Operational Quantum Reference Frame Transformations
We provide a general, operational, and rigorous basis for quantum reference frames and their transformations using covariant positive operator valued measures to represent frame observables. The framework holds for locally compact groups and differs from all prior proposals for frame changes, being built around the notion of operational equivalence, in which states that cannot be distinguished physically are identified. This allows for the construction of the space of (invariant) relative observables and the convex set of relative states as dual objects. By demanding a further equivalence relation on the relative states which takes into account the nature of the frames, we provide a quantum reference frame change map. We show that this map is invertible exactly when the initial and final frames admit states which are arbitrarily well localized. We compare the presented frame change with other constructions available in the literature, finding operational agreement on the domain of common applicability.