 
Summary: Controllability of quantum mechanical systems by root space decomposition of
su(N)
Claudio Altafini
SISSAISAS
International School for Advanced Studies
via Beirut 24, 34014 Trieste, Italy
The controllability property of the unitary propagator of an Nlevel quantum mechanical system
subject to a single control field is described using the structure theory of semisimple Lie algebras.
Sufficient conditions are provided for the vector fields in a generic configuration as well as in a few
degenerate cases.
I. INTRODUCTION
The question of controllability for a finite level quantum system, see Ref. [Dahlen 1996, Schirmer 2001a,
Turinici 2001], is studied in this paper by analyzing the structure of the semisimple Lie algebra of its time
evolution operator. The system dynamics is determined by its internal Hamiltonian and by an external
Hamiltonian describing the interaction with a control field. Of the several different aspects of controllability
that can be defined for a closed system of such type (overviewed in Ref. [Albertini 2001, Schirmer 2001b]),
we consider here the more direct and important in practical applications, namely the controllability of
its unitary propagator which, in control terms, is governed by a bilinear system with drift and a single
control input and evolves on SU(N). For a compact semisimple Lie group like SU(N), the testing of global
controllability is the simplest of all noncommutative Lie groups. In fact, compactness implies that the
