{
  "abstract": "A fundamental question in physics is: When does classical behavior emerge from quantum systems? Bosonic Gaussian states provide a natural setting to explore this quantum-classical boundary, as they capture both the classical field behavior and the intrinsic quantum nature of light. Here, we address this problem from a learning-theoretic perspective by asking: When are bosonic Gaussian states classical to learn? That is, under what conditions (if any) can an n-mode bosonic Gaussian state be learned with as few samples, and with operations as simple, as are needed to learn a classical 2n-variate Gaussian distribution? We establish a smooth crossover in learnability governed by the state's thermal fluctuations: - Cold Gaussian states are non-classical to learn: When the covariance matrix satisfies $Σ\\le(\\frac12+O(\\frac1n))I$, i.e. close to the vacuum covariance, tomography under single-copy (i.e., non-entangled) measurements fundamentally requires $Ω(n^3)$ copies, strictly exceeding the sample complexity $Θ(n^2)$ of learning classical Gaussian distributions. We show that this hardness persists even when few-copy entangled measurements are allowed. - Warm Gaussian states are classical to learn: When thermal fluctuations exceed the vacuum noise, parameterized by $Σ\\ge(\\frac12+ν)I$ for any parameter $ν>0$, we prove that single-copy tomography requires $N=Θ\\left(n^2\\min(n,1+ν^{-1})\\right)$ copies. This bound is tight and is achieved by simple, non-adaptive, unentangled heterodyne measurements. Crucially, for $ν=Ω(1)$, the sample complexity drops to $Θ(n^2)$, matching the classical case. Our results tightly characterize a quantum-to-classical crossover in the learnability of bosonic Gaussian states, reveal a novel connection between fundamental physics and statistical learning theory, and have implications for real-world sensing experiments.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.26705",
  "author_total": 4,
  "authors": [
    {
      "name": "Senrui Chen"
    },
    {
      "name": "Antonio Anna Mele"
    },
    {
      "name": "Francesco Anna Mele"
    },
    {
      "name": "John Preskill"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ege3sv776561h.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.26705",
  "fulltext_url": null,
  "id": "ege3sv776561h",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.26705",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "cs.LG"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "quant-ph"
    ],
    [
      "openalex",
      "1702|Quantum Information and Cryptography"
    ]
  ],
  "openalex": "W7214112794",
  "pdf_url": "https://arxiv.org/pdf/2609.26705",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ege3sv776561h.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:39Z",
      "native_id": "2609.26705",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.26705"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "physics",
      "origin": "arxiv:quant-ph",
      "subfield": "quantum"
    }
  ],
  "title": "When are bosonic Gaussian states classical to learn?",
  "title_text": "When are bosonic Gaussian states classical to learn?",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:43Z",
  "url": "https://research.draconitic.com/paper/ege3sv776561h",
  "venue": "arXiv",
  "versions": []
}