{
  "annotation_id": "9f95cd1f-a485-4507-97ff-030082bc053a",
  "date_created": "2026-03-02T18:01:38.614000Z",
  "date_modified": "2026-03-02T18:01:38.614000Z",
  "file_hash": "1537979da127e8852f6ca447c98d69e4b3f3cd15329872999c03c451e8f5476f",
  "private": false,
  "record": {
    "abstract": "We study the rank of a general tensor $u$ in a tensor product $H_1\\ot...\\ot\nH_k$. The rank of $u$ is the minimal number $p$ of pure states $v_1,...,v_p$\nsuch that $u$ is a linear combination of the $v_j$\u0027s. This rank is an algebraic\nmeasure of the degree of entanglement of $u$. Motivated by quantum computation,\nwe completely describe the rank of an arbitrary tensor in $(\\C^2)^{\\ot 3}$ and\ngive normal forms for tensor states up to local unitary transformations. We\nalso obtain partial results for $(\\C^2)^{\\ot 4}$; in particular, we show that\nthe maximal rank of a tensor in $(\\C^2)^{\\ot 4}$ is equal to 4.",
    "arxiv_id": "quant-ph/0008031",
    "authors": [
      "Jean-Luc Brylinski"
    ],
    "categories": [
      "quant-ph"
    ],
    "title": "Algebraic measures of entanglement",
    "url": "https://arxiv.org/abs/quant-ph/0008031"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "7e6778f7-f529-407a-9161-ff7f592f79aa",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}