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By identifying an appropriate choice of resource R and target subsystem T, our problems reduce, respectively, to the well-known\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                        <mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">marginal problems<\/mml:mtext>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    for quantum states and the problem of determining if a given quantum system is a resource. More generally, we say that a set of marginal states is\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                        <mml:mtext class=\"MJX-tex-mathit\" mathvariant=\"italic\">resource-free incompatible<\/mml:mtext>\n                      <\/mml:mrow>\n                    <\/mml:math>\n                    with a target subsystem T if all global states compatible with this set must result in a resourceful state in T of type R. We show that this incompatibility\n                    <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                      <mml:mi>i<\/mml:mi>\n                      <mml:mi>n<\/mml:mi>\n                      <mml:mi>d<\/mml:mi>\n                      <mml:mi>u<\/mml:mi>\n                      <mml:mi>c<\/mml:mi>\n                      <mml:mi>e<\/mml:mi>\n                      <mml:mi>s<\/mml:mi>\n                    <\/mml:math>\n                    a resource theory that can be quantified by a monotone and obtain necessary and sufficient conditions for this monotone to be computable as a conic program with finite optimum. We further show, via the corresponding witnesses, that (1) resource-free incompatibility is equivalent to an operational advantage in some channel-discrimination tasks, and (2) some specific cases of such tasks fully characterize the convertibility between marginal density matrices exhibiting resource-free incompatibility. Through our framework, one sees a clear connection between any marginal problem \u2013 which implicitly involves some notion of incompatibility \u2013 for quantum states and a resource theory for quantum states. We also establish a close connection between the physical relevance of resource marginal problems and the ground state properties of certain many-body Hamiltonians. In terms of application, the universality of our framework leads, for example, to a further quantitative understanding of the incompatibility associated with the recently-proposed entanglement marginal problems and entanglement transitivity problems.\n                  <\/jats:p>","DOI":"10.22331\/q-2024-05-22-1353","type":"journal-article","created":{"date-parts":[[2024,5,22]],"date-time":"2024-05-22T10:04:03Z","timestamp":1716372243000},"page":"1353","update-policy":"https:\/\/doi.org\/10.22331\/q-crossmark-policy-page","source":"Crossref","is-referenced-by-count":11,"title":["Resource Marginal Problems"],"prefix":"10.22331","volume":"8","author":[{"given":"Chung-Yun","family":"Hsieh","sequence":"first","affiliation":[{"name":"H. H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol, BS8 1TL, UK"},{"name":"ICFO - Institut de Ci\u00e8ncies Fot\u00f2niques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Gelo Noel M.","family":"Tabia","sequence":"additional","affiliation":[{"name":"Foxconn Quantum Computing Research Center, Taipei 114, Taiwan"},{"name":"Department of Physics and Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan"},{"name":"Physics Division, National Center for Theoretical Sciences, Taipei 106319, Taiwan"},{"name":"Center for Quantum Technology, National Tsing Hua University, Hsinchu 300, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yu-Chun","family":"Yin","sequence":"additional","affiliation":[{"name":"Institute of Communications Engineering, National Yang Ming Chiao Tung University, Hsinchu 300093, Taiwan"},{"name":"Department of Physics and Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yeong-Cherng","family":"Liang","sequence":"additional","affiliation":[{"name":"Department of Physics and Center for Quantum Frontiers of Research & Technology (QFort), National Cheng Kung University, Tainan 701, Taiwan"},{"name":"Physics Division, National Center for Theoretical Sciences, Taipei 106319, Taiwan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"9598","published-online":{"date-parts":[[2024,5,22]]},"reference":[{"key":"0","doi-asserted-by":"publisher","unstructured":"A. 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