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 $\varphi :A\to B$ is called a positive map if $\varphi$ maps positive elements to positive elements: $a\ge 0⟹\varphi (a)\ge 0$. Any linear map $\varphi :A\to B$ induces another map

$\text{<mrow data-mjx-texclass="ORD">id</mrow>}\otimes \varphi :{\mathbb{ℂ}}^{k\times k}\otimes A\to {\mathbb{ℂ}}^{k\times k}\otimes B$

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 $\text{<mrow data-mjx-texclass="ORD">id</mrow>}\otimes \varphi$ acts as

$\left(\begin{array}{ccc}{a}_{11}& \cdots & a_{1k}\\ ⋮& \ddots & ⋮\\ a_{k1}& \cdots & a_{kk}\end{array}\right)\mapsto \left(\begin{array}{ccc}\varphi (a_{11})& \cdots & \varphi (a_{1k})\\ ⋮& \ddots & ⋮\\ \varphi (a_{k1})& \cdots & \varphi (a_{kk})\end{array}\right).$

$\varphi$ is called k-positive if ${\text{<mrow data-mjx-texclass="ORD">id</mrow>}}_{{\mathbb{ℂ}}^{k\times k}}\otimes \varphi$ is a positive map and completely positive if $\varphi$ is k-positive for all k.

Properties

• Positive maps are monotone, i.e. $a_1\le a_2⟹\varphi (a_1)\le \varphi (a_2)$ for all self-adjoint elements $a_1,a_2\in A_{sa}$.
• Since $-\Vert a{\Vert }_A{1}_A\le a\le \Vert a{\Vert }_A{1}_A$ 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 $\Vert \varphi (1_A){\Vert }_B$. 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 $$\begin{gathered} L(H_1)\to L(H_2), \\ A\mapsto VA{V}^{\ast } \end{gathered}$$ is completely positive. Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
• Every positive functional $\varphi :A\to \mathbb{ℂ}$ (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 ${\mathbb{ℂ}}^{2\times 2}\otimes {\mathbb{ℂ}}^{2\times 2}$: $$\left[\begin{array}{cc}\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)& \left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)\\ \left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)& \left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 1\\ \end{array}\right].$$ The image of this matrix under $I_2\otimes T$ is $$\left[\begin{array}{cc}{\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)}^T& {\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)}^T\\ {\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)}^T& {\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)}^T\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ \end{array}\right],$$ 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.

See also

• Choi's theorem on completely positive maps