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

A Nonconventional Ergodic Theorem for quantum "diagonal measures"