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Restrictions on quantum operations give rise to resource theories. Total lackof a shared reference frame for transformations associated with a group Gbetween two parties is equivalent to having, in effect, an invariant channelbetween the parties and a corresponding superselection rule. The resourceassociated with the absence of the reference frame is known as "frameness" or"asymmetry." We show that any entanglement monotone for pure bipartite statescan be adapted as a pure-state frameness monotone for phase-invariant channels[equivalently U(1) superselection rules] and extended to the case of mixedstates via the convex-roof extension. As an application, we construct a familyof concurrence monotones for U(1) frameness for general finite-dimensionalHilbert spaces. Furthermore, we study "frameness of formation" for mixed statesanalogous to entanglement of formation. In the case of a qubit, we show that itcan be expressed as an analytical function of the concurrence analogously tothe Wootters formula for entanglement of formation. Our results highlight deeplinks between entanglement and frameness resource theories.

Constructing monotones for quantum phase references in totally dephasing channels