  Typical quantum communication schemes are such that to achieve perfect
decoding the receiver must share a reference frame with the sender. Indeed, if
the receiver only possesses a bounded-size quantum token of the sender's
reference frame, then the decoding is imperfect, and we can describe this
effect as a noisy quantum channel. We seek here to characterize the performance
of such schemes, or equivalently, to determine the effective decoherence
induced by having a bounded-size reference frame. We assume that the token is
prepared in a special state that has particularly nice group-theoretic
properties and that is near-optimal for transmitting information about the
sender's frame. We present a decoding operation, which can be proven to be
near-optimal in this case, and we demonstrate that there are two distinct ways
of implementing it (corresponding to two distinct Kraus decompositions). In
one, the receiver measures the orientation of the reference frame token and
reorients the system appropriately. In the other, the receiver extracts the
encoded information from the virtual subsystems that describe the relational
degrees of freedom of the system and token. Finally, we provide explicit
characterizations of these decoding schemes when the system is a single qubit
and for three standard kinds of reference frame: a phase reference, a Cartesian
frame (representing an orthogonal triad of spatial directions), and a reference
direction (representing a single spatial direction).
