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Relativization is naturally functorial

Jan Głowacki

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.

Relativization is naturally functorial

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.
Paper Structure (7 sections, 1 theorem, 46 equations)

This paper contains 7 sections, 1 theorem, 46 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basics
  4. Quantum reference frames
  5. Semi-quantum systems and frames
  6. Relativization functors
  7. Summary

Key Result

Theorem 4.1

For any morphism of principal semi-quantum reference frames $\psi: \mathcal{R} \to {\mathcal{R}'}$ and a semi-quantum channel $\phi: V_\mathcal{S} \to V_{\mathcal{S}'}$ the map $\yen(\psi,\phi)$, given by the ultraweakly continuous extension of from the dense subset $\yen^\mathcal{R}(V_\mathcal{S}) \subseteq V_\mathcal{S}^\mathcal{R}$ is a well-defined channel between invariant semi-quantum syste

Theorems & Definitions (2)

  • Theorem 4.1
  • proof