if P(B) is well defined without reference to A. In this case it's not. Because B is defined in terms of correlations between measurements made at T1 (noon) and measurements made at T2 (evening) then B cannot be said to have actually happened until T2.
No, it's an n-squared-minus-one-way street. It appears to be a two-way street in one (very common) special case (two macroscopic systems mutually entangled with each other), but weak measurements are interesting precisely because they do not conform to the conditions of that special case. When you go beyond the conditions of the common special case you can't keep using the rhetoric and intuitions that apply only to the special case and hope to come up with the right answer.
You're right. Good point.
Don't you mean n-factorial? Anyway, ... hmm, I need to think about this more.
Yeah, probably. It's actually probably N!-1 because you have to trace over one degree of freedom to obtain a classical universe. But the details don't really matter. What matters is that it's >>N.