  To implement any quantum operation (a.k.a. ``superoperator'' or ``CP map'')
on a d-dimensional quantum system, it is enough to apply a suitable overall
unitary transformation to the system and a d^2-dimensional environment which is
initialized in a fixed pure state. It has been suggested that a d-dimensional
environment might be enough if we could initialize the environment in a mixed
state of our choosing. In this note we show with elementary means that certain
explicit quantum operations cannot be realized in this way. Our counterexamples
map some pure states to pure states, giving strong and easily manageable
conditions on the overall unitary transformation. Everything works in the more
general setting of quantum operations from d-dimensional to d'-dimensional
spaces, so we place our counterexamples within this more general framework.
