## Decomposability of positive maps

By results of Størmer and Woronowicz, every positive map $\Phi \colon \mathcal{M}_{d \times d} \rightarrow \mathcal{M}_{d' \times d'}$ for $dd' \leq 6$ can be decomposed as a convex combination

$$\Phi = p \phi + (1-p) ~ T \circ \psi$$

where $\phi$, $\psi$ are completely positive maps and $T$ is the transposition map.

For higher dimensions, this is in general false. Does there however (for fixed $d$, $d'$) exist a finite set of positive maps $(P_i)$ such that every general positive map $\Phi$ is a convex combination

$$\Phi = \sum p_i P_i \circ \phi_i$$

where the $\phi_i$ are suitably chosen completely positive maps?

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Perhaps you should give the definition of positive map, as well as of completely positive map. – Denis Serre Feb 16 2011 at 9:29
Thank you for your suggestion. A linear map $\Phi$ of $C^*$-algebras is positive if sends the positive cone to the positive cone. It is $k$-positive if $\Phi$ tensored with the identity map on $k \times k$-matrices is positive. It is completely positive if is $k$-positive for all $k > 0$. – Michael Feb 16 2011 at 10:00
Maybe I'm worng, but I thought that is an open problem at the heart of quantum information theory (?) – Stefan Waldmann Feb 16 2011 at 10:07
Michael, would you please include a reference for the first decomposition you mention, or at least to the specific results of Stormer and Woronowicz that lead to it? – Jon Bannon Feb 16 2011 at 16:01
Jon, the results can be found in the articles "Positive maps of low dimensional matrix algebras" (Woronowicz) and "Positive linear maps of operator algebras" (Størmer). – Michael Feb 16 2011 at 17:37
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