Operational Quantum Reference Frame Transformations
Titouan Carette, Jan Głowacki, Leon Loveridge
TL;DR
This work builds an operational framework for quantum reference frames (QRFs) applicable to locally compact groups by employing operational equivalence and a relativization map to define relative observables and states. QRFs are modeled as systems of covariance, and framing couples a frame observable to the system, yielding framed relative observables whose invariants capture physically observable content. Relative states are defined as equivalence classes under framed observables, with a duality to the standard quantum description via the predual of the relativization map; conditioning on a localized frame recovers ordinary quantum mechanics in the appropriate limit. The authors construct invertible, composable frame transformations using lifting and an $ y^{\mathcal{R}}_*$ predual, show that localizable frames reproduce standard kinematics, and compare their approach to existing perspectival and perspective-neutral frameworks, arguing that operational equivalence leads to observational indistinguishability of frame changes. The theory provides a concrete pathway to internal QRFs, including composition across multiple frames, while highlighting the need for careful interpretation of entanglement claims within an operationally indistinguishable framework.
Abstract
Quantum reference frames are needed in quantum theory for much the same reasons that reference frames are in classical theories: to manifest invariance in line with fundamental relativity principles and to provide a basis for the definition of observable quantities. Though around since the 1960s, and used in a wide range of applications, only recently has the means for transforming descriptions between different quantum reference frames been tackled in detail. In this work, we provide a general, operationally motivated framework for quantum reference frames and their transformations, holding for locally compact groups. The work is built around the notion of operational equivalence, in which quantum states that cannot be physically distinguished are identified. For example, we describe the collection of relative observables as a subspace of the algebra of invariants on the composite of system and frame, and from here the set of relative states is constructed through the identification of states which cannot be distinguished by relative observables. Through the notion of framed observables -- the formation of joint observables of system and frame -- of which the relative observables can be understood as examples, quantum reference frame transformations are then maps between equivalence classes of relative states which respect the framing. We give an explicit realisation in the setting that the initial frame admits a highly localized state with respect to the frame observable. The transformations are invertible exactly when the final frame also has such a localizability property. The procedure we present is in operational agreement with other recent inequivalent constructions on the domain of common applicability, but extends them in a number of ways, and weakens claims of entanglement generation through frame changes.