Stratified Manifold of Quantum States, actions of the complex special linear group

We review the geometry of the space of quantum states mathscr{S}\(mathcal{H}\) of a finite-level quantum system with Hilbert space mathcal{H} from a group-theoretical point of view. This space carries two stratifications generated by the action of two different Lie groups: the special unitary group mathcal{SU}\(mathcal{H}\) and its complexification mathcal{SL}\(mathcal{H}\), the complex special linear group. A stratum of the stratification generated by mathcal{SU}\(mathcal{H}\) is composed of isospectral states, that is, density operators with the same spectrum, A stratum of the stratification generated by mathcal{SL}\(mathcal{H}\) is composed of quantum states with the same rank. We prove that on every submanifold of isospectral quantum states there is also a canonical left action of mathcal{SL}\(mathcal{H}\) which is related with the canonical Kähler structure on isospectral quantum states. The fundamental vector fields of this mathcal{SL}\(mathcal{H}\)-action are divided into Hamiltonian and gradient vector fields. The former give rise to invertible maps on mathscr{S}\(mathcal{H}\) that preserve the von Neumann entropy and the convex structure of mathscr{S}\(mathcal{H}\), while the latter give rise to invertible maps on mathscr{S}\(mathcal{H}\) that preserve the von Neumann entropy but not the convex structure of mathscr{S}\(mathcal{H}\). A similar decomposition is given for the mathcal{SL}\(mathcal{H}\)-action generating the stratification of mathscr{S}\(mathcal{H}\) into manifolds of quantum states with the same rank, where gradient vector fields preserve the rank but do not preserve entropy. Some comments on multipartite quantum systems are made. It is proved that the sets of product states of a multipartite quantum system are homogeneous manifolds for the action of the complex special linear group associated with the partition.