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?



Decomposability of positive maps
I'm quite late answering this question, but I figured it still deserves an answer. The answer to your question is "no": when $d,d^\prime \geq 3$ there does not exist such a finite (or even countable) set of positive maps.


In [arXiv:1209.0437](http://arxiv.org/abs/1209.0437), we considered the following problem: given a set of positive maps $\mathcal{Q}$ from $\mathcal{M}\_d$ to $\mathcal{M}\_{d^\prime}$, what is the set


$$\mathcal{C}\_\mathcal{Q} \stackrel{\text{def}}{=} \{\sum\_i \psi\_i \circ P\_i \circ \phi\_i : P\_i \in \mathcal{Q}, \psi\_i \text{ and } \phi\_i \text{ are completely positive for all $i$}\}?$$


As you noted, Størmer and Woronowicz showed that if $\mathcal{Q} = \{T\}$ (where $T$ is the transpose map) and $dd^\prime \leq 6$, then $\mathcal{C}\_\mathcal{Q}$ is the set of all positive maps.


We didn't prove it in the paper, but as an offshoot Łukasz Skowronek proved in the $d,d^\prime\geq 3$ case that if $\mathcal{C}\_\mathcal{Q}$ is the set of all positive maps then $\mathcal{Q}$ must be uncountable. I don't believe he ever did end up formally writing up the result, but he at least gave a talk about it (slides available [here](http://www.fizyka.umk.pl/zfmis/smp44/LNotes/Conference_talks/June_22/skowronek.pdf)).


The slides go through the proof in pretty good detail: the rough idea is that there is a known uncountably infinite set of positive maps on $\mathcal{M}\_3$ that are each exposed in the set of positive maps and are all indecomposable (i.e., can not be written in the form $\phi\_1 + T \circ \phi\_2$ for some completely positive $\phi\_1,\phi\_2$) due to Ha and Kye (see [arXiv:1108.0130](http://arxiv.org/abs/1108.0130)). Łukasz showed that there is no countable set $\mathcal{Q}$ such that $\mathcal{C}\_\mathcal{Q}$ contains even just this particular (relatively small) set of positive maps.


**Update [May 23, 2016]:** It seems that Łukasz finally got around to formally writing up this result, and it is now proved explicitly in [this new paper](http://arxiv.org/abs/1605.05254)).


