There are many places where to find the proof of the conservation of the quantum Shannon information under unitary evolution. However, I just could not find the equivalent proof that everybody talks in classical mechanics, so here it goes.
The Lioville equation can be expressed in terms of a self-adjoint operator as observed by Koopman and von Neumann, such that
Given that the Liouvillian follows the Leibniz rule, which is true for conservative systems
it is now easy to prove that
This fact is in complete contrast with respect to the thermodynamical entropy, which can be fundamentally irreversible.
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