  This work addresses a fundamental problem of controllability of open quantum
systems, meaning the ability to steer arbitrary initial system density matrix
into any final density matrix. We show that under certain general conditions
open quantum systems are completely controllable and propose the first, to the
best of our knowledge, deterministic method for a laboratory realization of
such controllability which allows for a practical engineering of arbitrary pure
and mixed quantum states. The method exploits manipulation by the system with a
laser field and a tailored nonequilibrium and time-dependent state of the
surrounding environment. As a physical example of the environment we consider
incoherent light, where control is its spectral density. The method has two
specifically important properties: it realizes the strongest possible degree of
quantum state control --- complete density matrix controllability which is the
ability to steer arbitrary pure and mixed initial states into any desired pure
or mixed final state, and is "all-to-one", i.e. such that each particular
control can transfer simultaneously all initial system states into one target
state.
