  In the absence of a reference frame for transformations associated with a
group G, any quantum state that is non-invariant under the action of G may
serve as a token of the missing reference frame. We here introduce a novel
measure of the quality of such a token: the relative entropy of frameness. This
is defined as the relative entropy distance between the state of interest and
the nearest G-invariant state. Unlike the relative entropy of entanglement,
this quantity is straightforward to calculate and we find it to be precisely
equal to the G-asymmetry, a measure of frameness introduced by Vaccaro et al.
It is shown to provide an upper bound on the mutual information between the
group element encoded into the token and the group element that may be
extracted from it by measurement. In this sense, it quantifies the extent to
which the token successfully simulates a full reference frame. We also show,
that despite a suggestive analogy from entanglement theory, the regularized
relative entropy of frameness is zero and therefore does not quantify the rate
of interconversion between the token and some standard form of quantum
reference frame. Finally, we show how these investigations yield a novel
approach to bounding the relative entropy of entanglement.
