  A powerful tool for studying geometrical problems in Hilbert space is
developed. In particular, we study the quantum pure state tomography problem in
finite dimensions from the point of view of dynamical systems and bifurcations
theory. First, we introduce a generalization of the Hellinger metric for
probability distributions which allows us to find a geometrical interpretation
of the quantum state tomography problem. Thereafter, we prove that every
solution to the state tomography problem is an attractive fixed point of the
so-called physical imposition operator. Additionally, we demonstrate that
multiple states corresponding to the same experimental data are associated to
bifurcations of this operator. Such a kind of bifurcations only occurs when
informationally incomplete set of observables are considered. Finally, we prove
that the physical imposition operator has a non-contractive Lipschitz constant
2 for the Bures metric. This value of the Lipschitz constant manifests the
existence of the quantum tomography problem for pure states.
