  To what extent does Noether's principle apply to quantum channels? Here, we
quantify the degree to which imposing a symmetry constraint on quantum channels
implies a conservation law, and show that this relates to physically impossible
transformations in quantum theory, such as time-reversal and spin-inversion. In
this analysis, the convex structure and extremal points of the set of quantum
channels symmetric under the action of a Lie group $G$ becomes essential. It
allows us to derive bounds on the deviation from conservation laws under any
symmetric quantum channel in terms of the deviation from closed dynamics as
measured by the unitarity of the channel. In particular, we investigate in
detail the $U(1)$ and $SU(2)$ symmetries related to energy and angular momentum
conservation laws. In the latter case, we provide fundamental limits on how
much a spin-$j_A$ system can be used to polarise a larger spin-$j_B$ system,
and on how much one can invert spin polarisation using a rotationally-symmetric
operation. Finally, we also establish novel links between unitarity,
complementary channels and purity that are of independent interest.
