We prove the quantum version of an ergodic result of H. Furstenberg relative to nonin-variant measures. The natural setting will be the case of the "quantum diagonal measure"relative to the product measure. Even if in all the interesting situations such diagonalmeasures are neither invariant nor normal with respect to the corresponding productones, we still provide an ergodic theorem for them, generalizing the classical case. As anatural application, we are able to prove the entangled ergodic theorem in some inter-esting situations out of the known ones, that is when the unitary is not almost periodic,or when the involved operators are not compact.
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