{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nLet $A$ and $B$ be C*-algebras, and let $\\varphi \\colon A \\to B$ be a linear map.\nFor $n \\ge 1$, let $M_n(A)$ be the [[matrix ring|0FHVPRAE60]] of $n \\times n$ matrices with entries in $A$, which is again a C\\*-algebra.\nLet $\\varphi_n \\colon M_n(A) \\to M_n(B)$ be the map $[a_{ij}] \\mapsto [\\varphi(a_{ij})]$.\nThe map $\\varphi$ is *$n$-positive* if $\\varphi_n$ sends [[positive elements|NA8DF5X3M4]] of $M_n(A)$ to positive elements of $M_n(B)$.\nIt is *completely positive* if it is $n$-positive for every $n \\ge 1$.\n\n## Remarks\n\nEquivalently, $\\varphi \\otimes \\id_{M_n}$ is positive for every $n$.\nEvery $*$-homomorphism $\\pi$ is completely positive, since $\\pi_n$ is a $*$-homomorphism and maps $x^* x$ to $\\pi_n(x)^* \\pi_n(x)$.\n\nExample: for a fixed $m \\times k$ complex matrix $V$, the map $\\varphi \\colon M_m(\\C) \\to M_k(\\C)$, $\\varphi(a) = V^* a V$, is completely positive.\nIndeed $\\varphi_n(x) = W^* x W$ with $W = \\operatorname{diag}(V, \\ldots, V)$, and $W^* y^* y W = (y W)^* (y W) \\ge 0$.\n\nNon-example: the transpose $\\theta(a) = a^T$ on $M_2(\\C)$ is positive, since $a$ and $a^T$ have the same eigenvalues.\nIt is not $2$-positive: the matrix $x = [E_{ij}]_{i,j=1}^2 \\in M_2(M_2(\\C))$, with $E_{ij}$ the matrix units, equals $v v^*$ with $v = e_1 \\otimes e_1 + e_2 \\otimes e_2$, so it is positive.\nBut $\\theta_2(x) = [E_{ji}]_{i,j}$ is the swap matrix, which has the eigenvalue $-1$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "CGY299QMC3",
    "kind": "definition",
    "msc": [
      "46L07",
      "46L05",
      "81P47"
    ],
    "name": "Completely positive map",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Completely_positive_map"
      }
    ],
    "tag": "NEZYP9HJB9",
    "uses": [
      "NA8DF5X3M4",
      "0FHVPRAE60"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/NEZYP9HJB9/",
  "links": [
    {
      "field": "defines",
      "target": "CGY299QMC3"
    },
    {
      "field": "uses",
      "target": "NA8DF5X3M4"
    },
    {
      "field": "uses",
      "target": "0FHVPRAE60"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/NEZYP9HJB9.md",
  "msc": [
    "46L07",
    "46L05",
    "81P47"
  ],
  "name": "Completely positive map",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/NEZ/NEZYP9HJB9.md",
  "status": null,
  "tag": "NEZYP9HJB9",
  "uri": "m2://NEZYP9HJB9",
  "url": "https://math2.tamnd.com/wiki/completely-positive-map-definition/",
  "writing": {
    "tag": "NEZYP9HJB9",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}