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Quantum reference frames are needed in quantum theory for much the samereasons as reference frames are in classical relativity theories: to manifestinvariance in line with fundamental relativity principles. Though around sincethe 1960s, and used in a wide range of applications, only recently has themeans for transforming descriptions between different frames been tackled indetail. Such transformations are needed for an internally consistent theory ofquantum reference frames. In this work, we provide a general, operationallymotivated framework for quantum reference frames and their transformations,holding for locally compact groups. The work is built around the notion ofoperational equivalence, in which theoretical objects that cannot be physicallydistinguished are identified. For example, we describe the collection ofobservables relative to a given frame as a subspace of the algebra ofinvariants on the composite of system and frame, and from here the set ofrelative states can be constructed as a convex subset of the predual. Besidesbeing invariant, the relative observables are also framed, meaning that theycan be realized with the chosen frame observable. The frame transformations arethen maps between equivalence classes of relative states that can bedistinguished by both initial and final frames. We give an explicit realisationin the setting that the initial frame admits a highly localized state withrespect to the frame observable. The transformations are invertible exactlywhen the final frame also has such a localizability property. The procedure wepresent is in operational agreement with other recent inequivalentconstructions on the domain of common applicability, but extends them in anumber of ways which we describe.

Operational Quantum Reference Frame Transformations