  We show that the set of not-completely-positive (NCP) maps is unbounded,
unless further assumptions are made. This is done by first proposing a
reasonable definition of a valid NCP map, which is nontrivial because NCP maps
may lack a full positivity domain. The definition is motivated by specific
examples. We prove that for valid NCP maps, the eigenvalue spectrum of the
corresponding dynamical matrix is not bounded. Based on this, we argue that in
general the volume measure of qubit maps, including NCP maps, is not well
defined.
