{
  "id": "2308.04627",
  "version": "v1",
  "published": "2023-08-08T23:37:02.000Z",
  "updated": "2023-08-08T23:37:02.000Z",
  "title": "Hilbert-Schmidt operators and the conjugate of a complex Hilbert space: Dirac's bra-ket formalism revisited",
  "authors": [
    "Frank Oertel"
  ],
  "comment": "Reference [6] looks very different (and quite odd) after compiling the TeX file in arXiv, compared to my original version before",
  "categories": [
    "math.FA",
    "quant-ph"
  ],
  "abstract": "We reveal in detail how the definition of the inner product on a given complex Hilbert space - usually used in mathematics (where linearity is assumed in the first component and semilinearity in the second) - directly links to Dirac's powerful bra-ket formalism in quantum physics. To this end, we just have to make use of the conjugate of a complex Hilbert space (by which an analysis of semilinear operators can be handled by means of linear operator theory) and re-apply the theorem of Fr\\'{e}chet-Riesz accordingly. Applications are specified, including a self-contained and simple description of the tensor product of two complex Hilbert spaces $H \\otimes K$ (answering a related question of B. K. Driver) and a purely linear algebraic description of the quantum teleportation process (Example 3.8). In doing so, we provide an explicit construction of a canonical isometric isomorphism between the Hilbert spaces $H \\otimes (K \\otimes L)$ and $(H \\otimes K) \\otimes L$ (Theorem 3.7).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-08-08T23:37:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46B10",
      "46C05",
      "47B10",
      "46M05",
      "81P45"
    ],
    "keywords": [
      "complex hilbert space",
      "diracs bra-ket formalism",
      "hilbert-schmidt operators",
      "purely linear algebraic description",
      "diracs powerful bra-ket formalism"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}