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If a system undergoes symmetric dynamics, then the final state of the systemcan only break the symmetry in ways in which it was broken by the initialstate, and its measure of asymmetry can be no greater than that of the initialstate. It follows that for the purpose of understanding the consequences ofsymmetries of dynamics, in particular, complicated and open-system dynamics, itis useful to introduce the notion of a state's asymmetry properties, whichincludes the type and measure of its asymmetry. We demonstrate and exploit thefact that the asymmetry properties of a state can also be understood in termsof information-theoretic concepts, for instance in terms of the state's abilityto encode information about an element of the symmetry group. We show that theasymmetry properties of a pure state psi relative to the symmetry group G arecompletely specified by the characteristic function of the state, defined aschi_psi(g)=<psi|U(g)|psi> where g\in G and U is the unitary representation ofinterest. For a symmetry described by a compact Lie group G, we show that twopure states can be reversibly interconverted one to the other by symmetricoperations if and only if their characteristic functions are equal up to a1-dimensional representation of the group. Characteristic functions also allowus to easily identify the conditions for one pure state to be converted toanother by symmetric operations (in general irreversibly) for the variousparadigms of single-copy transformations: deterministic, state-to-ensemble,stochastic and catalyzed.

The theory of manipulations of pure state asymmetry: basic tools and equivalence classes of states under symmetric operations