{
  "id": "1205.2747",
  "version": "v2",
  "published": "2012-05-12T07:51:22.000Z",
  "updated": "2017-03-23T09:41:41.000Z",
  "title": "Laplacian matrices of weighted digraphs represented as quantum states",
  "authors": [
    "Bibhas Adhikari",
    "Subhashish Banerjee",
    "Satyabrata Adhikari",
    "Atul Kumar"
  ],
  "comment": "19 pages, Modified version of quant-ph/1205.2747, title and abstract has been changed, One author has been added",
  "journal": "Quantum Information Processing 16, 1-22 (2017)",
  "categories": [
    "quant-ph",
    "math-ph",
    "math.MP"
  ],
  "abstract": "Representing graphs as quantum states is becoming an increasingly important approach to study entanglement of mixed states, alternate to the standard linear algebraic density matrix-based approach of study. In this paper, we propose a general weighted directed graph framework for investigating properties of a large class of quantum states which are defined by three types of Laplacian matrices associated with such graphs. We generalize the standard framework of defining density matrices from simple connected graphs to density matrices using both combinatorial and signless Laplacian matrices associated with weighted directed graphs with complex edge weights and with/without self-loops. We also introduce a new notion of Laplacian matrix, which we call signed Laplacian matrix associated with such graphs. We produce necessary and/or sufficient conditions for such graphs to correspond to pure and mixed quantum states. Using these criteria, we finally determine the graphs whose corresponding density matrices represent entangled pure states which are well known and important for quantum computation applications. We observe that all these entangled pure states share a common combinatorial structure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-05-12T07:51:22.000Z",
      "title": "Graph representation of quantum states",
      "abstract": "In this work we propose graphical representation of quantum states. Pure states require weighted digraphs with complex weights, while mixed states need, in general, edge weighted digraphs with loops; constructions which, to the best of our knowledge, are new in the theory of graphs. Both the combinatorial as well as the signless Laplacian are used for graph representation of quantum states. We also provide some interesting analogies between physical processes and graph representations. Entanglement between two qubits is approached by the development of graph operations that simulate quantum operations, resulting in the generation of Bell and Werner states. As a biproduct, the study also leads to separability criteria using graph operations. This paves the way for a study of genuine multipartite correlations using graph operations.",
      "comment": "28 pages",
      "journal": null,
      "doi": null,
      "authors": [
        "Bibhas Adhikari",
        "Satyabrata Adhikari",
        "Subhashish Banerjee"
      ]
    },
    {
      "version": "v2",
      "updated": "2017-03-23T09:41:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quantum states",
      "graph representation",
      "graph operations",
      "simulate quantum operations",
      "genuine multipartite correlations"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1205.2747A"
    }
  }
}