Quantum tomography meets dynamical systems and bifurcations theory

Abstract: 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.