{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "9MB4JD02YF"
    },
    {
      "field": "uses",
      "target": "EQG2K8ZFM4"
    },
    {
      "field": "uses",
      "target": "X8K0Z7484R"
    },
    {
      "field": "about",
      "target": "EQG2K8ZFM4"
    },
    {
      "field": "about",
      "target": "X8K0Z7484R"
    },
    {
      "field": "defines",
      "target": "NEZYP9HJB9"
    }
  ],
  "body": "\n## Summary\n\nA completely positive map between C*-algebras is a linear map that stays positive after tensoring with the identity on every matrix algebra $M_n(\\C)$.\nThese maps are the natural morphisms of operator algebras when one keeps only the order structure.\nThe [[Stinespring dilation theorem|X8K0Z7484R]] describes them all as compressions of $*$-homomorphisms.\nIn quantum information theory, completely positive trace-preserving maps are the quantum channels.\n\n## Explanations\n\n### L1 human\n\nA positive map sends [[positive elements|NA8DF5X3M4]] to positive elements.\nComplete positivity asks more: if the map is applied entrywise to matrices whose entries are in the algebra, positive block matrices must still go to positive block matrices.\nPhysically, an operation on one system must remain a valid operation when the system is part of a larger one.\nThe transpose of $2 \\times 2$ matrices is positive but not completely positive, which shows that the extra condition is real.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "46L07",
      "46L05",
      "81P47"
    ],
    "name": "Completely positive map",
    "same_as": {
      "wikidata": "Q24107961",
      "wikipedia": "Completely positive map"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Completely_positive_map"
      }
    ],
    "tag": "CGY299QMC3"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/CGY299QMC3/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/CGY299QMC3.md",
  "msc": [
    "46L07",
    "46L05",
    "81P47"
  ],
  "name": "Completely positive map",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/CGY/CGY299QMC3.md",
  "status": null,
  "tag": "CGY299QMC3",
  "uri": "m2://CGY299QMC3",
  "url": "https://math2.tamnd.com/wiki/completely-positive-map/",
  "writing": {
    "tag": "CGY299QMC3",
    "kind": "concept",
    "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
      }
    ]
  }
}