{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T02:05:21Z","timestamp":1760234721878,"version":"build-2065373602"},"reference-count":15,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2021,6,8]],"date-time":"2021-06-08T00:00:00Z","timestamp":1623110400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We consider three types of entities for quantum measurements. In order of generality, these types are observables, instruments and measurement models. If \u03b1 and \u03b2 are entities, we define what it means for \u03b1 to be a part of \u03b2. This relationship is essentially equivalent to \u03b1 being a function of \u03b2 and in this case \u03b2 can be employed to measure \u03b1. We then use the concept to define the coexistence of entities and study its properties. A crucial role is played by a map \u03b1^ which takes an entity of a certain type to one of a lower type. For example, if I is an instrument, then I^ is the unique observable measured by I. 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 L\u00fcders, 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.<\/jats:p>","DOI":"10.3390\/sym13061031","type":"journal-article","created":{"date-parts":[[2021,6,8]],"date-time":"2021-06-08T21:16:58Z","timestamp":1623187018000},"page":"1031","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Parts and Composites of Quantum Systems"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4681-0135","authenticated-orcid":false,"given":"Stanley P.","family":"Gudder","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Denver, Denver, CO 80208, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,6,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Busch, P., Grabowski, M., and Lahti, P. (1995). Operational Quantum Physics, Springer.","DOI":"10.1007\/978-3-540-49239-9"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Heinosaari, T., and Ziman, M. (2012). The Mathematical Language of Quantum Theory, Cambridge University Press.","DOI":"10.1017\/CBO9781139031103"},{"key":"ref_3","unstructured":"Nielson, M., and Chuang, I. (2000). Quantum Computation and Quantum Information, Cambridge University Press."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"062102","DOI":"10.1103\/PhysRevA.97.062102","article-title":"Simulability of observables in general probabilistic theories","volume":"A97","author":"Fillipov","year":"2018","journal-title":"Phys. Rev."},{"key":"ref_5","unstructured":"Kraus, K. (1983). States, Effects and Operations, Springer."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/S0034-4877(02)80007-6","article-title":"Sequential Products on effect algebras","volume":"49","author":"Gudder","year":"2002","journal-title":"Rep. Math. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5212","DOI":"10.1063\/1.1407837","article-title":"Sequential quantum measurements","volume":"42","author":"Gudder","year":"2001","journal-title":"J. Math. Phys."},{"key":"ref_8","unstructured":"Gudder, S. (2020). Conditioned observables in quantum mechanics. arXiv."},{"key":"ref_9","unstructured":"Gudder, S. (2020). Quantum instruments and conditioned observables. arXiv."},{"key":"ref_10","unstructured":"Gudder, S. (2020). Finite quantum instruments. arXiv."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"032109","DOI":"10.1103\/PhysRevA.63.032109","article-title":"Operations, disturbance, and simultaneous measurability","volume":"A63","author":"Ozawa","year":"2001","journal-title":"Phys. Rev."},{"key":"ref_12","first-page":"365302","article-title":"Coexistence of quantum operations","volume":"A42","author":"Heinosaari","year":"2009","journal-title":"J. Phys."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"34","DOI":"10.1007\/s10701-013-9761-1","article-title":"Strongly incompatible quantum devices","volume":"44","author":"Heinosaari","year":"2014","journal-title":"Found. Phys."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"893","DOI":"10.1023\/A:1025406103210","article-title":"Coexistence and joint measurability in quantum mechanics","volume":"42","author":"Lahti","year":"2003","journal-title":"Int. J. Theor. Phys."},{"key":"ref_15","first-page":"322","article-title":"\u00dcber due Zustands\u00e4nderung durch den Messprozess","volume":"6","year":"1951","journal-title":"Ann. Physik"}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/1031\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:12:11Z","timestamp":1760163131000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/13\/6\/1031"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,8]]},"references-count":15,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2021,6]]}},"alternative-id":["sym13061031"],"URL":"https:\/\/doi.org\/10.3390\/sym13061031","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2021,6,8]]}}}