A geometric approach to quantum control in projective Hilbert spaces

Abstract

A quantum theory in a finite-dimensional Hilbert space can be geometrically formulated as a proper Hamiltonian theory as explained in [2, 3, 7, 8]. From this point of view a quantum system can be described in a classical-like framework where quantum dynamics is represented by a Hamiltonian flow in the phase space given by projective Hilbert space. This paper is devoted to investigate how the notion of accessibility algebra from classical control theory can be applied within geometric Hamiltonian formulation of Quanum Mechanics to study controllability of a quantum system. A new characterization of quantum controllability in terms of Killing vector fields w.r.t. Fubini-Study metric on projective space is also discussed.