{
  "annotation_id": "317be135-af69-4a82-b3c6-826b5bd85a7c",
  "date_created": "2026-03-02T18:02:47.727000Z",
  "date_modified": "2026-03-02T18:02:47.727000Z",
  "file_hash": "728c39d1869b126284e550e9af68ef7ee86f473f8a8b6ac21ecc9dfa8531767d",
  "private": false,
  "record": {
    "abstract": "The optimal (pure state) ensemble length of a separable state, A, is the\nminimum number of (pure) product states needed in convex combination to\nconstruct A. We study the set of all separable states with optimal (pure state)\nensemble length equal to k or fewer. Lower bounds on k are found below which\nthese sets have measure 0 in the set of separable states. In the bipartite case\nand the multiparticle case where one of the particles has significantly more\nquantum numbers than the rest, the lower bound for non-pure state ensembles is\nsharp. A consequence of our results is that for all two particle systems,\nexcept possibly those with a qubit or those with a nine dimensional Hilbert\nspace, and for all systems with more than two particles the optimal pure state\nensemble length for a randomly picked separable state is with probability 1\ngreater than the state\u0027s rank. In bipartite systems with probability 1 it is\ngreater than 1/4 the rank raised to the 3/2 power and in a system with p qubits\nwith probability 1 it is greater than (2^2p)/(1+2p), which is almost the square\nof the rank.",
    "arxiv_id": "quant-ph/9908050",
    "authors": [
      "Robert B. Lockhart"
    ],
    "categories": [
      "quant-ph"
    ],
    "doi": "10.1063/1.1290055",
    "title": "Optimal Ensemble Length of Mixed Separable States",
    "url": "https://arxiv.org/abs/quant-ph/9908050"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "f21e6eaf-1cbd-474c-8f51-b29c8c353f1a",
    "id": "arXiv Dataset IDs",
    "type": "Model",
    "variant": "snapshot-2026-03-01",
    "version": "0.1.0"
  },
  "user_id": 1000002
}