Parts and Composites of Quantum Systems

TL;DR

This paper explores the relationships and structures of quantum measurement entities, including observables, instruments, and measurement models, focusing on parts, composites, and sequential products in finite-dimensional quantum systems.

Contribution

It introduces a formal framework for parts and composites of quantum measurement entities, including the concept of coexistence and the use of a map to relate different entity types.

Findings

Defined the part relationship for quantum measurement entities

Analyzed the properties of composite systems and local reductions

Illustrated concepts with examples like L"uders and Kraus instruments

Abstract

We consider three types of entities for quantum measurements. In order of generality, these types are: observables, instruments and measurement models. If $\alpha$ and $\beta$ are entities, we define what it means for $\alpha$ to be a part of $\beta$. This relationship is essentially equivalent to $\alpha$ being a function of $\beta$ and in this case $\beta$ can be employed to measure $\alpha$. We then use the concept to define coexistence of entities and study its properties. A crucial role is played by a map $\alphahat$ which takes an entity of a certain type to one of lower type. For example, if $\iscript$ is an instrument, then $\iscripthat$ is the unique observable measured by $\iscript$. Composite systems are discussed next. These are constructed by taking the tensor product of the Hilbert spaces of the systems being combined. Composites of the three types of measurements and their parts are studied. Reductions of types to their local components are discussed. We also consider sequential products of measurements. Specific examples of L\"uders, Kraus and trivial instruments are used to illustrate various concepts. We only consider finite-dimensional systems in this article.