  Quantum information geometry studies families of quantum states by means of
differential geometry. A new approach is followed with the intention to
facilitate the introduction of a more general theory in subsequent work. To
this purpose, the emphasis is shifted from a manifold of strictly positive
density matrices to a manifold of faithful quantum states on the C*-algebra of
bounded linear operators. In addition, ideas from the parameter-free approach
to information geometry are adopted. The underlying Hilbert space is assumed to
be finite-dimensional. In this way technicalities are avoided so that strong
results are obtained, which one can hope to prove later on in a more general
context. Two different atlases are introduced, one in which it is
straightforward to show that the quantum states form a Banach manifold, the
other which is compatible with the inner product of Bogoliubov and which yields
affine coordinates for the exponential connection.
