  In the quantization of gauge theories and quantum gravity, it is crucial to
treat reference frames such as rods or clocks not as idealized external
classical relata, but as internal quantum subsystems. In the Page-Wootters
formalism, for example, evolution of a quantum system S is described by a
stationary joint state of S and a quantum clock, where time-dependence of S
arises from conditioning on the value of the clock. Here, we consider (possibly
imperfect) internal quantum reference frames R for arbitrary compact symmetry
groups, and show that there is an exact quantitative correspondence between the
amount of entanglement in the invariant state on RS and the amount of asymmetry
in the corresponding conditional state on S. Surprisingly, this duality holds
exactly regardless of the choice of coherent state system used to condition on
the reference frame. Averaging asymmetry over all conditional states, we obtain
a simple representation-theoretic expression that admits the study of the
quality of imperfect quantum reference frames, quantum speed limits for
imperfect clocks, and typicality of asymmetry in a unified way. Our results
shed light on the role of entanglement for establishing asymmetry in a fully
symmetric quantum world.
