quantum mechanics - What is a completely positive map *physically*?

I am sure this question is really stupid, but I could not refrain from asking it in this forum. This can be considered as a continuation of this question.

What does it mean to be a completely positive map, from a Physics point of view. 

A positive map $h:\mathcal{B(H)}\rightarrow\mathcal{B(K)}$ is a map which takes states to states. However if we put an auxiliary space $\mathcal{B(A)}$ and take the natural extension $1\otimes h:\mathcal{B(A)}\otimes\mathcal{B(H)}\rightarrow\mathcal{B(A)}\otimes\mathcal{B(K)}$, then completely positive maps are the ones which preserves positivity whatever the dimension of $\mathcal{B(A)}$ may be. So they form what we know as quantum channel (and all its relations with Jamiołkowski isomorphism etc.). Obviously foor positive maps which are not completely positive, when extended, will not remain as a physical object. In a way the same thing is done by operator space theorists as well.

My question is, can we give a definition of complete positivity without involving auxiliary systems? After all positive maps sends a state to a state. So which physical process actually hinders them of being a valid quantum operation? Looking back, are all not completely maps are physically impossible to simulate? (This alone perhaps should be written as a different question all together)