Completely positive map - Imported Knowlpedia Content

Definition

Let $A$ and $B$ be $C^*$ -algebras. A bounded linear map $\Phi:A\to B$ is completely positive if it is $n$ -positive for every integer $n\geq1$ . Explicitly, for each $n$ , the entrywise amplification

\Phi^{(n)}:M_n(A)\longrightarrow M_n(B),\qquad [a_{ij}]\longmapsto[\Phi(a_{ij})],

must carry positive elements to positive elements. Complete positivity therefore controls positivity simultaneously at every matrix level, not only on $A$ itself. The definition does not require $\Phi$ to preserve the identity, be multiplicative, or be normal when the algebras are von Neumann algebras.

Stinespring characterization

When $B=B(H)$ , Stinespring's theorem characterizes complete positivity by a factorization

\Phi(a)=V^*\pi(a)V,

where $\pi$ is a $*$ -representation of $A$ on a Hilbert space $K$ and $V:H\to K$ is bounded. With the usual minimality condition the factorization is unique up to unitary equivalence Paulsen, Chapter 4.

Closure properties and examples

Every $*$ -homomorphism is completely positive. Compositions and nonnegative linear combinations of completely positive maps are completely positive, as are maps $a\mapsto V^*\pi(a)V$ . A positive linear functional $A\to\mathbb C$ is automatically completely positive. These facts make CP maps stable under the constructions used for representations and quantum operations.

Positivity is not enough

References

Vern Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge University Press, 2002. DOI record. Relevant: Chapters 2–4 on matrix positivity, completely positive maps, and Stinespring dilation.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter IV on completely positive maps.