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 a 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}. $$

$$\phi$$ is called k-positive if $$\textrm{id}_{\mathbb{C}^{k\times k}} \otimes \phi$$ is a positive map and 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 $$-\|a\|_A 1_A \leq a \leq \|a\|_A 1_A$$ for all self-adjoint elements $$a\in A_{sa}$$, every positive map is automatically continuous with respect to the C*-norms and its operator norm equals $$\|\phi(1_A)\|_B$$. A similar 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
\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} , $$ which is clearly not positive, having determinant −1. Moreover, the eigenvalues of this matrix are 1,1,1 and −1. (This matrix happens to be the Choi matrix of T, in fact.) Incidentally, a map Φ is said to be co-positive if the composition Φ $$\circ$$ T is positive. The transposition map itself is a co-positive map.
 * Every *-homomorphism is completely positive.
 * For every linear operator $$V:H_1\to H_2$$ between Hilbert spaces, the map $$L(H_1)\to L(H_2), \ A \mapsto V A V^\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}$$ (in particular every state) is automatically completely positive.
 * Given the algebras $$C(X)$$ and $$C(Y)$$ of complex-valued continuous functions on compact Hausdorff spaces $$X, Y$$, every positive map $$C(X)\to C(Y)$$ is completely positive.
 * The transposition of matrices is a standard example of a positive map that fails to be 2-positive. Let $T$ denote this map on $$\mathbb{C}^{n \times n}$$. The following is a positive matrix in $$\mathbb{C}^{2\times 2} \otimes \mathbb{C}^{2\times 2}$$: $$