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To echo Scott, a density matrix is a probability distribution over quantum states - over a set of basis states, specifically. But if you pick a new basis, the same density matrix resolves into a different probability distribution over a different set of quantum states. If you believe that reality does reduce to amplitude flows in configuration space, then that means that one basis, the position basis, is the real one (since it corresponds to different possible states of reality, i.e. distributions of amplitude in configuration space); you can think of a density matrix as a probability distribution, period, and you're done. Density matrices will mean trouble if and only if you want to think of various incompatible choices of basis as equally real.
I would suggest that understanding superdense coding is another test of whether one's explanation of the significance of entanglement works. There is no purely quantum communication at a distance, but there can be a quantum rider on an otherwise classical communication channel.
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I'm a bit puzzled by the problem here. What's wrong with the interpretation that the Born probabilities just are the limiting frequencies in infinite independent repetitions of the same experiment? Further, that these limiting frequencies really are defined because the universe really is spatially infinite, with infinitely many causally isolated regions. There is nothing hypothetical at all about the infinite repetition - it actually happens.
My understanding is that in such a universe model, the Everett-Wheeler version of quantum theory makes a precise prediction: the limiting frequencies will with certainty correspond to the Born probabilities because the amplitude vanishes completely over the subspace of Hilbert space where they don't. More formally, the wave function of the universe is in an eigenstate of the relative frequency operator with the eigenvalue equal to the Born probability. Job done, surely?
Is the objection here just that we don't want to believe that the universe is spatially infinite?
Well why on Earth(s) would a MWI fan have any problem with that at all? Is it really any harder to believe that each branch of the wave function describes a strictly infinite universe (but that these infinite universes are all essentially identical, because they all have the correct frequency limits) than to believe that each branch describes a finite universe, and that while some of the branches get the frequency limits right, most of them don't?
That gave me, if I am not mistaken, the last piece of the puzzle. Let's just take the naive definition of probability - the relative frequency of outcomes as N goes to infinity. Now prepare N systems independently in the state a|0>+b|1>. Now measure one after another - couple the measurement device to the system. At first we have (a|0>+b|1>)^N * |0>. Now the first one is measured: (a|0>+b|1>)^(N-1) * (a|0,0>+b|1,1>) where the number after the comma denotes the state of the measuring device, which just counts the number of measured ones. After the second measurement we have (a|0>+b|1>)^(N-2) * (a²|00,0>+ab|01,1>+ab|10,1>+b²|11,2>) Since the two states ab|01,1> and ab|10,1> are not distinguished by the measurement, the basis should be changed - and this is the crucial point: |01>+|10> has a length of sqrt(2), so if we change the basis to |+>=(|01>+|10>)/sqrt(2), we have (a|0>+b|1>)^(N-2) * (a²|00,0>+absqrt(2)|+,1>+b²|11,2>).
The coefficiants are like in the binomial theorem, but note the sqare root!
Continuing, we will get something similar to a binomial distribution:
sum(k=0..N: sqrt(N!/(k!(N-k)!))*a^k * b^(N-k) |...,k>).
Now it remains to prove that for j/N not equal to a² the amplitudes go to zero as N goes to infinity. This is equivalent to the square of the amplitude going to zero (this is just to make the calculation easier, it does not have anything to do with the Born rule). It is, for |...,k>,
ck² = N!/(k!(N-k)!) * a²^k * b²^(N-k)
which becomes a Gaussian distribution for large N, with mean at k=Na² and width Na²b². So at k/N=a²+d it has a value proportional to exp(-(Nd)²/(2Na²b²))=exp(-Nd²/(2a²b²)) --> 0 as N --> inf.
So a time capsule where the records indicate that some quantum experiment has been performed a great number of times and the Born rule is broken will have an amplitude that goes to zero (yeah, I just read Barbour's book).