Quantum reference frames for general symmetry groups

Anne-Catherine de la Hamette1,2 and Thomas D. Galley2

1Institute for Theoretical Physics, ETH Zürich, Wolfgang-Pauli-Str. 27, 8093 Zürich, Switzerland
2Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada

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A fully relational quantum theory necessarily requires an account of changes of quantum reference frames, where quantum reference frames are quantum systems relative to which other systems are described. By introducing a relational formalism which identifies coordinate systems with elements of a symmetry group $G$, we define a general operator for reversibly changing between quantum reference frames associated to a group $G$. This generalises the known operator for translations and boosts to arbitrary finite and locally compact groups, including non-Abelian groups. We show under which conditions one can uniquely assign coordinate choices to physical systems (to form reference frames) and how to reversibly transform between them, providing transformations between coordinate systems which are `in a superposition' of other coordinate systems. We obtain the change of quantum reference frame from the principles of relational physics and of coherent change of reference frame. We prove a theorem stating that the change of quantum reference frame consistent with these principles is unitary if and only if the reference systems carry the left and right regular representations of $G$. We also define irreversible changes of reference frame for classical and quantum systems in the case where the symmetry group $G$ is a semi-direct product $G = N \rtimes P$ or a direct product $G = N \times P$, providing multiple examples of both reversible and irreversible changes of quantum reference system along the way. Finally, we apply the relational formalism and changes of reference frame developed in this work to the Wigner's friend scenario, finding similar conclusions to those in relational quantum mechanics using an explicit change of reference frame as opposed to indirect reasoning using measurement operators.

Reference frames are essential in the description of physical phenomena.
Whenever a physical quantity is measured or a physical event is described, this is done with respect to a specified reference frame.
Treating reference frames themselves as physical objects and submitting them to the laws of quantum mechanics, they become quantum reference frames. Recently, there has been an increased interest in analysing spatial and temporal quantum reference frames and in establishing a formalism that allows to switch between different perspectives. In this work we construct a formalism to change between the descriptions assigned by different quantum reference frames for general symmetry groups. The change of perspective is quantum in the sense that the operator we construct allows to take on the perspective of an observer who is in superposition relative to the initial one. We apply the relational formalism developed in this work to the Wigner’s friend thought experiment providing an explicit change of perspective from Wigner to the Friend, arriving at a similar conclusion to that of relational quantum mechanics.

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