| Summary: | We consider three types of entities for quantum measurements. In order of generality, these types are observables, instruments and measurement models. If <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> and <inline-formula><math display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> are entities, we define what it means for <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> to be a part of <inline-formula><math display="inline"><semantics><mi>β</mi></semantics></math></inline-formula>. This relationship is essentially equivalent to <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> being a function of <inline-formula><math display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> and in this case <inline-formula><math display="inline"><semantics><mi>β</mi></semantics></math></inline-formula> can be employed to measure <inline-formula><math display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>. We then use the concept to define the coexistence of entities and study its properties. A crucial role is played by a map <inline-formula><math display="inline"><semantics><mover accent="true"><mi>α</mi><mo>^</mo></mover></semantics></math></inline-formula> which takes an entity of a certain type to one of a lower type. For example, if <inline-formula><math display="inline"><semantics><mi mathvariant="script">I</mi></semantics></math></inline-formula> is an instrument, then <inline-formula><math display="inline"><semantics><mover accent="true"><mi mathvariant="script">I</mi><mo>^</mo></mover></semantics></math></inline-formula> is the unique observable measured by <inline-formula><math display="inline"><semantics><mi mathvariant="script">I</mi></semantics></math></inline-formula>. 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 in types to their local components are discussed. We also consider sequential products of measurements. Specific examples of Lüders, Kraus and trivial instruments are used to illustrate various concepts. We only consider finite-dimensional systems in this article. Finally, we mention the role of symmetry representations for groups using quantum channels.
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