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. 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 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.

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.

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