Here's something neat: Here it is proved, by Sixia Yu, that (I think for in the finite-dimensional casealone but have not checked) , any positive map that isn't completely positive is can be written as the difference of two completely positive maps. There is The paper also considers a necessary and sufficient condition for a positive map to fail complete positivity (also in finite dimensions). This paper apparently provides a machine mechanism for generating examples answering your question in the finite-dimensional case.
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Here's something neat: Here it is proved that (I think for the finite-dimensional case alone but have not checked) any positive map that isn't completely positive is the difference of two completely positive maps. There is also a necessary and sufficient condition for a positive map to fail complete positivity. This paper apparently provides a machine for generating examples answering your question. |
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