The space of Quantum States, a Differential Geometric Setting

The subject of this thesis is the geometry of the space of quantum states. The aim of this thesis is to present a geometrical analysis of the structural properties of this space, being them of ``kinematical'' or ``dynamical'' character. We will see that the space of quantum states of finite-dimensional systems may be partitioned into the union of disjoint orbits of the complexification of the unitary group. These orbits are the manifolds of quantum states with fixed rank. On the one hand, we will compute the two-parameter family of quantum metric tensors associated with the two-parameter family of quantum q-z-Rényi relative entropies on the manifold of invertible quantum states (maximal rank). Using the powerful language of differential geometry we are able to perform all the computations in an arbitrary number of (finite) dimensions without the need to introduce explicit coordinate systems. On the other hand, we will develop a geometrization of the GKLS equation for the dynamical evolution of Markovian open quantum systems. Specifically, we will write the GKLS generator by means of an affine vector field on an affine space, and we will decompose this vector field into the sum of a Hamiltonian vector field, a gradient-like vector field, and a so-called Kraus vector field. This geometrization will be used in order to analyze and completely characterize the asymptotic behaviour of the dynamical evolutions known as quantum random unitary semigroups by means of the so-called purity function. Finally, we will comment on the possibility of extending the results presented to the infinite-dimensional case, and to the case of multipartite quantum systems.