Parts and Composites of Quantum Systems

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.