Internal quantum reference frames for finite Abelian groups

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".

22 pages, 1 figure. V2: close to published version

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• Inference of gravitational field superposition from quantum measurements
• Watching the Clocks: Interpreting the Page-Wootters Formalism and the Internal Quantum Reference Frame Programme
• Quantum Reference Frames for Lorentz Symmetry
• Quantum generalisation of Einstein's Equivalence Principle can be verified with entangled clocks as quantum reference frames
• Robustness of the Page-Wootters construction across different pictures, states of the universe and system-clock interactions
• Crossed products and quantum reference frames: on the observer-dependence of gravitational entropy
• The sum of entanglement and subsystem coherence is invariant under quantum reference frame transformations
• Minimal Areas from Entangled Matrices
• Long-time Quantum Scrambling and Generalized Tensor Product Structures
• Searching for a physical description relative to a quantum system
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• What can we do in a symmetry-constrained perspective? The importance of the total charge's status in quantum reference frame frameworks
• Gravitational null rays: Covariant Quantization and the Dressing Time
• Energy Cost of Localization of Relational Quantum Information