{
  "id": "1603.06199",
  "version": "v1",
  "published": "2016-03-20T10:29:19.000Z",
  "updated": "2016-03-20T10:29:19.000Z",
  "title": "Average position in quantum walks with an arbitrary initial state",
  "authors": [
    "Li Min",
    "Cheng ZaiJun",
    "Wang LingJie",
    "Huang HaiBo"
  ],
  "categories": [
    "quant-ph"
  ],
  "abstract": "We investigated discrete-time quantum walks with an arbitrary initial state \\mid\\Psi_{0}(\\theta,\\phi,\\varphi)\\rangle=\\cos\\theta ^{i\\phi}\\mid0L\\rangle+\\sin\\theta e^{i\\varphi}\\mid0R\\rangle with a U(2) coin U(\\alpha,\\beta,\\gamma) . We discover that the average position \\overline{x}=\\max(\\overline{x})\\cos(\\alpha+\\gamma+\\phi-\\varphi), with coin operator U_{C}(\\alpha,\\pi/4,\\gamma) and initial state \\mid\\Phi_{0}(\\pi/4,\\phi,\\varphi)\\rangle=(e^{i\\phi}\\mid0L\\rangle+e^{i\\varphi}\\mid0R\\rangle)\\sqrt{2}/2 .If we set initial state and coin operator to \\mid\\Phi_{0}\\rangle(\\theta,\\pi/2,0)=i\\cos\\theta\\mid0L\\rangle+\\sin\\theta\\mid0R\\rangle) and coin operators U_{C}(0,\\pi/4,0), for \\alpha+\\gamma+\\phi-\\varphi=\\pi/2, we discover that \\overline{x}=-\\max(\\overline{x})\\cos(2\\theta). Last we verify the result above, and obtain the summarize properties of quantum walks with an arbitrary state. We get that \\overline{x}\\theta,\\phi,\\varphi,\\alpha,\\beta,\\gamma,t)=\\cos2\\theta*\\overline{x}_{|0L\\rangle}(\\beta,t)+\\sin2\\theta*\\cos(\\alpha+\\gamma+\\phi-\\varphi)*\\overline{x}_{(\\mid0L\\rangle+\\mid0R\\rangle)\\sqrt{2}/2}(\\alpha=\\gamma=0,\\beta,t) . If the average positions \\overline{x} with initial state |0L\\rangle and \\mid\\Psi_{0}\\rangle=(\\mid0L\\rangle+\\mid0R\\rangle)\\sqrt{2}/2 are known, we can get the average position result for an arbitrary initial state.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-20T10:29:19.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arbitrary initial state",
      "coin operator",
      "average position result",
      "discrete-time quantum walks",
      "set initial state"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160306199M"
    }
  }
}