  We investigate a system of two coupled harmonic oscillators on the
non-commutative plane \RR^2_{\theta} by requiring that the spatial coordinates
do not commute. We show that the system can be diagonalized by a suitable
transformation, i.e. a rotation with a mixing angle \alpha. The obtained
eigenstates as well as the eigenvalues depend on the non-commutativity
parameter \theta. Focusing on the ground state wave function before the
transformation, we calculate the density matrix \rho_0(\theta) and find that
its traces {\rm Tr}(\rho_{0}(\theta)) and {\rm Tr}(\rho_0^2(\theta)) are not
affected by the non-commutativity. Evaluating the Wigner function on
\RR^2_{\theta} confirms this. The uncertainty relation is explicitly determined
and found to depend on \theta. For small values of \theta, the relation is
shifted by a \theta^2 term, which can be interpreted as a quantum correction.
The calculated entropy does not change with respect to the normal case. We
consider the limits \alpha=1 and \alpha={\pi\over 2}. In first case, by
identifying \theta to the squared magnetic length, one can recover basic
features of the Hall system.
