The paper's optimal CP approximation result is a restatement of known matrix projection facts, and its claimed minimal auxiliary dimension (the Choi rank) is false: two dimensions always suffice.
Ando,Positive map as difference of two completely positive or super-positive maps, Adv
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Hermitian Maps: Approximations and Completely Positive Extensions
The paper's optimal CP approximation result is a restatement of known matrix projection facts, and its claimed minimal auxiliary dimension (the Choi rank) is false: two dimensions always suffice.