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    "story_id": "mp-2026-09-14-014",
    "source_story_id": "tmp-story-hypercube-log-depth-entanglement",
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    "headline": "The Hypercube Went From Empty to Nearly Maximally Entangled",
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    "dek": "Below log₂ n depth an extensive subsystem stayed unentangled; at logarithmic depth every subsystem approached maximum entanglement within a constant factor.",
    "summary": "Below log₂ n depth an extensive subsystem stayed unentangled; at logarithmic depth every subsystem approached maximum entanglement within a constant factor.",
    "body_text": "A theoretical analysis of random hypercube linear-optical networks starts with all modes squeezed and tracks how subsystem entanglement develops with circuit depth. Below log₂ n, the authors prove that an extensive subsystem has no entanglement. At depth proportional to log n, ensemble-averaged entanglement in every subsystem comes within a constant factor of its maximum. The transition parallels recent arguments for logarithmic-depth average-case sampling hardness in Gaussian boson sampling. This is a mathematical result about a random network ensemble, not an experimental demonstration or a general statement about every optical circuit.",
    "why_it_matters": "Below log₂ n depth an extensive subsystem stayed unentangled; at logarithmic depth every subsystem approached maximum entanglement within a constant factor.",
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      "This is a mathematical result about a random network ensemble, not an experimental demonstration or a general statement about every optical circuit."
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    "first_published_at": "2026-09-14T09:00:00.000-04:00",
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    "tags": [
      "Gaussian boson sampling",
      "entanglement",
      "hypercube networks"
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  "sources": [
    {
      "source_id": "source-2026-09-14-014",
      "title": "arXiv preprint 2609.12043",
      "publisher": "arXiv",
      "url": "https://arxiv.org/abs/2609.12043",
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    "title": "The Hypercube Went From Empty to Nearly Maximally Entangled",
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    "published_at": "2026-09-14T09:00:00.000-04:00",
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