Transformation of Spin in Quantum Reference Frames

Abstract

In physical experiments, reference frames are standardly modelled through a specific choice of coordinates used to describe the physical systems, but they themselves are not considered as such. However, any reference frame is a physical system that ultimately behaves according to quantum mechanics. We develop a framework for rotational (i.e. spin) quantum reference frames, with respect to which quantum systems with spin degrees of freedom are described. We give an explicit model for such frames as systems composed of three spin coherent states of angular momentum j and introduce the transformations between them by upgrading the Euler angles occurring in classical SO(3) spin transformations to quantum mechanical operators acting on the states of the reference frames. To ensure that an arbitrary rotation can be applied on the spin we take the limit of infinitely large j, in which case the angle operator possesses a continuous spectrum. We prove that rotationally invariant Hamiltonian of the Heisenberg model is invariant under a larger group of quantum reference frame transformations. Finally, the application to physical examples shows that superposition and entanglement of spin states are frame-dependent notions.

Perspective-neutral approach to quantum frame covariance for general symmetry groups

“…It established reversible changes of quantum reference frame for ideal frames only and found ambiguous transformations for non-trivial isotropy groups, the latter of which were restricted to be normal subgroups. This is consistent with the unitary frame changes for effectively ideal spin frames proposed in [72] and for an Abelian subgroup of the Lorentz group exhibited in [65,66], both formulated within a purely perspective-dependent framework. The perspective-neutral approach, on the other hand, was so far restricted to Abelian groups (incl.…”

“…• We elucidate why the perspective-neutral approach to quantum frame covariance, further developed here, is not in general equivalent to the purely perspective-dependent approaches [48,[65][66][67][68]71,72]. The latter start with a fixed subsystem Hilbert space in a given internal quantum frame perspective that is not in general compatible with gauge-invariance under the symmetry group.…”

“…G acts regularly on itself and the frame orientations {|ψ((g) Ri } are perfectly distinguishable. 19 This also applies to the case of spin quantum reference frames proposed in [72], which effectively constitute an ideal frame (owing to j → ∞). By contrast, in the present work, the quantum coordinate changes are unitary regardless of whether the frames are ideal, complete or even incomplete.…”

“…The changes of quantum reference frames introduced in [48,65,67,68,72] did not make use of a perspectiveneutral structure (in particular, did not invoke a gauge symmetry principle), but rather mapped directly between what are here reduced descriptions, i.e. frame perspectives.…”

“…Let us elucidate, why the purely perspective-dependent approaches [48,65,67,68,72] are nonetheless not in general equivalent to the perspective-neutral approach [53-60, 62, 63] significantly generalized here:…”