Gently wandering through the space of quantum states, relating metric tensors with group actions

I will discuss some aspects of the geometry of the space of quantum states in finite dimensions (which means I will mainly focus on good ol' matrices).

After a brief introduction, I will try to exhibit (what I find to be) a curious link that exists between some of the most used Riemannian metric tensors in quantum information geometry (e.g., the Bures metric tensor, the Wigner-Yanase metric tensor, and the Bogoliubov-Kubo-Mori metric tensor) and some group actions on the space of quantum states which are associated with suitable extensions of the unitary group.