Relativization is naturally functorial

TL;DR

This paper explores the categorical structure of relativization in quantum measurement theory, framing it as a functorial construction that enhances understanding of quantum reference frames and their transformations.

Contribution

It introduces a functorial perspective on relativization maps in quantum systems, extending the framework to subspace models and defining a new type of frame transformation called external.

Findings

Relativization maps can be viewed as natural transformations.

The construction provides a deeper structural understanding of quantum reference frames.

Potential applications to algebraic quantum field theories.

Abstract

In this note, we provide some categorical perspectives on the relativization construction arising from quantum measurement theory in the presence of symmetries and occupying a central place in the operational approach to quantum reference frames. This construction provides, for any quantum system, a quantum channel from the system's algebra to the invariant algebra on the composite system also encompassing the chosen reference, contingent upon a choice of the pointer observable. These maps are understood as relativizing observables on systems upon the specification of a quantum reference frame. We begin by extending the construction to systems modelled on subspaces of algebras of operators to then define a functor taking a pair consisting of a reference frame and a system and assigning to them a subspace of relative operators defined in terms of an image of the corresponding relativization map. When a single frame and equivariant channels are considered, the relativization maps can be understood as a natural transformation. Upon fixing a system, the functor provides a novel kind of frame transformation that we call external. Results achieved provide a deeper structural understanding of the framework of interest and point towards its categorification and potential application to local systems of algebraic quantum field theories.