Conservation of information seems to be a deep physical principle. For instance, Unitarity is a key concept in Quantum Mechanics and Quantum Field Theory.
We may wonder if there is an underlying symmetry, in some space, which may explain this conservation of information.
1) If you want a Noether theorem for information, there is no such thing.
Trying to obtain it from a symmetry law, by Noether's theorem can't work, simply because information is not a quantity that can be obtained for instance by the derivative of the Lagrangian with respect to some variable. Information is not scalar, vector, tensor, spinor etc.
2) Another way to obtain conservation laws can be found in quantum mechanics. The observables that commute with the Hamiltonian are conserved. Again, you don't have an observable, in the sense of quantum mechanic, for information.
Trying to obtain conservation of information from commutation with Hamiltonian can't work, because there is no observable (hermitian operator on the Hilbert space) associated to information. Information is not the eigenvalue of such an operator.
3) The only way, which also is the simplest and the most direct, is the following: to have information conservation, when you reverse the evolution laws, you have to obtain evolution laws that are deterministic. This ensures conservation of information, in fact, they are equivalent. In particular, most classical laws are deterministic and reversible. Also, in quantum mechanics, unitary evolution is reversible, giving you the conservation of information.
I don't say that the evolution laws have to be deterministic, or that they have to be invariant to time reversal. Just that, when you apply time reversal, the evolution equations you obtain (which are allowed to be different than the original ones) are deterministic. Simplest way to think about this is by using dynamical systems. Trajectories in phase space are not allowed to merge, because if they merge, the information about what trajectory was before merging is lost. They are allowed to branch, because you can still go back and see what any previous state was. Branching breaks determinism, but not preservation of information. Old information is preserved at branching, but, as WetSavannaAnimal mentioned, new information is added. Therefore, if we want strict conservation, we should forbid both merging and branching, and in this case determinism is required.
CPT seems to imply it. You can reverse the system evolution by applying charge, parity and time conjugation, so the information about the past must be contained in the present state. That implies conservation of information by the evolution.
This may not be the answer you wanted, because it does not imply unitarity, but it is the only relationship between symmetry and information conservation that I can think of. Unitarity seems to be a very fundamental assumption though, and there is not much more fundamental mathematical structure you could use to argue about its necessity.
I realize I'm a bit late to this thread, but incase anyone stumbles upon this question, here's the answer:
The answer is that there is a symmetry associated with information conservation, but it doesn't come from the usual Lagrangian. Ordinarily for quantum or classical systems, we have a Lagrangian of the form $L=\frac{1}{2}m \dot x^2+V(x)$ and conserved quantities have to do with symmetries of this Lagrangian. For conservation of information, things are a little different. Instead of coming up with a Lagrangian that describes the motion of a particle, instead, we must treat the quantum wave function as a (classical) field. In this case the Action would look like $$S=\int dt\int dx \bigg[\frac{i\hbar}{2}[\dot \psi\psi^*-\psi\dot\psi^*]+\psi^*\hat H\psi \bigg],$$ where $\hat H$ is the Hamiltonian of the system. It is not hard to check that the Euler Lagrange equations coming from this action reproduce the Schrodinger equation.
Notice that under the transformation $$\psi\to e^{-i\alpha}\psi $$ $$\psi^*\to e^{i\alpha}\psi^* $$ for constant $\alpha$ leaves $S$ invariant, meaning that there is an associated conserved current. It turns out that this current is the probability current. (see https://en.wikipedia.org/wiki/Probability_current). As a result, $$\int dx~ \psi(x)\psi^*(x)= const.$$ for all time. In particular, for a normalized wave function, $$\int dx~ \psi(x)\psi^*(x)= 1,$$ meaning that the probability of $something$ happening is always 100%. So information (aka probability) is the conserved quantity corresponding the the fact that we can multiply wave functions by an overall complex phase without changing the physics.
In my answer, I focused on the case of ordinary non-relativistic quantum mechancis for the sake of clarity. But the above line of reasoning works (though in much more complicated ways) for any Unitary quantum theory eg QFT.
Conservation of information can be derived from Liouville theorem, which can be interpreted in terms of time-translation symmetries.
Unitarity is the symmetry you seek. Whats wrong with that?
Does the conservation of information not stem directly from the fact there exists equations of motion for a system? So the fact we can actually form a Lagrangian for a system implies information conservation? At least in a classical perspective. Unitary evolution would be the quantum mechanical version. Sorry if that is a naive suggestion.
In quantum physics, information is not usually taken to be an observable. It does not make sense to ask that it be conserved, if we take conservation to have its usual mathematical meaning.
If you want to insist that information be an observable, you can imagine that it is the dimension of the Hilbert space, or alternately the identity operator. Conservation of information is then a poetic way of saying that time evolution does not transform the identity into a projection.
If you are willing to grant that information is the identity observable, then it is clear what symmetry group it generates: it is the trivial group which acts identically on all states.
Entropy is used to quantify information and since disorder increases information decreases. I think you mean some specific information is conserved, not the general amount of information in the universe. In the same way, one can show that entropy doesn't always increase - only when you look at the entire universe ("generalized entropy")
Information understood as number of configurations being conserved means, in fact, that probability is conserved, as the answer above told you. A more intuitive way to say that is that QM evolution conserves the number of stacks, even when the number of particles (or the particle+antiparticles) is NOT conserved in QFT, the probability to find some number of particles in certain microstates or stacks must be conserved for a given energy or for a given configuration. Of course, the real classical world is a bit different since we have dissipation, something that we can include in QM not without some concerns and care.
In classical statistical mechanics, information conservation, comes in the form of Liouville's theorem, simply says that the object will not disappear or created (Or, the phase space density is constant along its trajectory). This do not correspond to any symmetry.
