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:
Solidly understand how to ground out the Born rule in the inner product. (https://arxiv.org/abs/1405.7907 might be a good place to start if you haven’t already done this? I didn’t personally like the narrative there, but I found some of the framework helpful.)
Recall the details of that one theorem that relates unitary evolution to conservation of information.
Meditate on the connection between unitary operators, orthonormal bases, and the inner product.
See if there’s a compelling link to be made that runs from the Born rule, through the inner product operator, through unitarity, to conservation of information.
Also, I find this question interesting and am also curious for an answer :-)
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:
Solidly understand how to ground out the Born rule in the inner product. (https://arxiv.org/abs/1405.7907 might be a good place to start if you haven’t already done this? I didn’t personally like the narrative there, but I found some of the framework helpful.)
Recall the details of that one theorem that relates unitary evolution to conservation of information.
Meditate on the connection between unitary operators, orthonormal bases, and the inner product.
See if there’s a compelling link to be made that runs from the Born rule, through the inner product operator, through unitarity, to conservation of information.
Also, I find this question interesting and am also curious for an answer :-)