  What can one infer about the dynamical evolution of quantum systems just by
symmetry considerations? For Markovian dynamics in finite dimensions, we
present a simple construction that assigns to each symmetry of the generator a
family of scalar functions over quantum states that are monotonic under the
time evolution. The aforementioned monotones can be utilized to identify states
that are non-reachable from an initial state by the time evolution and include
all constraints imposed by conserved quantities, providing a generalization of
Noether's theorem for this class of dynamics. As a special case, the generator
itself can be considered a symmetry, resulting in non-trivial constraints over
the time evolution, even if all conserved quantities trivialize. The
construction utilizes tools from quantum information-geometry, mainly the
theory of monotone Riemannian metrics. We analyze the prototypical cases of
dephasing and Davies generators.
