Beyond the established symmetries of physics - Charge, Spin, Color, Time, etc. - is there any such thing as symmetry of Information?

To narrow the scope and for ease, I take that information as "measurement" from quantum field.

If the many-world interpretation is real, we have multiverses branching off by different measurements as wave function collapse from the same universe (with all of its information). It seems to me that symmetry of information is broken continuously, as wave functions collapse continuously. This lead me to believe that it is beyond our comprehension to theorize and observe said symmetry.