{
  "annotation_id": "8d28feba-33a3-4200-a3e7-30482cdb518c",
  "date_created": "2026-03-02T18:02:06.215000Z",
  "date_modified": "2026-03-02T18:02:06.215000Z",
  "file_hash": "ce46b85ff3a9e23bef8a0d83e64a440acff61a0cab991e29ece4d6f310be08bd",
  "private": false,
  "record": {
    "abstract": "In this paper we study the problem of calculating the convex hull of certain\naffine algebraic varieties. As we explain, the motivation for considering this\nproblem is that certain pure-state measures of quantum entanglement, which we\ncall polynomial entanglement measures, can be represented as affine algebraic\nvarieties. We consider the evaluation of certain mixed-state extensions of\nthese polynomial entanglement measures, namely convex and concave roofs. We\nshow that the evaluation of a roof-based mixed-state extension is equivalent to\ncalculating a hyperplane which is multiply tangent to the variety in a number\nof places equal to the number of terms in an optimal decomposition for the\nmeasure. In this way we provide an implicit representation of optimal\ndecompositions for mixed-state entanglement measures based on the roof\nconstruction.",
    "arxiv_id": "quant-ph/0402055",
    "authors": [
      "Tobias J. Osborne"
    ],
    "categories": [
      "quant-ph"
    ],
    "journal_ref": "Quantum Information and Computation, vol. 7 (3), pp.209-227 (2007)",
    "title": "Convex Hulls of Varieties and Entanglement Measures Based on the Roof   Construction",
    "url": "https://arxiv.org/abs/quant-ph/0402055"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "977e1271-fe8a-4c32-8ec4-1acec075a86b",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}