Is the reduction map completely positive? [duplicate]

I am struggling with proving the complete positivity of a general map ( granted it is CP ). The reduction map is defined as $$\rho \rightarrow \mathrm{Tr}(\rho)I - \rho$$ It is a trivial job to show the positivity of this map. I don't know whether this map is CP at first place.

Is there some physically intuitive argument which explains why this map isn't (or is) completely positive (such as an entangled state on which it acts in a non-positive way)? (Which is something which is not discussed in the answers to the post marked as a duplicate on math.SE.)

• How is this any different from your previous question (that was migrated to math.stackexchange)? Apr 1, 2014 at 12:59
• @KyleKanos Hi , The question which was migrated to math section has been answered and it talks of Choi-Jamiolkowski-isomorphism . The issue is I don't understand why a special preference is given to E-bit in the algorithm. I want another method which is more convincing ? Apr 1, 2014 at 13:23
• Have you asked the answerer, Martin, if he could be more clear in his answer? Apr 1, 2014 at 13:26
• Okay , I would try there. Btw I would like someone to help me with more physical arguments ( as why E-bits pop up if this question is answered here in terms of Choi algorithm ). Apr 1, 2014 at 13:35
• I mean "does entropy of E-bits has anything to do with the algorithm ?" Apr 1, 2014 at 13:37