The space of quantum states, relative entropies and metric tensors

The habit does not make the monk....the algebraic dress of quantum mechanics hides a beautiful geometrical lingerie that I will try to uncover during the talk.
In this context, I will briefly outline how we may think of the space of quantum states S as being a non-commutative version of classical probability theory, that is, how to look at quantum states as non-commutative versions of probability distributions.
Then, we will see how the complex general linear group GL(n, C) and the unitary group U(n) act on S partitioning it into the disjoint union of orbits, and we will discover the beautiful and highly rich geometry of the manifolds of isospectral quantum states - the oribits of U(n) - using it as a point of departure in order to look for geometrical structures on the manifold of invertible quantum states - the orbit of GL(n,C) which is the primary object of quantum information theory.
These geometrical structures will be families of quantum metric tensors satisfying the monotonicity property, and we will see how it is possible to extract these families from families of quantum relative entropies satisfying the data processing inequality.
The explicit example of the (huge) two-parameter family of quantum relative entropies known as α-z-Rényi-relative will be fully worked out.
A covariant, coordinate-free, geometrical formalism will be the background spacetime in which we will move.