  Symmetries of both closed and open-system dynamics imply many significant
constraints. These generally have instantiations in both classical and quantum
dynamics (Noether's theorem, for instance, applies to both sorts of dynamics).
We here provide an example of such a constraint which has no counterpart for a
classical system, that is, a uniquely quantum consequence of symmetric
dynamics. Specifically, we demonstrate the impossibility of broadcasting
asymmetry (symmetry-breaking) relative to a continuous symmetry group, for
bounded-size quantum systems. The no-go theorem states that if two initially
uncorrelated systems interact by symmetric dynamics and asymmetry is created at
one subsystem, then the asymmetry of the other subsystem must be reduced. We
also find a quantitative relation describing the tradeoff between the
subsystems. These results cannot be understood in terms of additivity of
asymmetry, because, as we show here, any faithful measure of asymmetry violates
both sub-additivity and super-additivity. Rather, they must be understood as a
consequence of an (intrinsically quantum) information-disturbance principle.
Our result also implies that if a bounded-size quantum reference frame for the
symmetry group, or equivalently, a bounded-size reservoir of coherence (e.g., a
clock with coherence between energy eigenstates in quantum thermodynamics) is
used to implement any operation that is not symmetric, then the quantum state
of the frame/reservoir is necessarily disturbed in an irreversible fashion,
i.e., degraded.
