Operational Quantum Frames: An operational approach to quantum reference frames

The quantum reference frames program is based on the idea that reference frames should be treated as quantum physical systems. In this work, we combine these insights with the emphasis on operationality, understood as refraining from introducing into the framework objects not directly related to in principle verifiable probabilities of measurement outcomes, and identifying the setups indistinguishable as such. Based on intuitions from special relativity and gauge theory, we introduce an operational notion of a quantum reference frame -- which is defined as a quantum system equipped with a covariant positive operator-valued measure (POVM) -- and build a framework on the concept of operational equivalence that allows us to enforce operationality by quotienting the quantum state spaces with equivalence relation of indistinguishability by the available effects, assumed to be invariant under gauge transformations, and framed in the sense of respecting the choice of the frame's POVM. Such effects are accessed via the yen construction, which maps effects on the system to those on the composite system, satisfying gauge invariance and framing. They are called relative, and the classes of states indistinguishable by them are referred to as relative states. We show that when the frame is localizable, meaning that it allows for states that give rise to a highly localized probability distribution of the frame's observable, by restricting the relative description upon such localized frame preparation we recover the usual, non-relational formalism of quantum mechanics. We provide a consistent way of translating between different relative descriptions by means of frame-change maps and compare these with the corresponding notions in other approaches to QRF, establishing an operational agreement in the domain of common applicability.