Is there experimental evidence to support this idea? (I assume no, at least not directly, but correct me if I'm wrong.)
Also, if one subscribes to the many-worlds interpretation, then isn't the creation of another world the same as the loss of information from this world (that is, the other world takes a different eigenstate of the waveform, and therefore this world is left with just one eigenstate)?
The time evolution operator is unitary, so it is in particular inversible. So there is a one to one correspondence between the posible current states of the system/universe and the posible states of the universe one minute later. So all the information that is available now, must be preserved one minute later because the transformation can be (theoretically) reversed. If some information disappear, you could not reverse the changes due to time evolution.
The experimental evidence shows that the equations for the time evolution are correct (up to the current experimental precision), and these equations have a form that is unitary. I'm not sure that it is a direct proof, but it's quite strong evidence.
And about wavefunction collapse or the multiuniverse alternative ...
<personal opinion warning [1]> Nobody like neither of the options but they make predictions that agree with the experiments. If we assume that humans, dogs and measurement equipment are not magic then we could explain the apparent wavefunction collapse using a normal time evolution. IIRC the current betting favor decoherence, where the equipment somehow has an apparent effect of the wavefunction collapse, but the information that is apparently lost is spread in many small parts that evolve slightly different and after some time it's imposible to experimentally collect all the parts and reverse the transformation (in spite it is theoretically posible to collect all the parts and reverse the transformation)[2]. Nobody is sure, and it is an active research topic. </personal opinion warning>
[1] I took enough quantum mechanic courses in the university to give a good answer, but I'm not expert in (quantum) measurement theory.
[2] It's slightly similar to how a drop of ink in a glass of water spread after some time by diffusion. It's theoretical posible to reverse the system and get the original result, but in practice it's imposible. But the equations of both process are very different. I hate analogies that have very different underlaying equations because they are misleading sooner or later. So take this as a bad analogy and don't draw too many conclusions from it.