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 which satisfies a stronger, more robust condition.

Definition

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

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

$\textrm{id} \otimes \phi : \mathbb{C}^{k \times k} \otimes A \to \mathbb{C}^{k \times k} \otimes B$

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

$\begin{pmatrix} a_{11} & \cdots & a_{1k} \\ \vdots & \ddots & \vdots \\ a_{k1} & \cdots & a_{kk} \end{pmatrix} \mapsto \begin{pmatrix} \phi(a_{11}) & \cdots & \phi(a_{1k}) \\ \vdots & \ddots & \vdots \\ \phi(a_{k1}) & \cdots & \phi(a_{kk}) \end{pmatrix}.$

We say that $\phi$ is k-positive if $\textrm{id}_{\mathbb{C}^{k\times k}} \otimes \Phi$ is a positive map, and $\phi$ is called completely positive if $\phi$ is k-positive for all k.

Properties

• Positive maps are monotone, i.e. $a_1\leq a_2\implies \phi(a_1)\leq\phi(a_2)$ for all self-adjoint elements $a_1,a_2\in A_{sa}$.
• Since $-\|a\|_A 1_A \leq a \leq \|a\|_A 1_A$ every positive map is automatically continuous w.r.t. to the C*-norms and its operator norm equals $\|\phi(1_A)\|_B$. A similary statement with approximate units holds for non-unital algebras.
• The set of positive functionals $\to\mathbb{C}$ is the dual cone of the cone of positive elements of $A$.

Examples

• Every *-homomorphism is completely positive.
• For every operator $V:H_1\to H_2$ between Hilbert spaces, the map $L(H_1)\to L(H_2), A\mapsto VAV^\ast$ is completely positive. Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
• Every positive functional $\phi:A\to\mathbb{C}$ (in particular every state) is automatically completely positive.
• 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 $\mathbb{C}^{n\times n}$. The following is a positive matrix in $C^{2\times 2} \otimes \mathbb{C}^{2\times 2}$:

$\begin{bmatrix} \begin{pmatrix}1&0\\0&0\end{pmatrix}& \begin{pmatrix}0&1\\0&0\end{pmatrix}\\ \begin{pmatrix}0&0\\1&0\end{pmatrix}& \begin{pmatrix}0&0\\0&1\end{pmatrix} \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ \end{bmatrix}.$

The image of this matrix under $I_2 \otimes T$ is

$\begin{bmatrix} \begin{pmatrix}1&0\\0&0\end{pmatrix}^T& \begin{pmatrix}0&1\\0&0\end{pmatrix}^T\\ \begin{pmatrix}0&0\\1&0\end{pmatrix}^T& \begin{pmatrix}0&0\\0&1\end{pmatrix}^T \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} ,$

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 $\circ$ T is positive. The transposition map itself is a co-positive map.