Wikipedia / Completely positive map
Wikipedia / Completely positive map
In mathematics a positive map is a map between C*-algebras that sends positive elements to positive elements. A completely positive map is one that satisfies a stronger, more robust condition.
Definition
be C*-algebras is called a positive map maps positive elements.
induces another map
acts as
is k-positive is a positive map and completely positive is k-positive for all k.
Properties
for all self-adjoint.
for all self-adjoint, every positive map is automatically continuous with respect to the C*-norms and its operator norm. A similar statement with approximate units holds for non-unital algebras.
is the dual cone.
Examples
Every *-homomorphism is completely positive.
is completely positive. Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
(in particular every state) is automatically completely positive.
of complex-valued continuous functions on compact Hausdorff spaces is completely positive.
The transposition of matrices is a standard example of a positive map that fails to be 2-positive. Let which is clearly not positive, having determinant −1. Moreover, the eigenvalues of this matrix are 1,1,1 and −1. (This matrix happens to be the Choi matrix of T, in fact.) Incidentally, a map Φ is said to be co-positive T is positive. The transposition map itself is a co-positive map.