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