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?
