  Given any two quantum states $\rho$ and $\sigma$ in Hilbert spaces of equal
dimension satisfying the majorization condition $\rho \succ \sigma$, it is
always possible to transform $\rho \mapsto \sigma$ by a unital quantum map. In
fact, any such transformation can be achieved just by means of noisy
operations, i.e., by access to maximally mixed ancillary states and unitary
transformations that act jointly in the system-ancilla space. Here, we
investigate the possible transitions between states (i.e., the induced preorder
of states) when one restricts the unitary control to the quantum system alone
and replaces the maximally mixed ancillas with a Markovian master equation,
represented by a unital Lindbladian. As a main result, we find necessary and
sufficient conditions for the Lindbladian dissipation to have the same
converting power as that of noisy operations, i.e., any transformation $\rho
\mapsto \sigma$ is possible if and only if $\rho \succ \sigma$.
