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 E. 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-jA system
can be used to polarise a larger spin-jB 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.