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Symmetries of both closed and open-system dynamics imply many significantconstraints. These generally have instantiations in both classical and quantumdynamics (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 aclassical system, that is, a uniquely quantum consequence of symmetricdynamics. Specifically, we demonstrate the impossibility of broadcastingasymmetry (symmetry-breaking) relative to a continuous symmetry group, forbounded-size quantum systems. The no-go theorem states that if two initiallyuncorrelated systems interact by symmetric dynamics and asymmetry is created atone subsystem, then the asymmetry of the other subsystem must be reduced. Wealso find a quantitative relation describing the tradeoff between thesubsystems. These results cannot be understood in terms of additivity ofasymmetry, because, as we show here, any faithful measure of asymmetry violatesboth sub-additivity and super-additivity. Rather, they must be understood as aconsequence of an (intrinsically quantum) information-disturbance principle.Our result also implies that if a bounded-size quantum reference frame for thesymmetry group, or equivalently, a bounded-size reservoir of coherence (e.g., aclock with coherence between energy eigenstates in quantum thermodynamics) isused to implement any operation that is not symmetric, then the quantum stateof the frame/reservoir is necessarily disturbed in an irreversible fashion,i.e., degraded.

A no-broadcasting theorem for quantum asymmetry and coherence and a trade-off relation for approximate broadcasting