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

:$\textrm \otimes \phi : \mathbb^ \otimes A \to \mathbb^ \otimes B$

in a natural way. If $\mathbb^\otimes A$ is identified with the C*-algebra $A^$ of $k\times k$-matrices with entries in $A$, then $\textrm\otimes\phi$ acts as
:$\begin a_ & \cdots & a_ \\ \vdots & \ddots & \vdots \\ a_ & \cdots & a_ \end \mapsto \begin \phi(a_) & \cdots & \phi(a_) \\ \vdots & \ddots & \vdots \\ \phi(a_) & \cdots & \phi(a_) \end.$

We then say $\phi$ is k-positive if $\textrm_ \otimes \phi$ 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 $a_1,a_2\in A_$.
* Since $-\, a\, _A 1_A \leq a \leq \, a\, _A 1_A$ for all self-adjoint elements $a\in A_$, 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 $\to\mathbb$ 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 $L(H_1)\to L(H_2), \ A \mapsto V A V^\ast$ is completely positive. R.V. Kadison, J. R. Ringrose: ''Fundamentals of the Theory of Operator Algebras II'', Academic Press (1983), ISBN 0-1239-3302-1, Sect. 11.5.21 Stinespring's theorem says that all completely positive maps are compositions of *-homomorphisms and these special maps.
* Every positive functional $\phi:A \to \mathbb$ (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 denote this map on $\mathbb^$. The following is a positive matrix in $\mathbb^ \otimes \mathbb^$: $\begin \begin1&0\\0&0\end& \begin0&1\\0&0\end\\ \begin0&0\\1&0\end& \begin0&0\\0&1\end \end = \begin 1 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 1 \\ \end.$ The image of this matrix under $I_2 \otimes T$ is $\begin \begin1&0\\0&0\end^T& \begin0&1\\0&0\end^T\\ \begin0&0\\1&0\end^T& \begin0&0\\0&1\end^T \end = \begin 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ \end ,$ 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

References

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C*-algebras