  In this paper, we present a controllability analysis of the quantum Ising
periodic chain of n spin 1/2 particles where the interpolating parameter
between the two Hamiltonians plays the role of the control. A fundamental
result in the control theory of quantum systems states that the set of
achievable evolutions is (dense in) the Lie group corresponding to the Lie
algebra generated by the Hamiltonians of the system. Such a dynamical Lie
algebra therefore characterizes all the state transitions available for a given
system. For the Ising spin periodic chain we characterize such a dynamical Lie
algebra and therefore the set of all reachable states. In particular, we prove
that the dynamical Lie algebra is a (3n-1)-dimensional Lie sub-algebra of
su(2^n) which is a direct sum of a two dimensional center and a
(3n-3)-dimensional semisimple Lie subalgebra. This in turn is the direct sum of
n-1 Lie algebras isomorphic to su(2) parametrized by the eigenvalues of a fixed
matrix. We display the basis for each of these Lie subalgebras. Therefore the
problem of control for the Ising spin periodic chain is, modulo the two
dimensional center, a problem of simultaneous control of n-1 spin 1/2
particles. In the process of proving this result, we develop some tools which
are of general interest for the controllability analysis of quantum systems
with symmetry.
