Combinations of Quantum Observables and Instruments

TL;DR

This paper explores various ways to combine quantum observables and instruments, analyzing their mathematical properties and physical interpretations within finite-dimensional quantum systems.

Contribution

It systematically studies the mathematical properties of combining quantum observables and instruments, including new insights into special instrument types and their closure properties.

Findings

Analysis of parts of observables and their combinations

Introduction of four special types of instruments and their properties

Results on closure properties of these instruments under combinations

Abstract

This article points out that observables and instruments can be combined in many ways that have natural and physical interpretations. We shall mainly concentrate on the mathematical properties of these combinations. Section~1 reviews the basic definitions and observables are considered in Section~2. We study parts of observables, post-processing, generalized convex combinations, sequential products and tensor products. These combinations are extended to instruments in Section~3. We consider properties of observables measured by combinations of instruments. We introduce four special types of instruments, namely Kraus, L\"uders, trivial and semitrivial instruments. We study when these types are closed under various combinations. In this work, we only consider finite-dimensional quantum systems. A few of the results presented here have appeared in the author's previous articles.