Eliezer: Yeah, I understand. I was making a sort of meta-joke, that you shouldn't trust me over Gell-Mann about particle physics even after accounting for the fact that I say that and would be correspondingly reluctant to disagree...

Particle masses?? Definitely go with Gell-Mann.

When Einstein invented General Relativity, he had almost no experimental data to go on, except the precession of Mercury's perihelion. And (AFAIK) Einstein did not use that data, except at the end.

Eliezer, I'd love to believe that too, but from the accounts I've read I don't think it's quite right. Because of his "hole argument", Einstein took a long detour from the correct path in 1913-1915. During that time, he abandoned his principle of general covariance, and tried to find field equations that would "work well enough in practice anyway." Apparently, one of the main reasons he finally abandoned that line of thought, and returned to general covariance, is that he was getting a prediction for Mercury's perihelion motion that was too small by a factor of 2.

So is it possible that not even Einstein was a Bayesian superintelligence?

Incidentally, it looks to me like you should be able to test macroscopic decoherence. Eventually. You just need nanotechnological precision, very low temperatures, and perhaps a clear area of interstellar (intergalactic?) space.

Short of that, building a scalable quantum computer would be another (possibly easier!) way to experiment with macroscopic coherence. The difference is that with quantum computing, you wouldn't even try to isolate a quantum system perfectly from its environment. Instead you'd use really clever error-correction to encode quantum information in nonlocal degrees of freedom, in such a way that it can survive the decoherence of (say) any 1% of the qubits.

Inspired by this post, I was reading some of the history today, and I learned something that surprised me: in all of his writings, Bohr apparently never once talked about the "collapse of the wavefunction," or the disappearance of all but one measurement outcome, or any similar formulation. Indeed, Huve Erett's theory would have struck the historical Bohr as complete nonsense, since Bohr didn't believe that wavefunctions were real in the first place -- there was nothing to collapse!

So it might be that MWI proponents (and Bohmians, for that matter) underestimate just how non-realist Bohr really was. They ask themselves: "what would the world have to be like if Copenhagenism were true?" -- and the answer they come up with involves wavefunction collapse, which strikes them as absurd, so then that's what they criticize. But the whole point of Bohr's philosophy was that you don't even ask such questions. (Needless to say, this is not a ringing endorsement of his philosophy.)

Incidentally, I'm skeptical of the idea that MWI never even occurred to Bohr, Heisenberg, Schrรถdinger, or von Neumann. I conjecture that something like it must have occurred to them, as an obvious reductio ad absurdum -- further underscoring (in their minds) why one shouldn't regard the wavefunction as "real". Does anyone have any historical evidence either way?

Since the laws of probability and rationality are LAWS rather than "just good ideas", it isn't entirely shocking that there'd be some mathematical object th that would seem to act like the place where the territory and map meet. More to the point, the some mathematical object related to the physics that says "this is the most accurate your map can possibly be given the information of whatever is going on with this part/factor of reality."

That's a beautiful way of putting it, which expresses what I was trying to say much better than I did.

Mitchell: No, even if you want to think of the position basis as the only "real" one, how does that let you decompose any density matrix uniquely into pure states? Sure, it suggests a unique decomposition of the maximally mixed state, but how would you decompose (for example) ((1/2,1/4),(1/4,1/2))?

As for your pedagogical question, Eliezer -- well, the gift of explaining mathematical concepts verbally is an incredibly rare one (I wish every day I were better at it). I don't think most textbook writers are being deliberately obscure; I just think they're following the path of least resistance, which is to present the math and hope each individual reader (after working it through) will have his or her own forehead-slapping "aha!" moment. Often (as with your calculus textbook) that's a serious abdication of authorial responsibility, but in some cases there might really not be any faster way.

Psy-Kosh: TrA just means the operation that "traces out" (i.e., discards) the A subsystem, leaving only the B subsystem. So for example, if you applied TrA to the state |0〉|1〉, you would get |1〉. If you applied it to |0〉|0〉+|1〉|1〉, you would get a classical probability distribution that's half |0〉 and half |1〉. Mathematically, it means starting with a density matrix for the joint quantum state ρAB, and then producing a new density matrix ρB for B only by summing over the A-indices (sort of like tensor contraction in GR, if that helps).

Eliezer: The best way I can think of to explain a density matrix is, it's what you'd inevitably come up with if you tried to encode all information locally available to you about a quantum state (i.e., all information needed to calculate the probabilities of local measurement outcomes) in a succinct way. (In fact it's the most succinct possible way.)

You can see it as the quantum generalization of a probability distribution, where the diagonal entries represent the probabilities of various measurement outcomes if you measure in the "standard basis" (i.e., whatever basis the matrix happens to be presented in). If you measure in a different orthogonal basis, identified with some unitary matrix U, then you have to "rotate" the density matrix ρ to UρU* before measuring it (where U* is U's conjugate transpose). In that case, the "off-diagonal entries" of ρ (which intuitively encode different pairs of basis states' "potential for interfering with each other") become relevant.

If you understand (1) why density matrices give you back the usual Born rule when ρ=|ψ〉〈ψ| is a pure state, and (2) why an equal mixture of |0〉 and |1〉 leads to exactly the same density matrix as an equal mixture of |0〉+|1〉 and |0〉-|1〉, then you're a large part of the way to understanding density matrices.

One could argue that density matrices must reflect part of the "fundamental nature of QM," since they're too indispensable not to. Alas, as long as you insist on sharply distinguishing between the "really real" from the "merely mathematical," density matrices might always cause trouble, since (as we were discussing a while ago) a density matrix is a strange sort of hybrid of amplitude vector with probability distribution, and the way you pick apart the amplitude vector part from the probability distribution part is badly non-unique. Think of someone who says: "I understand what a complex number does -- how to add and multiply one, etc. -- but what does it mean?" It means what it does, and so too with density matrices.

Eliezer, I know your feelings about density matrices, but this is exactly the sort of thing they were designed for. Let ρ_AB be the joint quantum state of two systems A and B, and let U_A be a unitary operation that acts only on the A subsystem. Then the fact that U_A is trace-preserving implies that Tr_A[U_A ρ_AB U_A^*] = ρ_B, in other words U_A has no effect whatsoever on the quantum state at B. Intuitively, applying U_A to the joint density matrix ρ_AB can only scramble around matrix entries within each "block" of constant B-value. Since U_A is unitary, the trace of each of these blocks remains unchanged, so each entry (ρ_B)_ij of the local density matrix at B (obtained by tracing over a block) also remains unchanged. Since all we needed about U_A was that it was trace-preserving, this can readily be generalized from unitaries to arbitrary quantum operations including measurements. There, we just proved the no-communication theorem, without getting our hands dirty with a single concrete example! :-)