Is my "Tentative Note on the origin and the evolution of information in the universe" correct?

Question

As a teacher of physics I got many questions asked by my students when aboarding the subject of information in quantum mechanics and the content of information in the universe. Based on my reading of articles and papers and participating in blogs (including the PF-Blog) I wrote down the following tentative summary about this topic. Please can you as a specialist who knows details about this subject check if my statements are correct and/or if you need to improve the text. My aim is to forward this summary to my students.

Tentative Note on the origin and the evolution of information in the universe.

In a closed quantum system the quantum information is conserved (in Hilbert space) according to the unitarity principle and conformal to the Liouville Theorem. As soon as an irreversible interaction or transition occurs, ie a measurement or a computation operation is made on the system, or if a decoherence process sets in, the hidden quantum information is extracted and spread out to the outside macroscopic world. In information communication/transmission systems, the information acquired through info processing is evaluated by Shannon's Information Theory, ie by his formula for the Information Entropy H(X) as a sum of probabilities, where H(X) is a measure of uncertainty or of "hidden Information" in the system. The amount of emergent (classical) information increases locally and becomes available as knowledge to an observer according to the probabilistic Born Rule and at the cost of local thermodynamic entropy increase according to the Landauer Principle. As the emergent information gets entangled with the environment the increasing correlations result in a degradation of the information in the universe with time. Or one could say that the information is still there in a different form, but is no longer accessible to observers. As a result the complexity in the universe first increases, attains a maximum and then decreases with time. Thus quantum information must have already existed at the Big Bang, and could have been generated by the decay of the inflaton at the end of the inflation era, when due to reheating, the huge thermodynamic entropy of cosmic radiation was released. Prior to this, near the Planck time in the quantum spacetime nugget, thermodynamic and gravitational entropy were very small (Past Hypothesis) and (quantum-gravitationally) the info content could have been minimal too. A still open question is the conjecture by some recent authors that the info bit is physical, ie is equivalent to a very small mass-energy. Quantum mechanical experiments seem to point in this direction: quantum Szilard engine, fluctuation theorem, reversible dynamics.