Epistemic status: very curious non-physicist.
Here's what I find weird about the Born rule.
Eliezer very successfully thought about intelligence by asking "how would you program a computer to be intelligent?". I would frame the Born rule using the analogous question for physics: "if you had an enormous amount of compute, how would you simulate a universe?".
Here is how I would go about it:
- Simulate an Alternate Earth, using quantum mechanics. The simulation has discrete time. At each step in time, the state of the simulation is a wavefunction: a set of
(amplitude, world)
pairs. If you would have two pairs with the same world
in the same time step, you combine them into one pair by adding their amplitude
s together. Standard QM, except for making time discrete, which is just there to make this easier to think about and run on a computer.
- Seed the Alternate Earth with humans, and run it for 100 years.
- Select a world at random, from some distribution. (!)
- Scan that world for a physicist on Alternate Earth who speaks English, and interview them.
The distribution used in step (3) determines what the physicist will tell you. For example, you could use the Born rule: pick at random from the distribution on worlds given by P(amplitude)=|amplitude2|. If you do, the interview will go something like this:
Simulator: Hi, it's God.
Physicist: Oh wow.
Simulator: I just have a quick question. In quantum mechanics, what's the rule for the probability that an observer finds themselves in a particular world?
Physicist: The probability is proportional to the square of the magnitude of the amplitude. Why is that, anyways?
Simulator: Awkwardly, that's what I'm trying to find out.
Physicist: ...God, why did you make a universe with so much suffering in it? My child died of bone cancer.
Simulator: Uh, gotta go.
Remember that you (the simulator) were picking at random from an astronomically large set of possible worlds. For example, in one of those worlds, photons in double slit experiments happened to always go left, and the physicists were very confused. However, by the law of large numbers, the world you pick almost certainly looks from the inside like it obeyed the Born rule.
However, the Born rule isn't the only distribution you could pick from in step 3. You could also pick from the distribution given by P(α)=|α| (with normalization). And frankly that's more natural. In this case, you would (almost certainly, by the law of large numbers) pick a world in which the physicists thought that the Born rule said P(α)=|α|. By Everett's argument, in this world probability does not look additive between orthogonal states. I think that means that its physicists would have discovered QM a lot earlier: the non-linear effects would be a lot more obvious! But is there anything wrong with this world, that would make you as the simulator go "oops I should have picked from a different distribution"?
There's also a third reasonable distribution: ignore the amplitudes, and pick uniformly at random from among the (distinct) worlds. I don't know what this world looks like from the inside.
I tried to commentate, and accidentally a whole post. Short version: I think one or two of the many mysteries people tend to find swirling around the Born rule are washed away by the argument you mention (regardless of how tight the analogy to Liouville's theorem), but some others remain (including the one that I currently consider central).
Warning: the post doesn't attempt to answer your question (ie, "can we reduce the Born rule to conservation of information?"). I don't know the answer to that. Sorry.
My guess is that a line can be drawn between the two; I'm uncertain how strong it can be made.
This may be just reciting things that you already know (or a worse plan than your current one), but in case not, the way I'd attempt to answer this would be: