Sana speaks to me whenever I go to my mirror, she says that the algebraic dress of quantum mechanics hides a beautiful geometrical lingerie that I will try to uncover during the talk. First of all, I will introduce the main features of quantum mechanics and I will outline how we may think of it as 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 explore the geometry of the space S of quantum states by looking at 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.We will discover the beautiful and highly rich geometry of the manifolds of isospectral quantum states - the oribits of U(n) - and use 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. Specifically, we will see how to extract quantum metric tensors satisfying the monotonicity property from quantum relative entropies satisfying the data processing inequality, and we will consider the explicit example of the two-parameter family of quantum relative entropies known as α-z-Rényi-relative entropies obtaining a two-parameter family of quantum metric tensors satisfying the monotonicity property. By suitably varying the parameters, we obtain the quantum metric tensors associated with well-known quantum relative entropies like, for instance, the von Neumann-Umegaki relative entropy, the quantum Rényi relative entropy and the Wigner-Yanase-Dyson skew information. Finally, we will mention some perspectives regarding the generalization of this work to composite systems - an extremely relevant situation in quantum computation. A covariant, coordinate-free, geometrical formalism will be the background spacetime in which we will move.