  A common objective for quantum control is to force a quantum system,
initially in an unknown state, into a particular target subspace. We show that
if the subspace is required to be a decoherence-free subspace of dimension
greater than 1, then such control must be decoherent. That is, it will take
almost any pure state to a mixed state. We make no assumptions about the
control mechanism, but our result implies that for this purpose coherent
control offers no advantage, in principle, over the obvious measurement-based
feedback protocol.
