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 that satisfies a stronger, more robust condition.

Definition

Let $A$ and $B$ be C*-algebras. A linear map $\phi :A\to B$ is called a 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

${\displaystyle {\textrm {id}}\otimes \phi$ :\mathbb {C} ^{k\times k}\otimes A\to \mathbb {C} ^{k\times k}\otimes B}

in a natural way. If $\mathbb {C} ^{k\times k}\otimes A$ 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 then say $\phi$ is k-positive if ${\textrm {id}}_{\mathbb {C} ^{k\times k}}\otimes \phi$ is a positive map and 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.
• 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 $\mathbb {C} ^{2\times 2}\otimes \mathbb {C} ^{2\times 2}$: $${\begin{bmatrix}{\begin{pmatrix}1&0\\0&0\end{pmatrix}}&{\begin{pmatrix}0&1\\0&0\end{pmatrix}}\\{\begin{pmatrix}0&0\\1&0\end{pmatrix}}&{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\end{bmatrix}}={\begin{bmatrix}1&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&1\\\end{bmatrix}}.$$ 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.

See also

• Choi's theorem on completely positive maps