{
  "id": "1407.2926",
  "version": "v2",
  "published": "2014-07-10T19:54:40.000Z",
  "updated": "2014-12-10T21:17:52.000Z",
  "title": "An invariant of topologically ordered states under local unitary transformations",
  "authors": [
    "Jeongwan Haah"
  ],
  "comment": "revtex, 17 pages (v2) minor change",
  "categories": [
    "quant-ph",
    "cond-mat.str-el",
    "math-ph",
    "math.MP"
  ],
  "abstract": "For an anyon model in two spatial dimensions described by a modular tensor category, the topological S-matrix encodes the mutual braiding statistics, the quantum dimensions, and the fusion rules of anyons. It is nontrivial whether one can compute the S-matrix from a single ground state wave function. Here, we define a class of Hamiltonians consisting of local commuting projectors and an associated matrix that is invariant under local unitary transformations. We argue that the invariant is equivalent to the topological S-matrix. The definition does not require degeneracy of the ground state. We prove that the invariant depends on the state only, in the sense that it can be computed by any Hamiltonian in the class of which the state is a ground state. As a corollary, we prove that any local quantum circuit that connects two ground states of quantum double models (discrete gauge theories) with non-isomorphic abelian groups, must have depth that is at least linear in the system's diameter.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-10T19:54:40.000Z",
      "comment": "revtex, 17 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-10T21:17:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "local unitary transformations",
      "topologically ordered states",
      "single ground state wave function",
      "modular tensor category",
      "non-isomorphic abelian groups"
    ],
    "note": {
      "typesetting": "RevTeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.2926H"
    }
  }
}