Employing internal quantum systems as reference frames is a crucial concept in quantum gravity, gauge theories, and quantum foundations whenever external relata are unavailable. In this work, we give a comprehensive and self-contained treatment of such quantum reference frames (QRFs) for the case when the underlying configuration space is a finite Abelian group, significantly extending our previous work [M. Krumm, P. A. Höhn, and M. P. Müller, Quantum 5, 530 (2021)]. The simplicity of this setup admits a fully rigorous quantum information–theoretic analysis, while maintaining sufficient structure for exploring many of the conceptual and structural questions also pertinent to more complicated setups. We exploit this to derive several important structures of constraint quantization with quantum information–theoretic methods and to reveal the relation between different approaches to QRF covariance. In particular, we characterize the “physical Hilbert space”—the arena of the “perspective-neutral” approach—as the maximal subspace that admits frame-independent descriptions of purifications of states. We then demonstrate the kinematical equivalence and, surprising, dynamical inequivalence of the “perspective-neutral” and the “alignability” approach to QRFs. While the former admits unitaries generating transitions between arbitrary subsystem relations, the latter, remarkably, admits no such dynamics when requiring symmetry-preservation. We illustrate these findings by example of interacting discrete particles, including how dynamics can be described “relative to one of the subystems.”
Note that the operators generate the same subalgebra for every i, i.e., the definition of is independent of the choice of particle that is used as a reference. This is because Πinv(U•U†) = Πinv(•)for all .
Equivalence between the quantum coordinate transformations of the perspective-neutral approach and the QRF transformations of Ref. 37 has previously been shown for the continuum translation group in Refs. 38 and 40 , while equivalence between the QRF transformations of Refs. 37 and 39 for the same group was demonstrated in Ref. 39 . Here, we reveal the explicit equivalence between the transformations in the perspective-neutral approach and the ones in Ref. 39 for finite groups. Equivalence of the QRF transformations for general groups of Ref. 39 with those of the perspective-neutral approach will be demonstrated in Ref. 70.
In continuous systems, relational Dirac observables are typically defined also “off-shell” of the constraint surface, i.e., as elements of rather than , which means as incoherently rather than coherently group-averaged observables in contrast to here, e.g., see Refs. 43–45, 50–52, 70, and 82 within the context of constraint quantization and Refs. 18 and 32–34 within the context of quantum information theory and quantum foundations. In constraint quantization, one is usually interested in their actions on , and since the two group-averaging procedures agree on this space, 63 the choice of definition then does not make a difference. Here, for notational simplicity, we restrict their definition ab initio to as some of their algebraic properties only hold when acting on . This permits us to write all their algebraic relations without additional restrictions.
It is important to note that this entanglement refers to the tensor product structure of the kinematical Hilbert space which does not inherit. The entanglement we refer to here is thus only discernible relative to an external observer with access to an external frame, however, is not internally detectable with observables in ; see Refs. 43 and 45for further discussion.
The choice of sign differs from the usual convention in the continuous case. This will not change the physics, but it will slightly simplify the notation. The standard convention can be reproduced by setting .
In principle, it would be sufficient to demand symmetry-equivalence except of identity, but we can always absorb the corresponding symmetry into the definition of W.
For the reader familiar with gauge theories, note that there is a difference in nomenclature here. What we call symmetries would in gauge theories be called field-dependent gauge transformations, while the transformations changing the relation between the frame and the remaining degrees of freedom are called “symmetries”.73,80 For consistency with our quantum information based nomenclature in Ref. 63, we call these concepts differently here.