Quantum Reference Frames on Homogeneous Spaces

TL;DR

This paper develops a mathematical framework for quantum reference frames on homogeneous spaces using operator-valued integration, extending existing results to more general settings and characterizing their properties.

Contribution

It introduces relativization maps for quantum reference frames on arbitrary homogeneous spaces, generalizing previous models and analyzing their properties as quantum channels.

Findings

Relativization maps are contracting quantum channels.

Maps are injective for localizable frames.

Maps are multiplicative for sharp frames (PVMs).

Abstract

This paper initiates a systematic study of operators arising as integrals of operator-valued functions with respect to positive operator-valued measures and utilizes these tools to provide relativization maps (Yen) for quantum reference frames (QRFs) defined on general homogeneous spaces. Properties of operator-valued integration are first studied and then employed to define general relativization maps and show their properties. The relativization maps presented here are defined for QRFs (systems of covariance) based on arbitrary homogeneous spaces of locally compact second countable topological groups and are shown to be contracting quantum channels, injective for localizable (norm-1 property) frames and multiplicative for the sharp ones (PVMs), extending the existing results.