  Both completely positive and completely copositive maps stay decomposable
under tensor powers, i.e under tensoring the linear map with itself. But are
there other examples of maps with this property? We show that this is not the
case: Any decomposable map, that is neither completely positive nor completely
copositive, will lose decomposability eventually after taking enough tensor
powers. Moreover, we establish explicit bounds to quantify when this happens.
To prove these results we use a symmetrization technique from the theory of
entanglement distillation, and analyze when certain symmetric maps become
non-decomposable after taking tensor powers. Finally, we apply our results to
construct new examples of non-decomposable positive maps, and establish a
connection to the PPT squared conjecture.
