Compositionality in quantum reference frame perspectives

This paper addresses the challenge of compositionality in quantum reference frames (QRFs). In standard quantum theory, systems are composed via the tensor product, which allows for a straightforward description of subsystems. However, when reference frames are themselves quantum systems, this structure becomes problematic, leading to the 'paradox of the third particle'—an ambiguity where the description of a subsystem depends on whether it is considered part of a larger, extended setup. The authors utilize a recent formalism for QRFs to show how to consistently add and remove subsystems within a QRF perspective without encountering such paradoxes.

The authors show that the 'extra-particle' degrees of freedom—a subsystem that emerges naturally in the QRF formalism—are essential for maintaining consistency. By internalizing external observers, they establish a hierarchy of QRF perspectives where all observers agree on the state and operator representations, regardless of their position in the hierarchy. A key result is that composing subsystems in a QRF perspective is more restrictive than in standard quantum theory; not every state can be appended in a simple tensor-product form. The authors characterize the set of compatible states and introduce a 'classicalisation' procedure that recovers a classical-like perspective, allowing for an unrestricted adding rule at the cost of a different operational interpretation.

This work provides a robust framework for understanding how physical systems are described relative to quantum observers. By resolving the paradox of the third particle, the authors demonstrate that QRF transformations can be both reversible and compositional. This is a significant step toward a fully relational quantum theory, where the description of the world is not dependent on an external, absolute reference frame, but is instead built from the relative perspectives of quantum systems.