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

Let $A$ and $B$ be C*-algebras. A linear map $ϕ : A \to B$ is called a positive map if $ϕ$ maps positive elements to positive elements: $a \geq 0 ⟹ ϕ (a) \geq 0$ .

Any linear map $ϕ : A \to B$ induces another map

id \otimes ϕ : ℂ^{k \times k} \otimes A \to ℂ^{k \times k} \otimes B

in a natural way. If $ℂ^{k \times k} \otimes A$ is identified with the C*-algebra $A^{k \times k}$ of $k \times k$ -matrices with entries in $A$ , then $id \otimes ϕ$ acts as

(\begin{matrix} a_{11} & \dots & a_{1 k} \\ ⋮ & ⋱ & ⋮ \\ a_{k 1} & \dots & a_{k k} \end{matrix}) \mapsto (\begin{matrix} ϕ (a_{11}) & \dots & ϕ (a_{1 k}) \\ ⋮ & ⋱ & ⋮ \\ ϕ (a_{k 1}) & \dots & ϕ (a_{k k}) \end{matrix}) .

We then say $ϕ$ is k-positive if ${id}_{ℂ^{k \times k}} \otimes ϕ$ is a positive map and completely positive if $ϕ$ is k-positive for all k.

Properties

Positive maps are monotone, i.e. $a_{1} \leq a_{2} ⟹ ϕ (a_{1}) \leq ϕ (a_{2})$ for all self-adjoint elements $a_{1}, a_{2} \in A_{s a}$ .
Since $- ‖ a ‖_{A} 1_{A} \leq a \leq ‖ a ‖_{A} 1_{A}$ for all self-adjoint elements $a \in A_{s a}$ , every positive map is automatically continuous with respect to the C*-norms and its operator norm equals $‖ ϕ (1_{A}) ‖_{B}$ . A similar statement with approximate units holds for non-unital algebras.
The set of positive functionals $\to ℂ$ is the dual cone of the cone of positive elements of $A$ .

Examples

Every *-homomorphism is completely positive.
For every linear operator $V : H_{1} \to H_{2}$ between Hilbert spaces, the map $L (H_{1}) \to L (H_{2}), A \mapsto V A V^{*}$ is completely positive. Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
Every positive functional $ϕ : A \to ℂ$ (in particular every state) is automatically completely positive.
Given the algebras $C (X)$ and $C (Y)$ of complex-valued continuous functions on compact Hausdorff spaces $X, Y$ , every positive map $C (X) \to C (Y)$ 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 $ℂ^{n \times n}$ . The following is a positive matrix in $ℂ^{2 \times 2} \otimes ℂ^{2 \times 2}$ : $[\begin{matrix} (\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}) & (\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix}) \\ (\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix}) & (\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}) \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \end{matrix}] .$ The image of this matrix under $I_{2} \otimes T$ is $[\begin{matrix} {(\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix})}^{T} & {(\begin{matrix} 0 & 1 \\ 0 & 0 \end{matrix})}^{T} \\ {(\begin{matrix} 0 & 0 \\ 1 & 0 \end{matrix})}^{T} & {(\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix})}^{T} \end{matrix}] = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}],$ 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 if the composition Φ $\circ$ T is positive. The transposition map itself is a co-positive map.

References

↑ K. R. Davidson: C*-Algebras by Example, American Mathematical Society (1996), ISBN 0-821-80599-1, Thm. IX.4.1
↑ R.V. Kadison, J. R. Ringrose: Fundamentals of the Theory of Operator Algebras II, Academic Press (1983), ISBN 0-1239-3302-1, Sect. 11.5.21

[1] K. R. Davidson: C*-Algebras by Example, American Mathematical Society (1996), ISBN 0-821-80599-1, Thm. IX.4.1

[2] R.V. Kadison, J. R. Ringrose: Fundamentals of the Theory of Operator Algebras II, Academic Press (1983), ISBN 0-1239-3302-1, Sect. 11.5.21

Completely positive map

Definition

Properties

Examples

See also

References