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Let $S_+^n$ be the cone of psd (real symmetric positive semidefinite) matrices of size $n\times n$. A cone $K$ is said to be a cross-section of $S_+^n$ if $K=S_+^n\cap \Pi$ for some linear subspace $\Pi$ of $M_n$, the vector space of $n\times n$ real matrices.

Let a cross section $K$ of $S_+^n$ be given. Does there exists $m$ and a completely positive linear map $\Phi:M_m\to M_n$ such that $\Phi$ maps the psd cone $S_+^m$ onto $K$? If so, is there a constructive way to describe the completely positive linear map $\Phi$?

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