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  "slug": "concentratable-entanglement-spectator-counterexample",
  "title": "Concentratable entanglement: counterexamples and sharp repairs",
  "shortTitle": "When an entanglement bound undercounts",
  "url": "https://evidencepress.org/releases/concentratable-entanglement-spectator-counterexample/",
  "oneLine": "A mixed extra qubit exposes a failure in a proposed entanglement upper bound; exact examples and sharp repairs identify what the statistic can guarantee.",
  "abstract": "This candidate disproves a proposed universal upper bound on mixed-state total concentratable entanglement. A Bell pair with a maximally mixed spectator has both ensemble roofs 1/4, whereas the statistic is 3/16. The analysis extends to full-rank globally depolarised states and genuinely tripartite-entangled states, gives an operational failure certificate, establishes the minimal three-qubit convex-roof counterexample size, and determines sharp impurity corrections uniformly in qubit number and at fixed size for the concave roof.",
  "datePublished": "2026-10-13",
  "dateModified": "2026-10-13",
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  "doi": "10.5281/zenodo.23336463",
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  "authors": [
    "Anonymous"
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  "status": "unrefereed-candidate",
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      "state": "passed",
      "note": "Rational, alternative Pauli and symbolic checks passed normally and with Python optimization; seven semantic corruptions rejected. Universal claims also require the written proofs.",
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    "name": "A proposed upper bound on mixed-state total concentratable entanglement",
    "url": "https://arxiv.org/abs/2112.04333v8"
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  "keywords": [
    "concentratable entanglement",
    "convex roof",
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    "The Bell-pair spectator state has both roofs 1/4 but proposed upper statistic 3/16; three qubits are minimal for convex-roof failure.",
    "Full-rank globally depolarised and genuinely tripartite-entangled counterexamples show failure beyond the original product example.",
    "The optimal qubit-number-independent impurity coefficient is 1/2 for both roofs; at fixed n the concave-roof coefficient is exactly 1/2−2^(−n).",
    "A Bell-fidelity lower bound can certify failure using measured probabilities without assuming a product state."
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    {
      "citation": "Foulds, Prove and Kendon (2024), Generalizing multipartite concentratable entanglement for practical applications: mixed, qudit and optical states. Source conjecture; p. 15 also discusses possible underestimation.",
      "url": "https://arxiv.org/abs/2112.04333v8",
      "doi": "10.48550/arXiv.2112.04333"
    },
    {
      "citation": "Beckey, Gigena, Coles and Cerezo (2021), Computable and operationally meaningful multipartite entanglement measures. Pure-state CE.",
      "url": "https://doi.org/10.1103/PhysRevLett.127.140501",
      "doi": "10.1103/PhysRevLett.127.140501"
    },
    {
      "citation": "Beckey, Pelegrí, Foulds and Pearson (2023), Multipartite entanglement measures via Bell-basis measurements. Mixed-state convex roof.",
      "url": "https://doi.org/10.1103/PhysRevA.107.062425",
      "doi": "10.1103/PhysRevA.107.062425"
    },
    {
      "citation": "Schatzki, Liu, Cerezo and Chitambar (2024), Hierarchy of multipartite correlations based on concentratable entanglement. Product structure and supplement XI.",
      "url": "https://doi.org/10.1103/PhysRevResearch.6.023019",
      "doi": "10.1103/PhysRevResearch.6.023019"
    },
    {
      "citation": "Vairogs, Hermes and Leditzky (2026), Localizing multipartite entanglement with local and global measurements. CE assistance and its concavity bound.",
      "url": "https://doi.org/10.22331/q-2026-02-23-2007",
      "doi": "10.22331/q-2026-02-23-2007"
    },
    {
      "citation": "Wootters (1998), Entanglement of Formation of an Arbitrary State of Two Qubits. Concurrence formula and equal-concurrence decomposition.",
      "url": "https://doi.org/10.1103/PhysRevLett.80.2245",
      "doi": "10.1103/PhysRevLett.80.2245"
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