  We characterise the sensitivity of several additive tensor decompositions
with respect to perturbations of the original tensor. These decompositions
include canonical polyadic decompositions, block term decompositions, and sums
of tree tensor networks. Our main result shows that the condition number of all
these decompositions is invariant under Tucker compression. This result can
dramatically speed up the computation of the condition number in practical
applications. We give the example of an $265\times 371\times 7$ tensor of rank
$3$ from a food science application whose condition number was computed in
$6.9$ milliseconds by exploiting our new theorem, representing a speedup of
four orders of magnitude over the previous state of the art.
