A. Neumaier said:
Thus all large systems have mixed states. The only possible exception would be the whole universe, as its state is not a partial trace of something bigger. But why make an exception for the universe?
Yes, I think I understood this part already; this motivation makes sense to me and I agree with it. I realize now that my last post was too long and not very clear, so I'll try to say it with fewer words :).
It is commonly argued that the measurement problem can't be solved with ordinary quantum mechanics. People who argue this find a situation where, by the linearity of time evolution, the universe has to end up in a superposition of both possible measurement outcomes, so the final state doesn't tell us "which one actually happened".
You reject this picture in favor of one where what's important is the q-expectation of the observable ##X^E## that records the macroscopic result of the experiment. The presence of mixed terms like (in our earlier notation) ##\psi_1\psi_2^*\otimes\rho^E## mean that we can't extract the result of measuring ##\frac{1}{\sqrt{2}}(\psi_1+\psi_2)## from the results of measuring ##\psi_1## and ##\psi_2##.
This explanation depends on the fact that these mixed terms can contribute a large amount to the q-expectation of ##X^E##. Furthermore, it seems like we would also like, in all three states, the q-variance of ##X^E## to be small. I'm worried these two desires might be incompatible, that requiring the q-variances to be small might also force the mixed terms to be small. The intuition I'm working from comes from pure states: in the extreme case where the final states are
eigenstates of ##X^E##, the mixed terms are zero, and for pure states that are merely
close to being eigenstates we should be able to make the mixed terms small, which also seems bad.
The question I'm asking is more a mathematical one than a physical one; I realize we have good physical grounds to disallow pure states, but I'm interested in exactly how that prescription interacts with the picture of measurement you're advocating.
(a) If, hypothetically, the thermal interpretation did allow pure states, would the thing I just described be a problem, or did I make a mistake again?
(b) If the answer to (a) is yes, why does the problem go away when we disallow pure states?
I think what might help me is an explicit example. How hard is it to construct a ##\rho^E##, ##X^E##, ##\psi_1##, and ##\psi_2## that behave this way after applying some unitary map?