{"paper":{"title":"The set of separable states has no finite semidefinite representation except in dimension $3\\times 2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.OC"],"primary_cat":"quant-ph","authors_text":"Hamza Fawzi","submitted_at":"2019-05-04T20:10:49Z","abstract_excerpt":"Given integers n $\\geq$ m, let Sep(n,m) be the set of separable states on the Hilbert space $\\mathbb{C}^n \\otimes \\mathbb{C}^m$. It is well-known that for (n,m)=(3,2) the set of separable states has a simple description using semidefinite programming: it is given by the set of states that have a positive partial transpose. In this paper we show that for larger values of n and m the set Sep(n,m) has no semidefinite programming description of finite size. As Sep(n,m) is a semialgebraic set this provides a new counterexample to the Helton-Nie conjecture, which was recently disproved by Scheiderer"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1905.02575","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}