{
  "id": "2404.03627",
  "version": "v1",
  "published": "2024-04-04T17:49:23.000Z",
  "updated": "2024-04-04T17:49:23.000Z",
  "title": "Injective norm of real and complex random tensors I: From spin glasses to geometric entanglement",
  "authors": [
    "Stephane Dartois",
    "Benjamin McKenna"
  ],
  "comment": "43 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP",
    "quant-ph"
  ],
  "abstract": "The injective norm is a natural generalization to tensors of the operator norm of a matrix. In quantum information, the injective norm is one important measure of genuine multipartite entanglement of quantum states, where it is known as the geometric entanglement. In this paper, we give a high-probability upper bound on the injective norm of real and complex Gaussian random tensors, corresponding to a lower bound on the geometric entanglement of random quantum states, and to a bound on the ground-state energy of a particular multispecies spherical spin glass model. For some cases of our model, previous work used $\\epsilon$-net techniques to identify the correct order of magnitude; in the present work, we use the Kac--Rice formula to give a one-sided bound on the constant which we believe to be tight.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-04-04T17:49:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "81P45",
      "81P42",
      "82D30",
      "60B20",
      "15B52"
    ],
    "keywords": [
      "injective norm",
      "complex random tensors",
      "geometric entanglement",
      "quantum states",
      "complex gaussian random tensors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 43,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}