Completely positive map

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

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

${\displaystyle {\textrm {id}}\otimes \phi$ :\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 ${\displaystyle {\textrm {id}}\otimes \phi }$ acts as

${\displaystyle {\begin{pmatrix}a_{11}&\cdots &a_{1k}\\\vdots &\ddots &\vdots \\a_{k1}&\cdots &a_{kk}\end{pmatrix}}\mapsto {\begin{pmatrix}\phi (a_{11})&\cdots &\phi (a_{1k})\\\vdots &\ddots &\vdots \\\phi (a_{k1})&\cdots &\phi (a_{kk})\end{pmatrix}}.}$

${\displaystyle \phi }$ is called k-positive if ${\displaystyle {\textrm {id}}_{\mathbb {C} ^{k\times k}}\otimes \phi }$ is a positive map and completely positive if ${\displaystyle \phi }$ is k-positive for all k.

• Positive maps are monotone, i.e. ${\displaystyle a_{1}\leq a_{2}\implies \phi (a_{1})\leq \phi (a_{2})}$ for all self-adjoint elements ${\displaystyle a_{1},a_{2}\in A_{sa}}$.
• Since ${\displaystyle -\|a\|_{A}1_{A}\leq a\leq \|a\|_{A}1_{A}}$ for all self-adjoint elements ${\displaystyle a\in A_{sa}}$, every positive map is automatically continuous with respect to the C*-norms and its operator norm equals ${\displaystyle \|\phi (1_{A})\|_{B}}$. 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.[1]
• For every linear operator ${\displaystyle V:H_{1}\to H_{2}}$ between Hilbert spaces, the map ${\displaystyle L(H_{1})\to L(H_{2}),\ A\mapsto VAV^{\ast }}$ is completely positive.[2] Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
• Every positive functional ${\displaystyle \phi :A\to \mathbb {C} }$ (in particular every state) is automatically completely positive.
• Given the algebras ${\displaystyle C(X)}$ and ${\displaystyle C(X)}$ of complex-valued continuous functions on compact Hausdorff spaces ${\displaystyle X,Y}$, 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 ${\displaystyle \mathbb {C} ^{2\times 2}\otimes \mathbb {C} ^{2\times 2}}$:
  $${\displaystyle {\begin{bmatrix}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}&{\begin{pmatrix}0&1\\0&0\end{pmatrix}}\\{\begin{pmatrix}0&0\\1&0\end{pmatrix}}&{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\end{bmatrix}}={\begin{bmatrix}1&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&1\\\end{bmatrix}}.}$$

  
  The image of this matrix under ${\displaystyle I_{2}\otimes T}$ is
  $${\displaystyle {\begin{bmatrix}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}^{T}&{\begin{pmatrix}0&1\\0&0\end{pmatrix}}^{T}\\{\begin{pmatrix}0&0\\1&0\end{pmatrix}}^{T}&{\begin{pmatrix}0&0\\0&1\end{pmatrix}}^{T}\end{bmatrix}}={\begin{bmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\\\end{bmatrix}},}$$

  
  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 ${\displaystyle \circ }$ T is positive. The transposition map itself is a co-positive map.

• Choi's theorem on completely positive maps