  We extend the Nonconventional Ergodic Theorem for generic measures by
Furstenberg, to several situations of interest arising from quantum dynamical
systems. We deal with the diagonal state canonically associated to the product
state (i.e. quantum "diagonal measures"), or to convex combinations of diagonal
measures for non ergodic cases. For the sake of completeness, we treat also the
Nonconventional Ergodic Theorem for compact dynamical systems, that is when the
unitary generating the dynamics in the GNS representation is almost periodic.
The Nonconventional Ergodic Theorem allows in a natural way to determine the
limit of the three-point correlations, naturally relevant for the knowledge of
the ergodic properties of a dynamical system.
