{"id":1026,"date":"2015-05-05T14:59:07","date_gmt":"2015-05-05T12:59:07","guid":{"rendered":"https:\/\/webs.uab.cat\/giq\/seminar\/positivity-linear-maps-under-tensor-powers\/"},"modified":"2015-05-05T14:59:07","modified_gmt":"2015-05-05T12:59:07","slug":"positivity-linear-maps-under-tensor-powers","status":"publish","type":"seminar","link":"https:\/\/webs.uab.cat\/giq\/seminar\/positivity-linear-maps-under-tensor-powers\/","title":{"rendered":"Positivity of linear maps under tensor powers"},"content":{"rendered":"<p><span>Based on&nbsp;arxiv:1502.05630<\/span><\/p>\n<p>We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with $n$ copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every $n\\in\\mathbb{N}$ there exist non-trivial maps with this property and that for two-dimensional Hilbert spaces there is no non-trivial map for which this holds for all $n$. For higher dimensions we reduce the existence question of such non-trivial &#8220;tensor-stable positive maps&#8221; to a one-parameter family of maps and show that an affirmative answer would imply the existence of NPPT bound entanglement.&nbsp;<\/p>\n<p><span>As an application we show that any tensor-stable positive map that is not completely positive yields an upper bound on the quantum channel capacity, which for the transposition map gives the well-known cb-norm bound. We furthermore show that the latter is an upper bound even for the LOCC-assisted quantum capacity, and that moreover it is a strong converse rate for this task.<\/span><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Based on&nbsp;arxiv:1502.05630 We investigate linear maps between matrix algebras that remain positive under tensor powers, i.e., under tensoring with $n$ copies of themselves. Completely positive and completely co-positive maps are trivial examples of this kind. We show that for every $n\\in\\mathbb{N}$ there exist non-trivial maps with this property and that for two-dimensional Hilbert spaces there [&hellip;]<\/p>\n","protected":false},"author":20,"featured_media":0,"template":"","class_list":["post-1026","seminar","type-seminar","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/seminar\/1026","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/seminar"}],"about":[{"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/types\/seminar"}],"author":[{"embeddable":true,"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/users\/20"}],"wp:attachment":[{"href":"https:\/\/webs.uab.cat\/giq\/wp-json\/wp\/v2\/media?parent=1026"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}