{
  "id": "1110.1645",
  "version": "v2",
  "published": "2011-10-07T20:15:15.000Z",
  "updated": "2011-11-19T22:52:13.000Z",
  "title": "Classical and nonclassical randomness in quantum measurements",
  "authors": [
    "Douglas Farenick",
    "Sarah Plosker",
    "Jerrod Smith"
  ],
  "comment": "to appear in Journal of Mathematical Physics",
  "categories": [
    "quant-ph",
    "math.OA"
  ],
  "abstract": "The space of positive operator-valued measures on the Borel sets of a compact (or even locally compact) Hausdorff space with values in the algebra of linear operators acting on a d-dimensional Hilbert space is studied from the perspectives of classical and non-classical convexity through a transform $\\Gamma$ that associates any positive operator-valued measure with a certain completely positive linear map of the homogeneous C*-algebra $C(X)\\otimes B(H)$ into $B(H)$. This association is achieved by using an operator-valued integral in which non-classical random variables (that is, operator-valued functions) are integrated with respect to positive operator-valued measures and which has the feature that the integral of a random quantum effect is itself a quantum effect. A left inverse $\\Omega$ for $\\Gamma$ yields an integral representation, along the lines of the classical Riesz Representation Theorem for certain linear functionals on $C(X)$, of certain (but not all) unital completely positive linear maps $\\phi:C(X)\\otimes B(H) \\rightarrow B(H)$. The extremal and C*-extremal points of the space of POVMS are determined.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-11-19T22:52:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "03.65.Ta",
      "03.65.Aa",
      "02.50.Cw",
      "03.65.Db",
      "03.65.Fd"
    ],
    "keywords": [
      "quantum measurements",
      "nonclassical randomness",
      "positive operator-valued measure",
      "positive linear map",
      "d-dimensional hilbert space"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1063/1.3668081",
      "journal": "Journal of Mathematical Physics",
      "year": 2011,
      "month": "Dec",
      "volume": 52,
      "number": 12,
      "pages": 2204
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011JMP....52l2204F"
    }
  }
}