  In this paper, Wielandt's inequality for classical channels is extended to
quantum channels. That is, an upper bound to the number of times a channel must
be applied, so that it maps any density operator to one with full rank, is
found. Using this bound, dichotomy theorems for the zero--error capacity of
quantum channels and for the Matrix Product State (MPS) dimension of ground
states of frustration-free Hamiltonians are derived. The obtained inequalities
also imply new bounds on the required interaction-range of Hamiltonians with
unique MPS ground state.
