  We introduce and analyze a task that we call symmetrization, in which a state
of a quantum system, associated with a symmetry group, is transformed by a
random unitary operation to a symmetric state. Each element of the unitary
ensemble is required to be symmetry preserving, in the sense that it keeps the
set of symmetric states invariant. We consider an asymptotic limit of
infinitely many copies and vanishingly small error, and analyze the
symmetrizing cost, that is, the minimum cost of randomness per copy required
for symmetrization. We prove that the symmetrizing cost of an arbitrary quantum
state is equal to the relative entropy of frameness, thereby providing it with
a direct operational meaning.
