Generalized metric phase space for quantum systems and the uncertainty principle

We demonstrate that when the Gibbs entropy is an invariant of motion, following Information Theory procedures it is possible to define a generalized metric phase space for the temporal evolution of the mean values of a given Hamiltonian. The metric is positive definite and this fact leads to a metric tensor, K(t), whose properties are well defined. Working with these properties we shown that: (a) the Generalized Uncertainty Principle (GUP), is always the summation over the principal minors of order 2 belonging to K(t); (b) several invariants of motion can be derived from the metric tensor; and (c) particularly, under certain conditions, the GUP itself, is also a motion invariant.