  We prove a general theorem that the action of arbitrary classical noise or
random unitary channels can not increase the maximum population of any
eigenstate of an open quantum system, assuming initial system-environment
factorization. Such factorization is the conventional starting point for
descriptions of open system dynamics. In particular, our theorem implies that a
system can not be ideally cooled down unless it is initially prepared as a pure
state. The resultant inequality rigorously constrains the possibility of
cooling the system solely through temporal manipulation, i.e., dynamical
control over the system Hamiltonian without resorting to measurement based
cooling methods.
