{"paper":{"title":"Completely positive semidefinite rank","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["quant-ph"],"primary_cat":"math.OC","authors_text":"Antonios Varvitsiotis, Anupam Prakash, Jamie Sikora, Zhaohui Wei","submitted_at":"2016-04-25T10:57:58Z","abstract_excerpt":"An $n\\times n$ matrix $X$ is called completely positive semidefinite (cpsd) if there exist $d\\times d$ Hermitian positive semidefinite matrices $\\{P_i\\}_{i=1}^n$ (for some $d\\ge 1$) such that $X_{ij}= {\\rm Tr}(P_iP_j),$ for all $i,j \\in \\{ 1, \\ldots, n \\}$. The cpsd-rank of a cpsd matrix is the smallest $d\\ge 1$ for which such a representation is possible. In this work we initiate the study of the cpsd-rank which we motivate twofold. First, the cpsd-rank is a natural non-commutative analogue of the completely positive rank of a completely positive matrix. Second, we show that the cpsd-rank is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.07199","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"}