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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 $-\|a\|_{A}1_{A}\leq a\leq \|a\|_{A}1_{A)$ 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

• Every *-homomorphism is completely positive.
• For every linear operator $V:H_{1}\to H_{2)$ between Hilbert spaces, the map $L(H_{1})\to L(H_{2}),\ A\mapsto VAV^{\ast )$ 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 $\mathbb {C} ^{2\times 2}\otimes \mathbb {C} ^{2\times 2)$ :
${\begin{bmatrix}{\begin{pmatrix}1&0\\0&0\end{pmatrix))&{\begin{pmatrix}0&1\\0&0\end{pmatrix))\\{\begin{pmatrix}0&0\\1&0\end{pmatrix))&{\begin{pmatrix}0&0\\0&1\end{pmatrix))\end{bmatrix))={\begin{bmatrix}1&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&1\\\end{bmatrix)).$ 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. (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.