Internal quantum reference frames for finite Abelian groups

Employing internal quantum systems as reference frames is a crucial concept in gravity, gauge theories and foundations whenever external relata are unavailable. In this work, we give comprehensive self-contained treatment of such (QRFs) for the case when underlying configuration space finite Abelian group, significantly extending our previous work (Quantum 5, 530 (2021)). The simplicity setup admits fully rigorous information-theoretic analysis, while maintaining sufficient structure exploring many conceptual structural questions also pertinent to more complicated setups. We exploit derive several important structures constraint quantization with methods reveal relation between different approaches QRF covariance. particular, characterize "physical Hilbert space" -- arena "perspective-neutral" approach maximal subspace that frame-independent descriptions purifications states. then demonstrate kinematical equivalence and, surprising, dynamical inequivalence "alignability" QRFs. While former unitaries generating transitions arbitrary subsystem relations, latter, remarkably, no dynamics requiring symmetry-preservation. illustrate these findings by example interacting discrete particles, including how can be described "relative one subsystems".