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First, we show that separability in the case of DS in<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:msup><mml:mi>C<\/mml:mi><mml:mi>d<\/mml:mi><\/mml:msup><mml:mo>\u2297<\/mml:mo><mml:msup><mml:mi>C<\/mml:mi><mml:mi>d<\/mml:mi><\/mml:msup><\/mml:math>(symmetric qudits) can be reformulated as a quadratic conic optimization problem. This connection allows us to exchange concepts and ideas between quantum information and this field of mathematics. For instance, copositive matrices can be understood as indecomposable entanglement witnesses for DS states. As a consequence, we show that positivity of the partial transposition (PPT) is sufficient and necessary for separability of DS states for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><mml:mo>\u2264<\/mml:mo><mml:mn>4<\/mml:mn><\/mml:math>. Furthermore, for<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>d<\/mml:mi><mml:mo>\u2265<\/mml:mo><mml:mn>5<\/mml:mn><\/mml:math>, we provide analytic examples of PPT-entangled states. Second, we develop new sufficient separability conditions beyond the PPT criterion for bipartite DS states. Finally, we focus on<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>N<\/mml:mi><\/mml:math>-partite DS qubits, where PPT is known to be necessary and sufficient for separability. In this case, we present a family of almost DS states that are PPT with respect to each partition but nevertheless entangled.<\/jats:p>","DOI":"10.22331\/q-2018-01-12-45","type":"journal-article","created":{"date-parts":[[2018,1,12]],"date-time":"2018-01-12T21:41:28Z","timestamp":1515793288000},"page":"45","source":"Crossref","is-referenced-by-count":44,"title":["Separability of diagonal symmetric states: a quadratic conic optimization problem"],"prefix":"10.22331","volume":"2","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-6123-1422","authenticated-orcid":false,"given":"Jordi","family":"Tura","sequence":"first","affiliation":[{"name":"ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain"},{"name":"Max-Planck-Institut f\u00fcr Quantenoptik, Hans-Kopfermann-Stra\u00dfe 1, 85748 Garching, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1401-0184","authenticated-orcid":false,"given":"Albert","family":"Aloy","sequence":"additional","affiliation":[{"name":"ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1745-678X","authenticated-orcid":false,"given":"Ruben","family":"Quesada","sequence":"additional","affiliation":[{"name":"Departament de F\u00edsica, Universitat Aut\u00f2noma de Barcelona, E-08193 Bellaterra, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0210-7800","authenticated-orcid":false,"given":"Maciej","family":"Lewenstein","sequence":"additional","affiliation":[{"name":"ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels (Barcelona), Spain"},{"name":"ICREA, Pg. 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