A physics friend of mine asserted today that within the field of information theory, information can be thought of as a conserved quantity. this put me in mind of Noether's theorem and motivated the following question; if it is ill-posed or meaningless, please let me know before you downvote and I will delete it so it doth not offend:

If information within a defined dynamical system is a conserved quantity, can Noether's theorem be used to tell us anything about the lagrangian of that system, and what the symmetry is which is responsible for that conserved quantity?