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.

Let ${\displaystyle A}$ and ${\displaystyle B}$ be C*-algebras. A linear map ${\displaystyle \varphi :A\to B}$ is called positive map if ${\displaystyle \varphi }$ maps positive elements to positive elements: ${\displaystyle a\ge 0⟹\varphi (a)\ge 0}$.

Any linear map ${\displaystyle \varphi :A\to B}$ induces another map

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

in a natural way. If ${\displaystyle {\mathbb{ℂ}}^{k\times k}\otimes A}$ is identified with the C*-algebra ${\displaystyle A^{k\times k}}$ of ${\displaystyle k\times k}$-matrices with entries in ${\displaystyle A}$, then ${\displaystyle \text{<mrow data-mjx-texclass="ORD">id</mrow>}\otimes \varphi }$ acts as

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

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

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

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

${\displaystyle \left[\begin{array}{cc}\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)& \left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)\\ \left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)& \left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 1\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 0& 1\\ \end{array}\right].}$

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

${\displaystyle \left[\begin{array}{cc}{\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)}^T& {\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)}^T\\ {\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right)}^T& {\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)}^T\end{array}\right]=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 0& 1& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\\ \end{array}\right],}$

which is clearly not positive, having determinant -1. Moreover, the eigenvalues of this matrix are 1,1,1 and -1.

Incidentally, a map Φ is said to be co-positive if the composition Φ ${\displaystyle \circ }$ T is positive. The transposition map itself is a co-positive map.

• Choi's theorem on completely positive maps

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