Completely positive map

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In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one which satisfies a stronger, more robust condition.

Definition[edit]

Let and be C*-algebras. A linear map is called positive map if maps positive elements to positive elements: .

Any linear map induces another map

in a natural way. If is identified with the C*-algebra of -matrices with entries in , then acts as

We say that is k-positive if is a positive map, and is called completely positive if is k-positive for all k.

Properties[edit]

  • Positive maps are monotone, i.e. for all self-adjoint elements .
  • Since every positive map is automatically continuous w.r.t. the C*-norms and its operator norm equals . A similary statement with approximate units holds for non-unital algebras.
  • The set of positive functionals is the dual cone of the cone of positive elements of .

Examples[edit]

  • Every *-homomorphism is completely positive.
  • For every operator between Hilbert spaces, the map is completely positive. Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
  • Every positive functional (in particular every state) is automatically completely positive.
  • Every positive map is completely positive.
  • The transposition of matrices is a standard example of a positive map that fails to be 2-positive. Let T denote this map on . The following is a positive matrix in :

The image of this matrix under is

which is clearly not positive, having determinant -1. Moreover, the eigenvalues of this matrix are 1,1,1 and -1.
Incidentally, a map Φ is said to be co-positive if the composition Φ T is positive. The transposition map itself is a co-positive map.