  In the literature on quantum reference frames, the internal (relative)
properties of a system are defined as those which are preserved under an
arbitrary change of reference frame. For a system of quantum spins, these are
all properties preserved by proper spatial rotations of the laboratory.
However, this approach does not account for the hypothetical possibility of the
laboratory becoming entangled to the system, as described by a second
laboratory (the `Wigner's friend' scenario), in which case the relationship
between the two laboratories is not a rotation, but is fundamentally quantum.
To overcome this limitation, we re-define the reference frame transformations
to be those that preserve the fidelities between subsystems. This enables us to
derive U(2) as the correct symmetry group for transformations of a system of N
spin-half particles. Next, we propose that systems having the same internal
properties should be regarded as physically equivalent in the absence of an
external frame. Remarkably, this implies that a single spin in a superposition
relative to a spin magnet is equivalent to a macroscopic superposition of the
magnet relative to the spin. We discuss the implications of this result for the
Wigner's friend paradox.
