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Such quantities are present on all levels of quantum objects: states, measurements, quantum channels, and even higher order dynamics. In this manuscript, we show that two seemingly different aspects of quantum incompatibility: the quantum marginal problem of states and the incompatibility on the level of quantum channels are in many-to-one correspondence. Importantly, as incompatibility of measurements is a special case of the latter, it also forms an instance of the quantum marginal problem. The generality of the connection is harnessed by solving the marginal problem for Gaussian and Bell diagonal states, as well as for pure states under depolarizing noise. Furthermore, we derive entropic criteria for channel compatibility, and develop a converging hierarchy of semi-definite programs for quantifying the strength of quantum memories.<\/jats:p>","DOI":"10.22331\/q-2021-06-15-476","type":"journal-article","created":{"date-parts":[[2021,6,15]],"date-time":"2021-06-15T15:18:19Z","timestamp":1623770299000},"page":"476","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":22,"title":["Quantum marginal problem and incompatibility"],"prefix":"10.22331","volume":"5","author":[{"given":"Erkka","family":"Haapasalo","sequence":"first","affiliation":[{"name":"Centre for Quantum Technologies, National University of Singapore, Science Drive 2, Block S15-03-18, Singapore 117543"},{"name":"Department of Physics and Center for Field Theory and Particle Physics, Fudan University, Shanghai 200433, China"}]},{"given":"Tristan","family":"Kraft","sequence":"additional","affiliation":[{"name":"Naturwissenschaftlich-Technische Fakult\u00e4t, Universit\u00e4t Siegen, Walter-Flex-Str. 3, D-57068 Siegen, Germany"}]},{"given":"Nikolai","family":"Miklin","sequence":"additional","affiliation":[{"name":"Institute of Theoretical Physics and Astrophysics, National Quantum Information Center, Faculty of Mathematics, Physics and Informatics, University of Gdansk, 80-952 Gda\u0144sk, Poland"}]},{"given":"Roope","family":"Uola","sequence":"additional","affiliation":[{"name":"D\u00e9partement de Physique Appliqu\u00e9e, Universit\u00e9 de Gen\u00e8ve, CH-1211 Gen\u00e8ve, Switzerland"}]}],"member":"9598","published-online":{"date-parts":[[2021,6,15]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"Teiko Heinosaari, Takayuki Miyadera, and M\u00e1rio Ziman. 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