A relational approach to quantum reference frames for spins

TL;DR

The paper revisits quantum reference frames through a relational lens, treating internal properties as fidelities between subsystems and seeking transformations that preserve them. By imposing that all observers distinguish equal-sized subsystems with identical capabilities, it derives a fidelity-preserving symmetry group, showing it is $U(2)$ on the spin Hilbert space and $SO(3)$ on the projective space. A proposed Postulate 2 then identifies states with the same internal properties as physically equivalent in the absence of external frames, yielding the provocative result that microscopic and macroscopic superpositions can be frame-equivalent. However, the authors acknowledge that this approach does not fully resolve the Wigner–friend paradox, and further work is needed to classify equivalence classes and account for decoherence.

Abstract

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.