Noether's Theorem and Conservation of Information

[Feeble Wonk]

I'm not sure if this is the appropriate forum, but I'm trying to find out if there is a specific symmetry (according to Noether's Theorem) that is reflected in the conservation of information?

 

[Quine!]

I am bumping this thread because I was wondering it myself. I can't claim the mathematical chops for understanding Noether's theorem, but perhaps someone can give me some idea/well-informed speculation about this question, or even if it's sensible.

Leonard Susskind is perhaps the most reputable physicist to use the term "conservation of information" as a synonym for unitarity in QM. Wikipedia tells me unitarity implies "probabilities are numbers between 0 and 1 whose sum is conserved". So we seem to have some sort of conserved quantity here... although probabilities seem quite different a sort of thing than energy or momentum. So can we point to a symmetry in nature that reflects the fact that the square of the magnitude of a probability amplitude is conserved?

Hope I've understood this correctly, thanks