Internal quantum reference frames for finite Abelian groups

TL;DR

The paper develops a rigorous framework for internal quantum reference frames with finite Abelian symmetry, formulating $G$-systems and distinguishing invariant and relational observables. It proves a kinematic equivalence between relational (perspective-neutral) and alignable descriptions via explicit reduction maps and the relational trace, while revealing a dynamical inequivalence: relational states support nontrivial symmetry-preserving dynamics across subsystem relations, whereas alignable states are severely constrained and cannot generate superpositions of relations. The perspective-neutral approach thus provides a coherent link between Page-Wootters-type, relational observables, and quantum-coordinate transformations, with implications for constraint quantization and gauge-like structures in quantum gravity and lattice gauge theories. Overall, the work clarifies when and how internal QRFs can yield frame-independent physics and how different QRF frameworks relate under time evolution and composition.

Abstract

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 (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 subsystems".

Figures (1)

• Figure 1: As in Ref. QRF1, we consider a finite Abelian group $\mathcal{G}$ (in this picture $\mathcal{G}=\mathbb{Z}_n$) that plays a double role: first, as a classical configuration space that serves as a distinguished basis of a one-particle Hilbert space $\mathcal{H}=\mathbb{C}(\mathcal{G})$: second, as a group of "translations" that acts on this space. We consider $N$ distinguishable particles in some quantum state $\rho_S\in\mathcal{L}(\mathcal{H}^{\otimes N})$, potentially entangled with an unspecified purifying ancillary system $A$. Based on simple postulates, we study the resulting quantum symmetries and properties of such "$\mathcal{G}$-systems" $S$.

Theorems & Definitions (52)

• Definition 1: $\mathcal{G}$-system
• Lemma 2: Ref. QRF1, Lemma 5
• Example 3
• Definition 4: Ref. QRF1, Definition 7
• Example 5: Cyclic group; Ref. QRF1, Example 8
• Lemma 6: Ref. QRF1, Lemma 10
• Lemma 7
• proof
• Definition 8: Ref. QRF1 Definition 13
• Definition 9: Alignable states and observables