  The question of open-loop control in the Gaussian regime may be cast by
asking which Gaussian unitary transformations are reachable by turning on and
off a given set of quadratic Hamiltonians. For compact groups, including finite
dimensional unitary groups, the well known Lie algebra rank criterion provides
a sufficient and necessary condition for the reachable set to cover the whole
group. Because of the non-compact nature of the symplectic group, which
corresponds to Gaussian unitary transformations, this criterion turns out to be
still necessary but not sufficient for Gaussian systems. If the control
Hamiltonians are unstable, in a sense made rigorous in the main text, the
peculiar situation may arise where the rank criterion is satisfied and yet not
all symplectic transformations are reachable. Here, we address this situation
for one degree of freedom and study the properties of the reachable set under
unstable control Hamiltonians. First, we provide a partial analytical
characterisation of the reachable set and prove that no orthogonal
(`energy-preserving' or `passive' in the literature) symplectic operations may
be reached with such controls. Then, we apply numerical optimal control
algorithms to demonstrate a complete characterisation of the set in specific
cases.
